3.17.4 \(\int (d+e x)^4 (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2} \, dx\)

Optimal. Leaf size=461 \[ \frac {99 \left (c d^2-a e^2\right )^8 \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{32768 c^{13/2} d^{13/2} e^{5/2}}-\frac {99 \left (c d^2-a e^2\right )^6 \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{16384 c^6 d^6 e^2}+\frac {33 \left (c d^2-a e^2\right )^4 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2048 c^5 d^5 e}+\frac {33 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{640 c^4 d^4}+\frac {33 (d+e x) \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{448 c^3 d^3}+\frac {11 (d+e x)^2 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{112 c^2 d^2}+\frac {(d+e x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{8 c d} \]

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Rubi [A]  time = 0.55, antiderivative size = 461, 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 {99 \left (c d^2-a e^2\right )^6 \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}}{16384 c^6 d^6 e^2}+\frac {33 \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 )^{3/2}}{2048 c^5 d^5 e}+\frac {33 \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 )^{5/2}}{640 c^4 d^4}+\frac {33 (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 )^{5/2}}{448 c^3 d^3}+\frac {11 (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 )^{5/2}}{112 c^2 d^2}+\frac {99 \left (c d^2-a e^2\right )^8 \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 )}{32768 c^{13/2} d^{13/2} e^{5/2}}+\frac {(d+e x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{8 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)^(3/2),x]

[Out]

(-99*(c*d^2 - a*e^2)^6*(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])/(16384*c^6*d^6
*e^2) + (33*(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)^(3/2))/(2048
*c^5*d^5*e) + (33*(c*d^2 - a*e^2)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(640*c^4*d^4) + (33*(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)^(5/2))/(448*c^3*d^3) + (11*(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)^(5/2))/(112*c^2*d^2) + ((d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x +
c*d*e*x^2)^(5/2))/(8*c*d) + (99*(c*d^2 - a*e^2)^8*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])])/(32768*c^(13/2)*d^(13/2)*e^(5/2))

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 )^{3/2} \, 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 )^{5/2}}{8 c d}+\frac {\left (11 \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 )^{3/2} \, dx}{16 d}\\ &=\frac {11 \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 )^{5/2}}{112 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 )^{5/2}}{8 c d}+\frac {\left (99 \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 )^{3/2} \, dx}{224 d^2}\\ &=\frac {33 \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 )^{5/2}}{448 c^3 d^3}+\frac {11 \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 )^{5/2}}{112 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 )^{5/2}}{8 c d}+\frac {\left (33 \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 )^{3/2} \, dx}{128 d^3}\\ &=\frac {33 \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 )^{5/2}}{640 c^4 d^4}+\frac {33 \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 )^{5/2}}{448 c^3 d^3}+\frac {11 \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 )^{5/2}}{112 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 )^{5/2}}{8 c d}+\frac {\left (33 \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 )^{3/2} \, dx}{256 d^4}\\ &=\frac {33 \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 )^{3/2}}{2048 c^5 d^5 e}+\frac {33 \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 )^{5/2}}{640 c^4 d^4}+\frac {33 \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 )^{5/2}}{448 c^3 d^3}+\frac {11 \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 )^{5/2}}{112 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 )^{5/2}}{8 c d}-\frac {\left (99 \left (c d^2-a e^2\right )^6\right ) \int \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{4096 c^5 d^5 e}\\ &=-\frac {99 \left (c d^2-a e^2\right )^6 \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}}{16384 c^6 d^6 e^2}+\frac {33 \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 )^{3/2}}{2048 c^5 d^5 e}+\frac {33 \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 )^{5/2}}{640 c^4 d^4}+\frac {33 \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 )^{5/2}}{448 c^3 d^3}+\frac {11 \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 )^{5/2}}{112 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 )^{5/2}}{8 c d}+\frac {\left (99 \left (c d^2-a e^2\right )^8\right ) \int \frac {1}{\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 {99 \left (c d^2-a e^2\right )^6 \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}}{16384 c^6 d^6 e^2}+\frac {33 \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 )^{3/2}}{2048 c^5 d^5 e}+\frac {33 \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 )^{5/2}}{640 c^4 d^4}+\frac {33 \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 )^{5/2}}{448 c^3 d^3}+\frac {11 \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 )^{5/2}}{112 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 )^{5/2}}{8 c d}+\frac {\left (99 \left (c d^2-a e^2\right )^8\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 )}{16384 c^6 d^6 e^2}\\ &=-\frac {99 \left (c d^2-a e^2\right )^6 \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}}{16384 c^6 d^6 e^2}+\frac {33 \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 )^{3/2}}{2048 c^5 d^5 e}+\frac {33 \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 )^{5/2}}{640 c^4 d^4}+\frac {33 \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 )^{5/2}}{448 c^3 d^3}+\frac {11 \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 )^{5/2}}{112 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 )^{5/2}}{8 c d}+\frac {99 \left (c d^2-a e^2\right )^8 \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 )}{32768 c^{13/2} d^{13/2} e^{5/2}}\\ \end {align*}

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Mathematica [B]  time = 6.42, size = 1276, normalized size = 2.77 \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))^{3/2} \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 {5}{16} \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}{14 \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 {33}{56 \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}{80 \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}{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 )^5}+\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 )^6}\right )-\frac {495 \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 \left (-\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 )}{65536 c^3 d^3 e^3 (a e+c d x)^3 \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 )}{5 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) \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)^(3/2),x]

[Out]

(2*(c*d^2 - a*e^2)^5*(a*e + c*d*x)*((a*e + c*d*x)*(d + e*x))^(3/2)*(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)*((5*(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))))^6) + 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))))^5) + 33/(80*(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) + 33/(56*(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/(14*(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)))/16 -
(495*(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*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) - (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)))])))/(65536*c^3*d^3*e^3*(a*e + c*d*x)^3*(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)))/(5*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)*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])

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IntegrateAlgebraic [F]  time = 180.01, 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)^(3/2),x]

[Out]

$Aborted

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fricas [A]  time = 0.55, size = 1520, normalized size = 3.30

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)^(3/2),x, algorithm="fricas")

[Out]

[1/2293760*(3465*(c^8*d^16 - 8*a*c^7*d^14*e^2 + 28*a^2*c^6*d^12*e^4 - 56*a^3*c^5*d^10*e^6 + 70*a^4*c^4*d^8*e^8
- 56*a^5*c^3*d^6*e^10 + 28*a^6*c^2*d^4*e^12 - 8*a^7*c*d^2*e^14 + a^8*e^16)*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*(71680*c^8*d^8*e^8*x^7 - 3465*c^8*d^15*e + 26565*a*c^7*d^13*e
^3 + 140903*a^2*c^6*d^11*e^5 - 193699*a^3*c^5*d^9*e^7 + 166749*a^4*c^4*d^7*e^9 - 88473*a^5*c^3*d^5*e^11 + 2656
5*a^6*c^2*d^3*e^13 - 3465*a^7*c*d*e^15 + 5120*(81*c^8*d^9*e^7 + 17*a*c^7*d^7*e^9)*x^6 + 1280*(769*c^8*d^10*e^6
+ 406*a*c^7*d^8*e^8 + a^2*c^6*d^6*e^10)*x^5 + 128*(9461*c^8*d^11*e^5 + 10067*a*c^7*d^9*e^7 + 83*a^2*c^6*d^7*e
^9 - 11*a^3*c^5*d^5*e^11)*x^4 + 16*(49251*c^8*d^12*e^4 + 105748*a*c^7*d^10*e^6 + 2450*a^2*c^6*d^8*e^8 - 748*a^
3*c^5*d^6*e^10 + 99*a^4*c^4*d^4*e^12)*x^3 + 8*(28441*c^8*d^13*e^3 + 153301*a*c^7*d^11*e^5 + 10642*a^2*c^6*d^9*
e^7 - 5742*a^3*c^5*d^7*e^9 + 1749*a^4*c^4*d^5*e^11 - 231*a^5*c^3*d^3*e^13)*x^2 + 2*(1155*c^8*d^14*e^2 + 220598
*a*c^7*d^12*e^4 + 61709*a^2*c^6*d^10*e^6 - 53900*a^3*c^5*d^8*e^8 + 28941*a^4*c^4*d^6*e^10 - 8778*a^5*c^3*d^4*e
^12 + 1155*a^6*c^2*d^2*e^14)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^7*d^7*e^3), -1/1146880*(3465*(
c^8*d^16 - 8*a*c^7*d^14*e^2 + 28*a^2*c^6*d^12*e^4 - 56*a^3*c^5*d^10*e^6 + 70*a^4*c^4*d^8*e^8 - 56*a^5*c^3*d^6*
e^10 + 28*a^6*c^2*d^4*e^12 - 8*a^7*c*d^2*e^14 + a^8*e^16)*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*(71680*c^8*d^8*e^8*x^7 - 3465*c^8*d^15*e + 26565*a*c^7*d^13*e^3 + 140903*a^2*c^6*d^11*e^5 - 193699
*a^3*c^5*d^9*e^7 + 166749*a^4*c^4*d^7*e^9 - 88473*a^5*c^3*d^5*e^11 + 26565*a^6*c^2*d^3*e^13 - 3465*a^7*c*d*e^1
5 + 5120*(81*c^8*d^9*e^7 + 17*a*c^7*d^7*e^9)*x^6 + 1280*(769*c^8*d^10*e^6 + 406*a*c^7*d^8*e^8 + a^2*c^6*d^6*e^
10)*x^5 + 128*(9461*c^8*d^11*e^5 + 10067*a*c^7*d^9*e^7 + 83*a^2*c^6*d^7*e^9 - 11*a^3*c^5*d^5*e^11)*x^4 + 16*(4
9251*c^8*d^12*e^4 + 105748*a*c^7*d^10*e^6 + 2450*a^2*c^6*d^8*e^8 - 748*a^3*c^5*d^6*e^10 + 99*a^4*c^4*d^4*e^12)
*x^3 + 8*(28441*c^8*d^13*e^3 + 153301*a*c^7*d^11*e^5 + 10642*a^2*c^6*d^9*e^7 - 5742*a^3*c^5*d^7*e^9 + 1749*a^4
*c^4*d^5*e^11 - 231*a^5*c^3*d^3*e^13)*x^2 + 2*(1155*c^8*d^14*e^2 + 220598*a*c^7*d^12*e^4 + 61709*a^2*c^6*d^10*
e^6 - 53900*a^3*c^5*d^8*e^8 + 28941*a^4*c^4*d^6*e^10 - 8778*a^5*c^3*d^4*e^12 + 1155*a^6*c^2*d^2*e^14)*x)*sqrt(
c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^7*d^7*e^3)]

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giac [A]  time = 0.63, size = 726, normalized size = 1.57 \begin {gather*} \frac {1}{573440} \, \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 (10 \, {\left (4 \, {\left (14 \, c d x e^{5} + \frac {{\left (81 \, c^{8} d^{9} e^{11} + 17 \, a c^{7} d^{7} e^{13}\right )} e^{\left (-7\right )}}{c^{7} d^{7}}\right )} x + \frac {{\left (769 \, c^{8} d^{10} e^{10} + 406 \, a c^{7} d^{8} e^{12} + a^{2} c^{6} d^{6} e^{14}\right )} e^{\left (-7\right )}}{c^{7} d^{7}}\right )} x + \frac {{\left (9461 \, c^{8} d^{11} e^{9} + 10067 \, a c^{7} d^{9} e^{11} + 83 \, a^{2} c^{6} d^{7} e^{13} - 11 \, a^{3} c^{5} d^{5} e^{15}\right )} e^{\left (-7\right )}}{c^{7} d^{7}}\right )} x + \frac {{\left (49251 \, c^{8} d^{12} e^{8} + 105748 \, a c^{7} d^{10} e^{10} + 2450 \, a^{2} c^{6} d^{8} e^{12} - 748 \, a^{3} c^{5} d^{6} e^{14} + 99 \, a^{4} c^{4} d^{4} e^{16}\right )} e^{\left (-7\right )}}{c^{7} d^{7}}\right )} x + \frac {{\left (28441 \, c^{8} d^{13} e^{7} + 153301 \, a c^{7} d^{11} e^{9} + 10642 \, a^{2} c^{6} d^{9} e^{11} - 5742 \, a^{3} c^{5} d^{7} e^{13} + 1749 \, a^{4} c^{4} d^{5} e^{15} - 231 \, a^{5} c^{3} d^{3} e^{17}\right )} e^{\left (-7\right )}}{c^{7} d^{7}}\right )} x + \frac {{\left (1155 \, c^{8} d^{14} e^{6} + 220598 \, a c^{7} d^{12} e^{8} + 61709 \, a^{2} c^{6} d^{10} e^{10} - 53900 \, a^{3} c^{5} d^{8} e^{12} + 28941 \, a^{4} c^{4} d^{6} e^{14} - 8778 \, a^{5} c^{3} d^{4} e^{16} + 1155 \, a^{6} c^{2} d^{2} e^{18}\right )} e^{\left (-7\right )}}{c^{7} d^{7}}\right )} x - \frac {{\left (3465 \, c^{8} d^{15} e^{5} - 26565 \, a c^{7} d^{13} e^{7} - 140903 \, a^{2} c^{6} d^{11} e^{9} + 193699 \, a^{3} c^{5} d^{9} e^{11} - 166749 \, a^{4} c^{4} d^{7} e^{13} + 88473 \, a^{5} c^{3} d^{5} e^{15} - 26565 \, a^{6} c^{2} d^{3} e^{17} + 3465 \, a^{7} c d e^{19}\right )} e^{\left (-7\right )}}{c^{7} d^{7}}\right )} - \frac {99 \, {\left (c^{8} d^{16} - 8 \, a c^{7} d^{14} e^{2} + 28 \, a^{2} c^{6} d^{12} e^{4} - 56 \, a^{3} c^{5} d^{10} e^{6} + 70 \, a^{4} c^{4} d^{8} e^{8} - 56 \, a^{5} c^{3} d^{6} e^{10} + 28 \, a^{6} c^{2} d^{4} e^{12} - 8 \, a^{7} c d^{2} e^{14} + a^{8} e^{16}\right )} e^{\left (-\frac {5}{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 )}{32768 \, \sqrt {c d} c^{6} d^{6}} \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)^(3/2),x, algorithm="giac")

[Out]

1/573440*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*(2*(4*(2*(8*(10*(4*(14*c*d*x*e^5 + (81*c^8*d^9*e^11 + 17*
a*c^7*d^7*e^13)*e^(-7)/(c^7*d^7))*x + (769*c^8*d^10*e^10 + 406*a*c^7*d^8*e^12 + a^2*c^6*d^6*e^14)*e^(-7)/(c^7*
d^7))*x + (9461*c^8*d^11*e^9 + 10067*a*c^7*d^9*e^11 + 83*a^2*c^6*d^7*e^13 - 11*a^3*c^5*d^5*e^15)*e^(-7)/(c^7*d
^7))*x + (49251*c^8*d^12*e^8 + 105748*a*c^7*d^10*e^10 + 2450*a^2*c^6*d^8*e^12 - 748*a^3*c^5*d^6*e^14 + 99*a^4*
c^4*d^4*e^16)*e^(-7)/(c^7*d^7))*x + (28441*c^8*d^13*e^7 + 153301*a*c^7*d^11*e^9 + 10642*a^2*c^6*d^9*e^11 - 574
2*a^3*c^5*d^7*e^13 + 1749*a^4*c^4*d^5*e^15 - 231*a^5*c^3*d^3*e^17)*e^(-7)/(c^7*d^7))*x + (1155*c^8*d^14*e^6 +
220598*a*c^7*d^12*e^8 + 61709*a^2*c^6*d^10*e^10 - 53900*a^3*c^5*d^8*e^12 + 28941*a^4*c^4*d^6*e^14 - 8778*a^5*c
^3*d^4*e^16 + 1155*a^6*c^2*d^2*e^18)*e^(-7)/(c^7*d^7))*x - (3465*c^8*d^15*e^5 - 26565*a*c^7*d^13*e^7 - 140903*
a^2*c^6*d^11*e^9 + 193699*a^3*c^5*d^9*e^11 - 166749*a^4*c^4*d^7*e^13 + 88473*a^5*c^3*d^5*e^15 - 26565*a^6*c^2*
d^3*e^17 + 3465*a^7*c*d*e^19)*e^(-7)/(c^7*d^7)) - 99/32768*(c^8*d^16 - 8*a*c^7*d^14*e^2 + 28*a^2*c^6*d^12*e^4
- 56*a^3*c^5*d^10*e^6 + 70*a^4*c^4*d^8*e^8 - 56*a^5*c^3*d^6*e^10 + 28*a^6*c^2*d^4*e^12 - 8*a^7*c*d^2*e^14 + a^
8*e^16)*e^(-5/2)*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)

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maple [B]  time = 0.08, size = 2065, normalized size = 4.48

result too large to display

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)^(3/2),x)

[Out]

297/4096*e*d^5*a*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x+33/2048/e*d^5*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/
2)+33/1024*d^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x+223/640/c*d^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)
+495/16384*d^6*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a+99/512*e^4/c^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2
)*x*a^2-99/16384/e^2*c*d^8*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)-1793/4480*e^2/c^2*(c*d*e*x^2+a*d*e+(a*e^2+c
*d^2)*x)^(5/2)*a+495/16384*e^6/c^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^4+53/112*e^2/c*x^2*(c*d*e*x^2+a*d
*e+(a*e^2+c*d^2)*x)^(5/2)-693/4096*e^8/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^5+693/8192*e^2*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^2+33/1024*e^5/c^3/d*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*
a^3-33/640*e^6/c^4/d^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*a^3+99/32768/e^2*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)+1/8*e^3*x^3*(c*d*e*x^2+a*d*e+
(a*e^2+c*d^2)*x)^(5/2)/c/d-99/2048*e/c*d^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a-891/16384*e^2/c*d^4*(c*d*
e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^2-99/4096*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+289/448*e/c*d*x*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)+495/163
84*e^4/c^2*d^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^3+33/2048*e^9/c^5/d^5*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*
x)^(3/2)*a^5-99/2048*e^7/c^4/d^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a^4+1023/4480*e^4/c^3/d^2*(c*d*e*x^2+
a*d*e+(a*e^2+c*d^2)*x)^(5/2)*a^2+33/1024*e^3/c^2*d*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a^2-99/8192/e*c*d^7
*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x-99/16384*e^12/c^6/d^6*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^7+4
95/16384*e^10/c^5/d^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^6-891/16384*e^8/c^4/d^2*(c*d*e*x^2+a*d*e+(a*e^
2+c*d^2)*x)^(1/2)*a^5+33/448*e^5/c^3/d^3*x*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*a^2-11/112*e^4/c^2/d^2*x^2*
(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*a-99/4096*e^12/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^7-1485/8192*e^7/c^3/d*a^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d
^2)*x)^(1/2)*x-1485/8192*e^3/c*d^3*a^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x+693/8192*e^10/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^6-33/256*e^6/
c^3/d^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x*a^3+3465/16384*e^6/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^4+33/1024*e^8/c^4/d^4*(c*d*e*x^2+a*d*e+
(a*e^2+c*d^2)*x)^(3/2)*x*a^4+297/4096*e^9/c^4/d^3*a^5*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x+99/32768*e^14/
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^8+495/2048*e^5/c^2*d*a^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x-693/4096*e^4/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^3-99/8192*e^11/c^5/d^5*(c*d*
e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*a^6-33/256*e^2/c*d^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x*a-11/32*e^
3/c^2/d*x*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*a

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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)^(3/2),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 )}^{3/2} \,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)^(3/2),x)

[Out]

int((d + e*x)^4*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2), 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 {3}{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)**(3/2),x)

[Out]

Integral(((d + e*x)*(a*e + c*d*x))**(3/2)*(d + e*x)**4, x)

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