3.17.17 \(\int (d+e x)^4 (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2} \, dx\)

Optimal. Leaf size=534 \[ -\frac {143 \left (c d^2-a e^2\right )^{10} \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{262144 c^{15/2} d^{15/2} e^{7/2}}+\frac {143 \left (c d^2-a e^2\right )^8 \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{131072 c^7 d^7 e^3}-\frac {143 \left (c d^2-a e^2\right )^6 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{49152 c^6 d^6 e^2}+\frac {143 \left (c d^2-a e^2\right )^4 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{15360 c^5 d^5 e}+\frac {143 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{4480 c^4 d^4}+\frac {143 (d+e x) \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{2880 c^3 d^3}+\frac {13 (d+e x)^2 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{180 c^2 d^2}+\frac {(d+e x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{10 c d} \]

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Rubi [A]  time = 0.68, antiderivative size = 534, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {670, 640, 612, 621, 206} \begin {gather*} \frac {143 \left (c d^2-a e^2\right )^8 \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{131072 c^7 d^7 e^3}-\frac {143 \left (c d^2-a e^2\right )^6 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{49152 c^6 d^6 e^2}+\frac {143 \left (c d^2-a e^2\right )^4 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{15360 c^5 d^5 e}+\frac {143 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{4480 c^4 d^4}+\frac {143 (d+e x) \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{2880 c^3 d^3}+\frac {13 (d+e x)^2 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{180 c^2 d^2}-\frac {143 \left (c d^2-a e^2\right )^{10} \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{262144 c^{15/2} d^{15/2} e^{7/2}}+\frac {(d+e x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{10 c d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

(143*(c*d^2 - a*e^2)^8*(c*d^2 + a*e^2 + 2*c*d*e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(131072*c^7*d^
7*e^3) - (143*(c*d^2 - a*e^2)^6*(c*d^2 + a*e^2 + 2*c*d*e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(49
152*c^6*d^6*e^2) + (143*(c*d^2 - a*e^2)^4*(c*d^2 + a*e^2 + 2*c*d*e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^
(5/2))/(15360*c^5*d^5*e) + (143*(c*d^2 - a*e^2)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(4480*c^4*d^4
) + (143*(c*d^2 - a*e^2)^2*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(2880*c^3*d^3) + (13*(c*d^
2 - a*e^2)*(d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(180*c^2*d^2) + ((d + e*x)^3*(a*d*e + (c
*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(10*c*d) - (143*(c*d^2 - a*e^2)^10*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*
Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(262144*c^(15/2)*d^(15/2)*e^(7/2))

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 640

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(a + b*x + c*x^2)^(p +
1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 670

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 1)), x] + Dist[((m + p)*(2*c*d - b*e))/(c*(m + 2*p + 1)), Int[(d + e
*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p]

Rubi steps

\begin {align*} \int (d+e x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx &=\frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{10 c d}+\frac {\left (13 \left (d^2-\frac {a e^2}{c}\right )\right ) \int (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx}{20 d}\\ &=\frac {13 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{180 c^2 d^2}+\frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{10 c d}+\frac {\left (143 \left (d^2-\frac {a e^2}{c}\right )^2\right ) \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx}{360 d^2}\\ &=\frac {143 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{2880 c^3 d^3}+\frac {13 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{180 c^2 d^2}+\frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{10 c d}+\frac {\left (143 \left (d^2-\frac {a e^2}{c}\right )^3\right ) \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx}{640 d^3}\\ &=\frac {143 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{4480 c^4 d^4}+\frac {143 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{2880 c^3 d^3}+\frac {13 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{180 c^2 d^2}+\frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{10 c d}+\frac {\left (143 \left (d^2-\frac {a e^2}{c}\right )^4\right ) \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx}{1280 d^4}\\ &=\frac {143 \left (c d^2-a e^2\right )^4 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{15360 c^5 d^5 e}+\frac {143 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{4480 c^4 d^4}+\frac {143 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{2880 c^3 d^3}+\frac {13 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{180 c^2 d^2}+\frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{10 c d}-\frac {\left (143 \left (c d^2-a e^2\right )^6\right ) \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{6144 c^5 d^5 e}\\ &=-\frac {143 \left (c d^2-a e^2\right )^6 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{49152 c^6 d^6 e^2}+\frac {143 \left (c d^2-a e^2\right )^4 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{15360 c^5 d^5 e}+\frac {143 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{4480 c^4 d^4}+\frac {143 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{2880 c^3 d^3}+\frac {13 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{180 c^2 d^2}+\frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{10 c d}+\frac {\left (143 \left (c d^2-a e^2\right )^8\right ) \int \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{32768 c^6 d^6 e^2}\\ &=\frac {143 \left (c d^2-a e^2\right )^8 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{131072 c^7 d^7 e^3}-\frac {143 \left (c d^2-a e^2\right )^6 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{49152 c^6 d^6 e^2}+\frac {143 \left (c d^2-a e^2\right )^4 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{15360 c^5 d^5 e}+\frac {143 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{4480 c^4 d^4}+\frac {143 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{2880 c^3 d^3}+\frac {13 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{180 c^2 d^2}+\frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{10 c d}-\frac {\left (143 \left (c d^2-a e^2\right )^{10}\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{262144 c^7 d^7 e^3}\\ &=\frac {143 \left (c d^2-a e^2\right )^8 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{131072 c^7 d^7 e^3}-\frac {143 \left (c d^2-a e^2\right )^6 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{49152 c^6 d^6 e^2}+\frac {143 \left (c d^2-a e^2\right )^4 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{15360 c^5 d^5 e}+\frac {143 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{4480 c^4 d^4}+\frac {143 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{2880 c^3 d^3}+\frac {13 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{180 c^2 d^2}+\frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{10 c d}-\frac {\left (143 \left (c d^2-a e^2\right )^{10}\right ) \operatorname {Subst}\left (\int \frac {1}{4 c d e-x^2} \, dx,x,\frac {c d^2+a e^2+2 c d e x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{131072 c^7 d^7 e^3}\\ &=\frac {143 \left (c d^2-a e^2\right )^8 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{131072 c^7 d^7 e^3}-\frac {143 \left (c d^2-a e^2\right )^6 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{49152 c^6 d^6 e^2}+\frac {143 \left (c d^2-a e^2\right )^4 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{15360 c^5 d^5 e}+\frac {143 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{4480 c^4 d^4}+\frac {143 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{2880 c^3 d^3}+\frac {13 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{180 c^2 d^2}+\frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{10 c d}-\frac {143 \left (c d^2-a e^2\right )^{10} \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{262144 c^{15/2} d^{15/2} e^{7/2}}\\ \end {align*}

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Mathematica [B]  time = 6.79, size = 1439, normalized size = 2.69 \begin {gather*} \frac {2 \left (c d^2-a e^2\right )^6 (a e+c d x) ((a e+c d x) (d+e x))^{5/2} \left (\frac {c d e (a e+c d x)}{\left (c d^2-a e^2\right ) \left (\frac {c^2 d^3}{c d^2-a e^2}-\frac {a c d e^2}{c d^2-a e^2}\right )}+1\right )^{15/2} \left (\frac {1001 \left (c d^2-a e^2\right )^4 \left (\frac {16 c^3 d^3 e^3 (a e+c d x)^3}{15 \left (c d^2-a e^2\right )^3 \left (\frac {c^2 d^3}{c d^2-a e^2}-\frac {a c d e^2}{c d^2-a e^2}\right )^3}-\frac {4 c^2 d^2 e^2 (a e+c d x)^2}{3 \left (c d^2-a e^2\right )^2 \left (\frac {c^2 d^3}{c d^2-a e^2}-\frac {a c d e^2}{c d^2-a e^2}\right )^2}+\frac {2 c d e (a e+c d x)}{\left (c d^2-a e^2\right ) \left (\frac {c^2 d^3}{c d^2-a e^2}-\frac {a c d e^2}{c d^2-a e^2}\right )}-\frac {2 \sqrt {c} \sqrt {d} \sqrt {e} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2} \sqrt {\frac {c^2 d^3}{c d^2-a e^2}-\frac {a c d e^2}{c d^2-a e^2}}}\right ) \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2} \sqrt {\frac {c^2 d^3}{c d^2-a e^2}-\frac {a c d e^2}{c d^2-a e^2}} \sqrt {\frac {c d e (a e+c d x)}{\left (c d^2-a e^2\right ) \left (\frac {c^2 d^3}{c d^2-a e^2}-\frac {a c d e^2}{c d^2-a e^2}\right )}+1}}\right ) \left (\frac {c^2 d^3}{c d^2-a e^2}-\frac {a c d e^2}{c d^2-a e^2}\right )^4}{524288 c^4 d^4 e^4 (a e+c d x)^4 \left (\frac {c d e (a e+c d x)}{\left (c d^2-a e^2\right ) \left (\frac {c^2 d^3}{c d^2-a e^2}-\frac {a c d e^2}{c d^2-a e^2}\right )}+1\right )^7}+\frac {7}{20} \left (\frac {1}{\frac {c d e (a e+c d x)}{\left (c d^2-a e^2\right ) \left (\frac {c^2 d^3}{c d^2-a e^2}-\frac {a c d e^2}{c d^2-a e^2}\right )}+1}+\frac {13}{18 \left (\frac {c d e (a e+c d x)}{\left (c d^2-a e^2\right ) \left (\frac {c^2 d^3}{c d^2-a e^2}-\frac {a c d e^2}{c d^2-a e^2}\right )}+1\right )^2}+\frac {143}{288 \left (\frac {c d e (a e+c d x)}{\left (c d^2-a e^2\right ) \left (\frac {c^2 d^3}{c d^2-a e^2}-\frac {a c d e^2}{c d^2-a e^2}\right )}+1\right )^3}+\frac {143}{448 \left (\frac {c d e (a e+c d x)}{\left (c d^2-a e^2\right ) \left (\frac {c^2 d^3}{c d^2-a e^2}-\frac {a c d e^2}{c d^2-a e^2}\right )}+1\right )^4}+\frac {143}{768 \left (\frac {c d e (a e+c d x)}{\left (c d^2-a e^2\right ) \left (\frac {c^2 d^3}{c d^2-a e^2}-\frac {a c d e^2}{c d^2-a e^2}\right )}+1\right )^5}+\frac {143}{1536 \left (\frac {c d e (a e+c d x)}{\left (c d^2-a e^2\right ) \left (\frac {c^2 d^3}{c d^2-a e^2}-\frac {a c d e^2}{c d^2-a e^2}\right )}+1\right )^6}+\frac {143}{4096 \left (\frac {c d e (a e+c d x)}{\left (c d^2-a e^2\right ) \left (\frac {c^2 d^3}{c d^2-a e^2}-\frac {a c d e^2}{c d^2-a e^2}\right )}+1\right )^7}\right )\right )}{7 c^7 d^7 \left (\frac {c d}{\frac {c^2 d^3}{c d^2-a e^2}-\frac {a c d e^2}{c d^2-a e^2}}\right )^{13/2} (d+e x)^2 \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

(2*(c*d^2 - a*e^2)^6*(a*e + c*d*x)*((a*e + c*d*x)*(d + e*x))^(5/2)*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)
*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^(15/2)*((7*(143/(4096*(1 + (c*d*e*(a*e + c*d*x))/
((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^7) + 143/(1536*(1 + (c*d*e*(a*e +
c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^6) + 143/(768*(1 + (c*d*
e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^5) + 143/(448*(1
+ (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^4) + 143
/(288*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^
3) + 13/(18*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^
2))))^2) + (1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2
))))^(-1)))/20 + (1001*(c*d^2 - a*e^2)^4*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))^4*((2*c*d*e
*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))) - (4*c^2*d^2*e^2*(
a*e + c*d*x)^2)/(3*(c*d^2 - a*e^2)^2*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))^2) + (16*c^3*d^
3*e^3*(a*e + c*d*x)^3)/(15*(c*d^2 - a*e^2)^3*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))^3) - (2
*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d^2 - a
*e^2]*Sqrt[(c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2)])])/(Sqrt[c*d^2 - a*e^2]*Sqrt[(c^2*d^3)/(c*
d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2)]*Sqrt[1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2
- a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2)))])))/(524288*c^4*d^4*e^4*(a*e + c*d*x)^4*(1 + (c*d*e*(a*e + c*d*x))/((
c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^7)))/(7*c^7*d^7*((c*d)/((c^2*d^3)/(
c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2)))^(13/2)*(d + e*x)^2*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])

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IntegrateAlgebraic [F]  time = 180.03, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(d + e*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

$Aborted

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fricas [B]  time = 0.66, size = 2114, normalized size = 3.96

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="fricas")

[Out]

[1/165150720*(45045*(c^10*d^20 - 10*a*c^9*d^18*e^2 + 45*a^2*c^8*d^16*e^4 - 120*a^3*c^7*d^14*e^6 + 210*a^4*c^6*
d^12*e^8 - 252*a^5*c^5*d^10*e^10 + 210*a^6*c^4*d^8*e^12 - 120*a^7*c^3*d^6*e^14 + 45*a^8*c^2*d^4*e^16 - 10*a^9*
c*d^2*e^18 + a^10*e^20)*sqrt(c*d*e)*log(8*c^2*d^2*e^2*x^2 + c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4 - 4*sqrt(c*d*e*x
^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*e*x + c*d^2 + a*e^2)*sqrt(c*d*e) + 8*(c^2*d^3*e + a*c*d*e^3)*x) + 4*(41
28768*c^10*d^10*e^10*x^9 + 45045*c^10*d^19*e - 435435*a*c^9*d^17*e^3 + 1885884*a^2*c^8*d^15*e^5 + 6983100*a^3*
c^7*d^13*e^7 - 9035650*a^4*c^6*d^11*e^9 + 8003710*a^5*c^5*d^9*e^11 - 4813380*a^6*c^4*d^7*e^13 + 1885884*a^7*c^
3*d^5*e^15 - 435435*a^8*c^2*d^3*e^17 + 45045*a^9*c*d*e^19 + 229376*(121*c^10*d^11*e^9 + 41*a*c^9*d^9*e^11)*x^8
+ 14336*(5503*c^10*d^12*e^8 + 4482*a*c^9*d^10*e^10 + 383*a^2*c^8*d^8*e^12)*x^7 + 1024*(119055*c^10*d^13*e^7 +
182129*a*c^9*d^11*e^9 + 37489*a^2*c^8*d^9*e^11 + 15*a^3*c^7*d^7*e^13)*x^6 + 256*(424895*c^10*d^14*e^6 + 11577
40*a*c^9*d^12*e^8 + 448938*a^2*c^8*d^10*e^10 + 620*a^3*c^7*d^8*e^12 - 65*a^4*c^6*d^6*e^14)*x^5 + 128*(419983*c
^10*d^15*e^5 + 2149035*a*c^9*d^13*e^7 + 1490630*a^2*c^8*d^11*e^9 + 5830*a^3*c^7*d^9*e^11 - 1365*a^4*c^6*d^7*e^
13 + 143*a^5*c^5*d^5*e^15)*x^4 + 16*(735993*c^10*d^16*e^4 + 9023498*a*c^9*d^14*e^6 + 11825815*a^2*c^8*d^12*e^8
+ 132300*a^3*c^7*d^10*e^10 - 52585*a^4*c^6*d^8*e^12 + 12298*a^5*c^5*d^6*e^14 - 1287*a^6*c^4*d^4*e^16)*x^3 + 8
*(3003*c^10*d^17*e^3 + 4394937*a*c^9*d^15*e^5 + 13885683*a^2*c^8*d^13*e^7 + 508825*a^3*c^7*d^11*e^9 - 310375*a
^4*c^6*d^9*e^11 + 123123*a^5*c^5*d^7*e^13 - 28743*a^6*c^4*d^5*e^15 + 3003*a^7*c^3*d^3*e^17)*x^2 - 2*(15015*c^1
0*d^18*e^2 - 144144*a*c^9*d^16*e^4 - 17075244*a^2*c^8*d^14*e^6 - 2878000*a^3*c^7*d^12*e^8 + 2579850*a^4*c^6*d^
10*e^10 - 1567280*a^5*c^5*d^8*e^12 + 619476*a^6*c^4*d^6*e^14 - 144144*a^7*c^3*d^4*e^16 + 15015*a^8*c^2*d^2*e^1
8)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^8*d^8*e^4), 1/82575360*(45045*(c^10*d^20 - 10*a*c^9*d^18
*e^2 + 45*a^2*c^8*d^16*e^4 - 120*a^3*c^7*d^14*e^6 + 210*a^4*c^6*d^12*e^8 - 252*a^5*c^5*d^10*e^10 + 210*a^6*c^4
*d^8*e^12 - 120*a^7*c^3*d^6*e^14 + 45*a^8*c^2*d^4*e^16 - 10*a^9*c*d^2*e^18 + a^10*e^20)*sqrt(-c*d*e)*arctan(1/
2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*e*x + c*d^2 + a*e^2)*sqrt(-c*d*e)/(c^2*d^2*e^2*x^2 + a*c*
d^2*e^2 + (c^2*d^3*e + a*c*d*e^3)*x)) + 2*(4128768*c^10*d^10*e^10*x^9 + 45045*c^10*d^19*e - 435435*a*c^9*d^17*
e^3 + 1885884*a^2*c^8*d^15*e^5 + 6983100*a^3*c^7*d^13*e^7 - 9035650*a^4*c^6*d^11*e^9 + 8003710*a^5*c^5*d^9*e^1
1 - 4813380*a^6*c^4*d^7*e^13 + 1885884*a^7*c^3*d^5*e^15 - 435435*a^8*c^2*d^3*e^17 + 45045*a^9*c*d*e^19 + 22937
6*(121*c^10*d^11*e^9 + 41*a*c^9*d^9*e^11)*x^8 + 14336*(5503*c^10*d^12*e^8 + 4482*a*c^9*d^10*e^10 + 383*a^2*c^8
*d^8*e^12)*x^7 + 1024*(119055*c^10*d^13*e^7 + 182129*a*c^9*d^11*e^9 + 37489*a^2*c^8*d^9*e^11 + 15*a^3*c^7*d^7*
e^13)*x^6 + 256*(424895*c^10*d^14*e^6 + 1157740*a*c^9*d^12*e^8 + 448938*a^2*c^8*d^10*e^10 + 620*a^3*c^7*d^8*e^
12 - 65*a^4*c^6*d^6*e^14)*x^5 + 128*(419983*c^10*d^15*e^5 + 2149035*a*c^9*d^13*e^7 + 1490630*a^2*c^8*d^11*e^9
+ 5830*a^3*c^7*d^9*e^11 - 1365*a^4*c^6*d^7*e^13 + 143*a^5*c^5*d^5*e^15)*x^4 + 16*(735993*c^10*d^16*e^4 + 90234
98*a*c^9*d^14*e^6 + 11825815*a^2*c^8*d^12*e^8 + 132300*a^3*c^7*d^10*e^10 - 52585*a^4*c^6*d^8*e^12 + 12298*a^5*
c^5*d^6*e^14 - 1287*a^6*c^4*d^4*e^16)*x^3 + 8*(3003*c^10*d^17*e^3 + 4394937*a*c^9*d^15*e^5 + 13885683*a^2*c^8*
d^13*e^7 + 508825*a^3*c^7*d^11*e^9 - 310375*a^4*c^6*d^9*e^11 + 123123*a^5*c^5*d^7*e^13 - 28743*a^6*c^4*d^5*e^1
5 + 3003*a^7*c^3*d^3*e^17)*x^2 - 2*(15015*c^10*d^18*e^2 - 144144*a*c^9*d^16*e^4 - 17075244*a^2*c^8*d^14*e^6 -
2878000*a^3*c^7*d^12*e^8 + 2579850*a^4*c^6*d^10*e^10 - 1567280*a^5*c^5*d^8*e^12 + 619476*a^6*c^4*d^6*e^14 - 14
4144*a^7*c^3*d^4*e^16 + 15015*a^8*c^2*d^2*e^18)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^8*d^8*e^4)]

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giac [B]  time = 0.80, size = 1026, normalized size = 1.92

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="giac")

[Out]

1/41287680*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*(2*(4*(2*(8*(2*(4*(14*(16*(18*c^2*d^2*x*e^6 + (121*c^11
*d^12*e^14 + 41*a*c^10*d^10*e^16)*e^(-9)/(c^9*d^9))*x + (5503*c^11*d^13*e^13 + 4482*a*c^10*d^11*e^15 + 383*a^2
*c^9*d^9*e^17)*e^(-9)/(c^9*d^9))*x + (119055*c^11*d^14*e^12 + 182129*a*c^10*d^12*e^14 + 37489*a^2*c^9*d^10*e^1
6 + 15*a^3*c^8*d^8*e^18)*e^(-9)/(c^9*d^9))*x + (424895*c^11*d^15*e^11 + 1157740*a*c^10*d^13*e^13 + 448938*a^2*
c^9*d^11*e^15 + 620*a^3*c^8*d^9*e^17 - 65*a^4*c^7*d^7*e^19)*e^(-9)/(c^9*d^9))*x + (419983*c^11*d^16*e^10 + 214
9035*a*c^10*d^14*e^12 + 1490630*a^2*c^9*d^12*e^14 + 5830*a^3*c^8*d^10*e^16 - 1365*a^4*c^7*d^8*e^18 + 143*a^5*c
^6*d^6*e^20)*e^(-9)/(c^9*d^9))*x + (735993*c^11*d^17*e^9 + 9023498*a*c^10*d^15*e^11 + 11825815*a^2*c^9*d^13*e^
13 + 132300*a^3*c^8*d^11*e^15 - 52585*a^4*c^7*d^9*e^17 + 12298*a^5*c^6*d^7*e^19 - 1287*a^6*c^5*d^5*e^21)*e^(-9
)/(c^9*d^9))*x + (3003*c^11*d^18*e^8 + 4394937*a*c^10*d^16*e^10 + 13885683*a^2*c^9*d^14*e^12 + 508825*a^3*c^8*
d^12*e^14 - 310375*a^4*c^7*d^10*e^16 + 123123*a^5*c^6*d^8*e^18 - 28743*a^6*c^5*d^6*e^20 + 3003*a^7*c^4*d^4*e^2
2)*e^(-9)/(c^9*d^9))*x - (15015*c^11*d^19*e^7 - 144144*a*c^10*d^17*e^9 - 17075244*a^2*c^9*d^15*e^11 - 2878000*
a^3*c^8*d^13*e^13 + 2579850*a^4*c^7*d^11*e^15 - 1567280*a^5*c^6*d^9*e^17 + 619476*a^6*c^5*d^7*e^19 - 144144*a^
7*c^4*d^5*e^21 + 15015*a^8*c^3*d^3*e^23)*e^(-9)/(c^9*d^9))*x + (45045*c^11*d^20*e^6 - 435435*a*c^10*d^18*e^8 +
1885884*a^2*c^9*d^16*e^10 + 6983100*a^3*c^8*d^14*e^12 - 9035650*a^4*c^7*d^12*e^14 + 8003710*a^5*c^6*d^10*e^16
- 4813380*a^6*c^5*d^8*e^18 + 1885884*a^7*c^4*d^6*e^20 - 435435*a^8*c^3*d^4*e^22 + 45045*a^9*c^2*d^2*e^24)*e^(
-9)/(c^9*d^9)) + 143/262144*(c^10*d^20 - 10*a*c^9*d^18*e^2 + 45*a^2*c^8*d^16*e^4 - 120*a^3*c^7*d^14*e^6 + 210*
a^4*c^6*d^12*e^8 - 252*a^5*c^5*d^10*e^10 + 210*a^6*c^4*d^8*e^12 - 120*a^7*c^3*d^6*e^14 + 45*a^8*c^2*d^4*e^16 -
10*a^9*c*d^2*e^18 + a^10*e^20)*e^(-7/2)*log(abs(-c*d^2 - 2*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x +
a*x*e^2 + a*d*e))*sqrt(c*d)*e^(1/2) - a*e^2))/(sqrt(c*d)*c^7*d^7)

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maple [B]  time = 0.08, size = 2973, normalized size = 5.57 \begin {gather*} \text {output too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2),x)

[Out]

1137/4480/c*d^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(7/2)+143/7680*d^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*x
+715/49152*d^6*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a+143/15360/e*d^5*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/
2)+1/10*e^3*x^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(7/2)/c/d-1001/131072/e*c*d^9*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)
*x)^(1/2)*a+715/32768*e*d^7*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^2+143/131072/e^3*c^2*d^11*(c*d*e*x^2+a*d
*e+(a*e^2+c*d^2)*x)^(1/2)-143/49152/e^2*c*d^8*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)+67/180*e^2/c*x^2*(c*d*e*
x^2+a*d*e+(a*e^2+c*d^2)*x)^(7/2)-10777/40320*e^2/c^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(7/2)*a+715/49152*e^6/c
^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a^4-143/262144/e^3*c^3*d^13*ln((c*d*e*x+1/2*a*e^2+1/2*c*d^2)/(c*d*e
)^(1/2)+(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2))/(c*d*e)^(1/2)-1001/131072*e^13/c^6/d^5*(c*d*e*x^2+a*d*e+(a*e^
2+c*d^2)*x)^(1/2)*a^8+715/32768*e^11/c^5/d^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^7+143/7680*e^3/c^2*d*(c
*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*a^2-143/5120*e/c*d^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*a-429/16384
*e^2/c*d^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a^2+143/4096*e*d^5*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*
x*a-143/8192*c*d^8*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*a-1001/32768*e^3/c*d^5*(c*d*e*x^2+a*d*e+(a*e^2+c*
d^2)*x)^(1/2)*a^3+143/7680*e^5/c^3/d*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*a^3-1001/32768*e^9/c^4/d*(c*d*e*x
^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^6+1001/65536*e^7/c^3*d*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^5-143/4480*
e^6/c^4/d^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(7/2)*a^3+5863/40320*e^4/c^3/d^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*
x)^(7/2)*a^2+1423/2880*e/c*d*x*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(7/2)+143/131072*e^15/c^7/d^7*(c*d*e*x^2+a*d*
e+(a*e^2+c*d^2)*x)^(1/2)*a^9+1001/65536*e^5/c^2*d^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^4-143/49152*e^12
/c^6/d^6*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a^7+715/49152*e^10/c^5/d^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^
(3/2)*a^6-429/16384*e^8/c^4/d^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a^5-143/24576/e*c*d^7*(c*d*e*x^2+a*d*e
+(a*e^2+c*d^2)*x)^(3/2)*x+143/65536/e^2*c^2*d^10*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x+715/49152*e^4/c^2*d
^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a^3+143/15360*e^9/c^5/d^5*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*a
^5-143/5120*e^7/c^4/d^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*a^4-1001/8192*e^8/c^3*(c*d*e*x^2+a*d*e+(a*e^2+
c*d^2)*x)^(1/2)*x*a^5+1001/16384*e^2*d^6*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*a^2+143/1280*e^4/c^2*(c*d*e
*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*x*a^2+2145/32768*e^3*d^7*ln((c*d*e*x+1/2*a*e^2+1/2*c*d^2)/(c*d*e)^(1/2)+(c*d
*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2))/(c*d*e)^(1/2)*a^3-15015/131072*e^5/c*d^5*ln((c*d*e*x+1/2*a*e^2+1/2*c*d^2)
/(c*d*e)^(1/2)+(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2))/(c*d*e)^(1/2)*a^4-13/180*e^4/c^2/d^2*x^2*(c*d*e*x^2+a*
d*e+(a*e^2+c*d^2)*x)^(7/2)*a+715/131072/e*c^2*d^11*ln((c*d*e*x+1/2*a*e^2+1/2*c*d^2)/(c*d*e)^(1/2)+(c*d*e*x^2+a
*d*e+(a*e^2+c*d^2)*x)^(1/2))/(c*d*e)^(1/2)*a+1001/16384*e^10/c^4/d^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x
*a^6+5005/32768*e^6/c^2*d^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*a^4-1001/8192*e^4/c*d^4*(c*d*e*x^2+a*d*e
+(a*e^2+c*d^2)*x)^(1/2)*x*a^3-15015/131072*e^9/c^3*d*ln((c*d*e*x+1/2*a*e^2+1/2*c*d^2)/(c*d*e)^(1/2)+(c*d*e*x^2
+a*d*e+(a*e^2+c*d^2)*x)^(1/2))/(c*d*e)^(1/2)*a^6+9009/65536*e^7/c^2*d^3*ln((c*d*e*x+1/2*a*e^2+1/2*c*d^2)/(c*d*
e)^(1/2)+(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2))/(c*d*e)^(1/2)*a^5-6435/262144*e^13/c^5/d^3*ln((c*d*e*x+1/2*a
*e^2+1/2*c*d^2)/(c*d*e)^(1/2)+(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2))/(c*d*e)^(1/2)*a^8-143/24576*e^11/c^5/d^
5*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x*a^6+143/4096*e^9/c^4/d^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x
*a^5-715/8192*e^7/c^3/d*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x*a^4-143/262144*e^17/c^7/d^7*ln((c*d*e*x+1/2*
a*e^2+1/2*c*d^2)/(c*d*e)^(1/2)+(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2))/(c*d*e)^(1/2)*a^10+715/131072*e^15/c^6
/d^5*ln((c*d*e*x+1/2*a*e^2+1/2*c*d^2)/(c*d*e)^(1/2)+(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2))/(c*d*e)^(1/2)*a^9
+143/2880*e^5/c^3/d^3*x*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(7/2)*a^2+715/6144*e^5/c^2*d*(c*d*e*x^2+a*d*e+(a*e^2
+c*d^2)*x)^(3/2)*x*a^3+143/7680*e^8/c^4/d^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*x*a^4-143/1920*e^6/c^3/d^2
*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*x*a^3-6435/262144*e*c*d^9*ln((c*d*e*x+1/2*a*e^2+1/2*c*d^2)/(c*d*e)^(1
/2)+(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2))/(c*d*e)^(1/2)*a^2+143/65536*e^14/c^6/d^6*(c*d*e*x^2+a*d*e+(a*e^2+
c*d^2)*x)^(1/2)*x*a^8-143/8192*e^12/c^5/d^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*a^7-39/160*e^3/c^2/d*x*(
c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(7/2)*a-143/1920*e^2/c*d^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*x*a-715/81
92*e^3/c*d^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x*a^2+2145/32768*e^11/c^4/d*ln((c*d*e*x+1/2*a*e^2+1/2*c*d
^2)/(c*d*e)^(1/2)+(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2))/(c*d*e)^(1/2)*a^7

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*e^2-c*d^2>0)', see `assume?`
for more details)Is a*e^2-c*d^2 zero or nonzero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (d+e\,x\right )}^4\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^4*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2),x)

[Out]

int((d + e*x)^4*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {5}{2}} \left (d + e x\right )^{4}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**4*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)

[Out]

Integral(((d + e*x)*(a*e + c*d*x))**(5/2)*(d + e*x)**4, x)

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