3.17.18 \(\int (d+e x)^3 (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2} \, dx\)

Optimal. Leaf size=474 \[ -\frac {55 \left (c d^2-a e^2\right )^9 \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{65536 c^{13/2} d^{13/2} e^{7/2}}+\frac {55 \left (c d^2-a e^2\right )^7 \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{32768 c^6 d^6 e^3}-\frac {55 \left (c d^2-a e^2\right )^5 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{12288 c^5 d^5 e^2}+\frac {11 \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{768 c^4 d^4 e}+\frac {11 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{224 c^3 d^3}+\frac {11 (d+e x) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{144 c^2 d^2}+\frac {(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 c d} \]

________________________________________________________________________________________

Rubi [A]  time = 0.49, antiderivative size = 474, normalized size of antiderivative = 1.00, number of steps used = 8, 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 {55 \left (c d^2-a e^2\right )^7 \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}}{32768 c^6 d^6 e^3}-\frac {55 \left (c d^2-a e^2\right )^5 \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}}{12288 c^5 d^5 e^2}+\frac {11 \left (c d^2-a e^2\right )^3 \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}}{768 c^4 d^4 e}+\frac {11 \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}}{224 c^3 d^3}+\frac {11 (d+e x) \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}}{144 c^2 d^2}-\frac {55 \left (c d^2-a e^2\right )^9 \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 )}{65536 c^{13/2} d^{13/2} e^{7/2}}+\frac {(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 c d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

(55*(c*d^2 - a*e^2)^7*(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])/(32768*c^6*d^6*
e^3) - (55*(c*d^2 - a*e^2)^5*(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))/(12288
*c^5*d^5*e^2) + (11*(c*d^2 - a*e^2)^3*(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
))/(768*c^4*d^4*e) + (11*(c*d^2 - a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(224*c^3*d^3) + (11*
(c*d^2 - a*e^2)*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(144*c^2*d^2) + ((d + e*x)^2*(a*d*e +
(c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(9*c*d) - (55*(c*d^2 - a*e^2)^9*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])])/(65536*c^(13/2)*d^(13/2)*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)^3 \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)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{9 c d}+\frac {\left (11 \left (d^2-\frac {a e^2}{c}\right )\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}{18 d}\\ &=\frac {11 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{144 c^2 d^2}+\frac {(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}}{9 c d}+\frac {\left (11 \left (d^2-\frac {a e^2}{c}\right )^2\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}{32 d^2}\\ &=\frac {11 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{224 c^3 d^3}+\frac {11 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{144 c^2 d^2}+\frac {(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}}{9 c d}+\frac {\left (11 \left (d^2-\frac {a e^2}{c}\right )^3\right ) \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx}{64 d^3}\\ &=\frac {11 \left (c d^2-a e^2\right )^3 \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}}{768 c^4 d^4 e}+\frac {11 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{224 c^3 d^3}+\frac {11 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{144 c^2 d^2}+\frac {(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}}{9 c d}-\frac {\left (55 \left (c d^2-a e^2\right )^5\right ) \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{1536 c^4 d^4 e}\\ &=-\frac {55 \left (c d^2-a e^2\right )^5 \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}}{12288 c^5 d^5 e^2}+\frac {11 \left (c d^2-a e^2\right )^3 \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}}{768 c^4 d^4 e}+\frac {11 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{224 c^3 d^3}+\frac {11 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{144 c^2 d^2}+\frac {(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}}{9 c d}+\frac {\left (55 \left (c d^2-a e^2\right )^7\right ) \int \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{8192 c^5 d^5 e^2}\\ &=\frac {55 \left (c d^2-a e^2\right )^7 \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}}{32768 c^6 d^6 e^3}-\frac {55 \left (c d^2-a e^2\right )^5 \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}}{12288 c^5 d^5 e^2}+\frac {11 \left (c d^2-a e^2\right )^3 \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}}{768 c^4 d^4 e}+\frac {11 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{224 c^3 d^3}+\frac {11 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{144 c^2 d^2}+\frac {(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}}{9 c d}-\frac {\left (55 \left (c d^2-a e^2\right )^9\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{65536 c^6 d^6 e^3}\\ &=\frac {55 \left (c d^2-a e^2\right )^7 \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}}{32768 c^6 d^6 e^3}-\frac {55 \left (c d^2-a e^2\right )^5 \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}}{12288 c^5 d^5 e^2}+\frac {11 \left (c d^2-a e^2\right )^3 \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}}{768 c^4 d^4 e}+\frac {11 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{224 c^3 d^3}+\frac {11 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{144 c^2 d^2}+\frac {(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}}{9 c d}-\frac {\left (55 \left (c d^2-a e^2\right )^9\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 )}{32768 c^6 d^6 e^3}\\ &=\frac {55 \left (c d^2-a e^2\right )^7 \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}}{32768 c^6 d^6 e^3}-\frac {55 \left (c d^2-a e^2\right )^5 \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}}{12288 c^5 d^5 e^2}+\frac {11 \left (c d^2-a e^2\right )^3 \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}}{768 c^4 d^4 e}+\frac {11 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{224 c^3 d^3}+\frac {11 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{144 c^2 d^2}+\frac {(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}}{9 c d}-\frac {55 \left (c d^2-a e^2\right )^9 \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 )}{65536 c^{13/2} d^{13/2} e^{7/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B]  time = 6.50, size = 1359, normalized size = 2.87 \begin {gather*} \frac {2 \left (c d^2-a e^2\right )^5 (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 )^{13/2} \left (\frac {385 \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}{131072 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 )^6}+\frac {7}{18} \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 {11}{16 \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 {99}{224 \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 {33}{128 \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 {33}{256 \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 {99}{2048 \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}\right )\right )}{7 c^6 d^6 \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 )^{11/2} (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)^3*(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)^5*(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))))^(13/2)*((7*(99/(2048*(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) + 33/(256*(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) + 33/(128*(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) + 99/(224*(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) + 11/(16*(
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)))/1
8 + (385*(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)))])))/(131072*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))))^6)))/(7*c^6*d^6*((c*d)/((c^2*d^3)/(c*d^2 - a*e^2)
- (a*c*d*e^2)/(c*d^2 - a*e^2)))^(11/2)*(d + e*x)^2*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 180.16, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

$Aborted

________________________________________________________________________________________

fricas [B]  time = 0.59, size = 1806, normalized size = 3.81

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="fricas")

[Out]

[1/8257536*(3465*(c^9*d^18 - 9*a*c^8*d^16*e^2 + 36*a^2*c^7*d^14*e^4 - 84*a^3*c^6*d^12*e^6 + 126*a^4*c^5*d^10*e
^8 - 126*a^5*c^4*d^8*e^10 + 84*a^6*c^3*d^6*e^12 - 36*a^7*c^2*d^4*e^14 + 9*a^8*c*d^2*e^16 - a^9*e^18)*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*(229376*c^9*d^9*e^9*x^8 + 3465*c^9*d
^17*e - 30030*a*c^8*d^15*e^3 + 115038*a^2*c^7*d^13*e^5 + 334602*a^3*c^6*d^11*e^7 - 360448*a^4*c^5*d^9*e^9 + 25
5222*a^5*c^4*d^7*e^11 - 115038*a^6*c^3*d^5*e^13 + 30030*a^7*c^2*d^3*e^15 - 3465*a^8*c*d*e^17 + 14336*(91*c^9*d
^10*e^8 + 37*a*c^8*d^8*e^10)*x^7 + 1024*(2955*c^9*d^11*e^7 + 3008*a*c^8*d^9*e^9 + 309*a^2*c^7*d^7*e^11)*x^6 +
256*(14075*c^9*d^12*e^6 + 28695*a*c^8*d^10*e^8 + 7401*a^2*c^7*d^8*e^10 + 5*a^3*c^6*d^6*e^12)*x^5 + 128*(17419*
c^9*d^13*e^5 + 71074*a*c^8*d^11*e^7 + 36864*a^2*c^7*d^9*e^9 + 94*a^3*c^6*d^7*e^11 - 11*a^4*c^5*d^5*e^13)*x^4 +
16*(36765*c^9*d^14*e^4 + 373583*a*c^8*d^12*e^6 + 390018*a^2*c^7*d^10*e^8 + 3198*a^3*c^6*d^8*e^10 - 847*a^4*c^
5*d^6*e^12 + 99*a^5*c^4*d^4*e^14)*x^3 + 8*(231*c^9*d^15*e^3 + 219204*a*c^8*d^13*e^5 + 572739*a^2*c^7*d^11*e^7
+ 16384*a^3*c^6*d^9*e^9 - 7491*a^4*c^5*d^7*e^11 + 1980*a^5*c^4*d^5*e^13 - 231*a^6*c^3*d^3*e^15)*x^2 - 2*(1155*
c^9*d^16*e^2 - 9933*a*c^8*d^14*e^4 - 847017*a^2*c^7*d^12*e^6 - 115609*a^3*c^6*d^10*e^8 + 82841*a^4*c^5*d^8*e^1
0 - 37719*a^5*c^4*d^6*e^12 + 9933*a^6*c^3*d^4*e^14 - 1155*a^7*c^2*d^2*e^16)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2
+ a*e^2)*x))/(c^7*d^7*e^4), 1/4128768*(3465*(c^9*d^18 - 9*a*c^8*d^16*e^2 + 36*a^2*c^7*d^14*e^4 - 84*a^3*c^6*d
^12*e^6 + 126*a^4*c^5*d^10*e^8 - 126*a^5*c^4*d^8*e^10 + 84*a^6*c^3*d^6*e^12 - 36*a^7*c^2*d^4*e^14 + 9*a^8*c*d^
2*e^16 - a^9*e^18)*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*(229376*c^9*d^9*e^9*x^8 + 3
465*c^9*d^17*e - 30030*a*c^8*d^15*e^3 + 115038*a^2*c^7*d^13*e^5 + 334602*a^3*c^6*d^11*e^7 - 360448*a^4*c^5*d^9
*e^9 + 255222*a^5*c^4*d^7*e^11 - 115038*a^6*c^3*d^5*e^13 + 30030*a^7*c^2*d^3*e^15 - 3465*a^8*c*d*e^17 + 14336*
(91*c^9*d^10*e^8 + 37*a*c^8*d^8*e^10)*x^7 + 1024*(2955*c^9*d^11*e^7 + 3008*a*c^8*d^9*e^9 + 309*a^2*c^7*d^7*e^1
1)*x^6 + 256*(14075*c^9*d^12*e^6 + 28695*a*c^8*d^10*e^8 + 7401*a^2*c^7*d^8*e^10 + 5*a^3*c^6*d^6*e^12)*x^5 + 12
8*(17419*c^9*d^13*e^5 + 71074*a*c^8*d^11*e^7 + 36864*a^2*c^7*d^9*e^9 + 94*a^3*c^6*d^7*e^11 - 11*a^4*c^5*d^5*e^
13)*x^4 + 16*(36765*c^9*d^14*e^4 + 373583*a*c^8*d^12*e^6 + 390018*a^2*c^7*d^10*e^8 + 3198*a^3*c^6*d^8*e^10 - 8
47*a^4*c^5*d^6*e^12 + 99*a^5*c^4*d^4*e^14)*x^3 + 8*(231*c^9*d^15*e^3 + 219204*a*c^8*d^13*e^5 + 572739*a^2*c^7*
d^11*e^7 + 16384*a^3*c^6*d^9*e^9 - 7491*a^4*c^5*d^7*e^11 + 1980*a^5*c^4*d^5*e^13 - 231*a^6*c^3*d^3*e^15)*x^2 -
2*(1155*c^9*d^16*e^2 - 9933*a*c^8*d^14*e^4 - 847017*a^2*c^7*d^12*e^6 - 115609*a^3*c^6*d^10*e^8 + 82841*a^4*c^
5*d^8*e^10 - 37719*a^5*c^4*d^6*e^12 + 9933*a^6*c^3*d^4*e^14 - 1155*a^7*c^2*d^2*e^16)*x)*sqrt(c*d*e*x^2 + a*d*e
+ (c*d^2 + a*e^2)*x))/(c^7*d^7*e^4)]

________________________________________________________________________________________

giac [B]  time = 0.69, size = 875, normalized size = 1.85 \begin {gather*} \frac {1}{2064384} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (4 \, {\left (14 \, {\left (16 \, c^{2} d^{2} x e^{5} + \frac {{\left (91 \, c^{10} d^{11} e^{12} + 37 \, a c^{9} d^{9} e^{14}\right )} e^{\left (-8\right )}}{c^{8} d^{8}}\right )} x + \frac {{\left (2955 \, c^{10} d^{12} e^{11} + 3008 \, a c^{9} d^{10} e^{13} + 309 \, a^{2} c^{8} d^{8} e^{15}\right )} e^{\left (-8\right )}}{c^{8} d^{8}}\right )} x + \frac {{\left (14075 \, c^{10} d^{13} e^{10} + 28695 \, a c^{9} d^{11} e^{12} + 7401 \, a^{2} c^{8} d^{9} e^{14} + 5 \, a^{3} c^{7} d^{7} e^{16}\right )} e^{\left (-8\right )}}{c^{8} d^{8}}\right )} x + \frac {{\left (17419 \, c^{10} d^{14} e^{9} + 71074 \, a c^{9} d^{12} e^{11} + 36864 \, a^{2} c^{8} d^{10} e^{13} + 94 \, a^{3} c^{7} d^{8} e^{15} - 11 \, a^{4} c^{6} d^{6} e^{17}\right )} e^{\left (-8\right )}}{c^{8} d^{8}}\right )} x + \frac {{\left (36765 \, c^{10} d^{15} e^{8} + 373583 \, a c^{9} d^{13} e^{10} + 390018 \, a^{2} c^{8} d^{11} e^{12} + 3198 \, a^{3} c^{7} d^{9} e^{14} - 847 \, a^{4} c^{6} d^{7} e^{16} + 99 \, a^{5} c^{5} d^{5} e^{18}\right )} e^{\left (-8\right )}}{c^{8} d^{8}}\right )} x + \frac {{\left (231 \, c^{10} d^{16} e^{7} + 219204 \, a c^{9} d^{14} e^{9} + 572739 \, a^{2} c^{8} d^{12} e^{11} + 16384 \, a^{3} c^{7} d^{10} e^{13} - 7491 \, a^{4} c^{6} d^{8} e^{15} + 1980 \, a^{5} c^{5} d^{6} e^{17} - 231 \, a^{6} c^{4} d^{4} e^{19}\right )} e^{\left (-8\right )}}{c^{8} d^{8}}\right )} x - \frac {{\left (1155 \, c^{10} d^{17} e^{6} - 9933 \, a c^{9} d^{15} e^{8} - 847017 \, a^{2} c^{8} d^{13} e^{10} - 115609 \, a^{3} c^{7} d^{11} e^{12} + 82841 \, a^{4} c^{6} d^{9} e^{14} - 37719 \, a^{5} c^{5} d^{7} e^{16} + 9933 \, a^{6} c^{4} d^{5} e^{18} - 1155 \, a^{7} c^{3} d^{3} e^{20}\right )} e^{\left (-8\right )}}{c^{8} d^{8}}\right )} x + \frac {{\left (3465 \, c^{10} d^{18} e^{5} - 30030 \, a c^{9} d^{16} e^{7} + 115038 \, a^{2} c^{8} d^{14} e^{9} + 334602 \, a^{3} c^{7} d^{12} e^{11} - 360448 \, a^{4} c^{6} d^{10} e^{13} + 255222 \, a^{5} c^{5} d^{8} e^{15} - 115038 \, a^{6} c^{4} d^{6} e^{17} + 30030 \, a^{7} c^{3} d^{4} e^{19} - 3465 \, a^{8} c^{2} d^{2} e^{21}\right )} e^{\left (-8\right )}}{c^{8} d^{8}}\right )} + \frac {55 \, {\left (c^{9} d^{18} - 9 \, a c^{8} d^{16} e^{2} + 36 \, a^{2} c^{7} d^{14} e^{4} - 84 \, a^{3} c^{6} d^{12} e^{6} + 126 \, a^{4} c^{5} d^{10} e^{8} - 126 \, a^{5} c^{4} d^{8} e^{10} + 84 \, a^{6} c^{3} d^{6} e^{12} - 36 \, a^{7} c^{2} d^{4} e^{14} + 9 \, a^{8} c d^{2} e^{16} - a^{9} e^{18}\right )} e^{\left (-\frac {7}{2}\right )} \log \left ({\left | -c d^{2} - 2 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} \sqrt {c d} e^{\frac {1}{2}} - a e^{2} \right |}\right )}{65536 \, \sqrt {c d} c^{6} d^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="giac")

[Out]

1/2064384*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*c^2*d^2*x*e^5 + (91*c^10*d^11*
e^12 + 37*a*c^9*d^9*e^14)*e^(-8)/(c^8*d^8))*x + (2955*c^10*d^12*e^11 + 3008*a*c^9*d^10*e^13 + 309*a^2*c^8*d^8*
e^15)*e^(-8)/(c^8*d^8))*x + (14075*c^10*d^13*e^10 + 28695*a*c^9*d^11*e^12 + 7401*a^2*c^8*d^9*e^14 + 5*a^3*c^7*
d^7*e^16)*e^(-8)/(c^8*d^8))*x + (17419*c^10*d^14*e^9 + 71074*a*c^9*d^12*e^11 + 36864*a^2*c^8*d^10*e^13 + 94*a^
3*c^7*d^8*e^15 - 11*a^4*c^6*d^6*e^17)*e^(-8)/(c^8*d^8))*x + (36765*c^10*d^15*e^8 + 373583*a*c^9*d^13*e^10 + 39
0018*a^2*c^8*d^11*e^12 + 3198*a^3*c^7*d^9*e^14 - 847*a^4*c^6*d^7*e^16 + 99*a^5*c^5*d^5*e^18)*e^(-8)/(c^8*d^8))
*x + (231*c^10*d^16*e^7 + 219204*a*c^9*d^14*e^9 + 572739*a^2*c^8*d^12*e^11 + 16384*a^3*c^7*d^10*e^13 - 7491*a^
4*c^6*d^8*e^15 + 1980*a^5*c^5*d^6*e^17 - 231*a^6*c^4*d^4*e^19)*e^(-8)/(c^8*d^8))*x - (1155*c^10*d^17*e^6 - 993
3*a*c^9*d^15*e^8 - 847017*a^2*c^8*d^13*e^10 - 115609*a^3*c^7*d^11*e^12 + 82841*a^4*c^6*d^9*e^14 - 37719*a^5*c^
5*d^7*e^16 + 9933*a^6*c^4*d^5*e^18 - 1155*a^7*c^3*d^3*e^20)*e^(-8)/(c^8*d^8))*x + (3465*c^10*d^18*e^5 - 30030*
a*c^9*d^16*e^7 + 115038*a^2*c^8*d^14*e^9 + 334602*a^3*c^7*d^12*e^11 - 360448*a^4*c^6*d^10*e^13 + 255222*a^5*c^
5*d^8*e^15 - 115038*a^6*c^4*d^6*e^17 + 30030*a^7*c^3*d^4*e^19 - 3465*a^8*c^2*d^2*e^21)*e^(-8)/(c^8*d^8)) + 55/
65536*(c^9*d^18 - 9*a*c^8*d^16*e^2 + 36*a^2*c^7*d^14*e^4 - 84*a^3*c^6*d^12*e^6 + 126*a^4*c^5*d^10*e^8 - 126*a^
5*c^4*d^8*e^10 + 84*a^6*c^3*d^6*e^12 - 36*a^7*c^2*d^4*e^14 + 9*a^8*c*d^2*e^16 - a^9*e^18)*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^6*d^6)

________________________________________________________________________________________

maple [B]  time = 0.06, size = 2368, normalized size = 5.00

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2),x)

[Out]

11/224*e^4/c^3/d^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(7/2)*a^2+55/12288*e^10/c^5/d^5*(c*d*e*x^2+a*d*e+(a*e^2+c
*d^2)*x)^(3/2)*a^6+53/224/c*d*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(7/2)+11/384*d^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2
)*x)^(5/2)*x+55/3072*d^5*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a+11/768/e*d^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)
*x)^(5/2)-55/12288/e^2*c*d^7*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)+55/32768/e^3*c^2*d^10*(c*d*e*x^2+a*d*e+(a
*e^2+c*d^2)*x)^(1/2)+385/16384*e^7/c^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^5+43/144*e/c*x*(c*d*e*x^2+a*d
*e+(a*e^2+c*d^2)*x)^(7/2)+385/16384*e*d^6*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^2-55/3072*e^8/c^4/d^3*(c*d
*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a^5+275/12288*e^6/c^3/d*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a^4+1/9*e^
2*x^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(7/2)/c/d+1155/16384*e^3*d^6*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+275/3072*e^5/c^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^
2)*x)^(3/2)*x*a^3+495/65536/e*c^2*d^10*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-3465/32768*e^5/c*d^4*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+385/16384*e^10/c^4/d^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/
2)*x*a^6-11/144*e^3/c^2/d^2*x*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(7/2)*a-11/384*e^6/c^3/d^3*(c*d*e*x^2+a*d*e+(a
*e^2+c*d^2)*x)^(5/2)*x*a^3+3465/32768*e^7/c^2*d^2*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+11/128*e^4/c^2/d*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*x*a^2-2
75/6144*e^7/c^3/d^2*a^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x+55/6144*e^9/c^4/d^4*(c*d*e*x^2+a*d*e+(a*e^2+
c*d^2)*x)^(3/2)*x*a^5-495/65536*e^13/c^5/d^4*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+55/65536*e^15/c^6/d^6*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+495/16384*e^11/c^4/d^2*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^7-1925/16384*e^4/c*d^3*(c*d*e*x^2+
a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*a^3-55/16384*e^12/c^5/d^5*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*a^7-1155/16
384*e^8/c^3/d*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*a^5-11/128*e^2/c*d*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(
5/2)*x*a+1925/16384*e^6/c^2*d*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*a^4-495/16384*e*c*d^8*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+275/6144*e*d^4*(c*d*
e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x*a-1155/16384*e^9/c^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^6+1155/16384*e^2*d^5*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/
2)*x*a^2-11/768*e^7/c^4/d^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*a^4+11/384*e^5/c^3/d^2*(c*d*e*x^2+a*d*e+(a
*e^2+c*d^2)*x)^(5/2)*a^3-165/16384/e*c*d^8*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a-55/65536/e^3*c^3*d^12*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)-55/32768*e^
13/c^6/d^6*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^8+165/16384*e^11/c^5/d^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x
)^(1/2)*a^7-385/16384*c*d^7*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*a+55/16384/e^2*c^2*d^9*(c*d*e*x^2+a*d*e+
(a*e^2+c*d^2)*x)^(1/2)*x-275/12288*e^2/c*d^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a^2-385/16384*e^3/c*d^4*(
c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^3-11/384*e/c*d^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*a-11/63*e^2/
c^2/d*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(7/2)*a-55/6144/e*c*d^6*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x-385/
16384*e^9/c^4/d^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^6-275/3072*e^3/c*d^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2
)*x)^(3/2)*x*a^2

________________________________________________________________________________________

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)^3*(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?

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (d+e\,x\right )}^3\,{\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)^3*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2),x)

[Out]

int((d + e*x)^3*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2), x)

________________________________________________________________________________________

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 )^{3}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3*(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)**3, x)

________________________________________________________________________________________