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

Optimal. Leaf size=305 \[ -\frac {5 \left (c d^2-a e^2\right )^6 \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 c^{7/2} d^{7/2} e^{7/2}}+\frac {5 \left (c d^2-a e^2\right )^4 \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 c^3 d^3 e^3}-\frac {5 \left (c d^2-a e^2\right )^2 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{192 c^2 d^2 e^2}+\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{12 c d e} \]

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Rubi [A]  time = 0.15, antiderivative size = 305, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {612, 621, 206} \begin {gather*} \frac {5 \left (c d^2-a e^2\right )^4 \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}}{512 c^3 d^3 e^3}-\frac {5 \left (c d^2-a e^2\right )^2 \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}}{192 c^2 d^2 e^2}-\frac {5 \left (c d^2-a e^2\right )^6 \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 )}{1024 c^{7/2} d^{7/2} e^{7/2}}+\frac {\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}}{12 c d e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5*(c*d^2 - a*e^2)^4*(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])/(512*c^3*d^3*e^3
) - (5*(c*d^2 - a*e^2)^2*(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))/(192*c^2*d
^2*e^2) + ((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))/(12*c*d*e) - (5*(c*d^2 -
a*e^2)^6*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])])/(1024*c^(7/2)*d^(7/2)*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]

Rubi steps

\begin {align*} \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx &=\frac {\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}}{12 c d e}-\frac {\left (5 \left (c d^2-a e^2\right )^2\right ) \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{24 c d e}\\ &=-\frac {5 \left (c d^2-a e^2\right )^2 \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}}{192 c^2 d^2 e^2}+\frac {\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}}{12 c d e}+\frac {\left (5 \left (c d^2-a e^2\right )^4\right ) \int \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{128 c^2 d^2 e^2}\\ &=\frac {5 \left (c d^2-a e^2\right )^4 \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}}{512 c^3 d^3 e^3}-\frac {5 \left (c d^2-a e^2\right )^2 \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}}{192 c^2 d^2 e^2}+\frac {\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}}{12 c d e}-\frac {\left (5 \left (c d^2-a e^2\right )^6\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{1024 c^3 d^3 e^3}\\ &=\frac {5 \left (c d^2-a e^2\right )^4 \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}}{512 c^3 d^3 e^3}-\frac {5 \left (c d^2-a e^2\right )^2 \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}}{192 c^2 d^2 e^2}+\frac {\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}}{12 c d e}-\frac {\left (5 \left (c d^2-a e^2\right )^6\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 )}{512 c^3 d^3 e^3}\\ &=\frac {5 \left (c d^2-a e^2\right )^4 \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}}{512 c^3 d^3 e^3}-\frac {5 \left (c d^2-a e^2\right )^2 \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}}{192 c^2 d^2 e^2}+\frac {\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}}{12 c d e}-\frac {5 \left (c d^2-a e^2\right )^6 \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 )}{1024 c^{7/2} d^{7/2} e^{7/2}}\\ \end {align*}

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Mathematica [B]  time = 6.11, size = 1119, normalized size = 3.67 \begin {gather*} \frac {2 \left (c d^2-a e^2\right )^2 (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 )^{7/2} \left (\frac {35 \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}{2048 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 )^3}+\frac {7}{12} \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 {1}{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 )^2}+\frac {3}{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 )^3}\right )\right )}{7 c^3 d^3 \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 )^{5/2} (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[(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)^2*(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))))^(7/2)*((7*(3/(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))))^3) + 1/(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))))^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)))/12 + (35*(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)))])
))/(2048*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))))^3)))/(7*c^3*d^3*((c*d)/((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2)
))^(5/2)*(d + e*x)^2*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])

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IntegrateAlgebraic [F]  time = 180.07, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(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 [A]  time = 0.51, size = 1034, normalized size = 3.39 \begin {gather*} \left [\frac {15 \, {\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\right )} \sqrt {c d e} \log \left (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 + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {c d e} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) + 4 \, {\left (256 \, c^{6} d^{6} e^{6} x^{5} + 15 \, c^{6} d^{11} e - 85 \, a c^{5} d^{9} e^{3} + 198 \, a^{2} c^{4} d^{7} e^{5} + 198 \, a^{3} c^{3} d^{5} e^{7} - 85 \, a^{4} c^{2} d^{3} e^{9} + 15 \, a^{5} c d e^{11} + 640 \, {\left (c^{6} d^{7} e^{5} + a c^{5} d^{5} e^{7}\right )} x^{4} + 16 \, {\left (27 \, c^{6} d^{8} e^{4} + 106 \, a c^{5} d^{6} e^{6} + 27 \, a^{2} c^{4} d^{4} e^{8}\right )} x^{3} + 8 \, {\left (c^{6} d^{9} e^{3} + 159 \, a c^{5} d^{7} e^{5} + 159 \, a^{2} c^{4} d^{5} e^{7} + a^{3} c^{3} d^{3} e^{9}\right )} x^{2} - 2 \, {\left (5 \, c^{6} d^{10} e^{2} - 28 \, a c^{5} d^{8} e^{4} - 594 \, a^{2} c^{4} d^{6} e^{6} - 28 \, a^{3} c^{3} d^{4} e^{8} + 5 \, a^{4} c^{2} d^{2} e^{10}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{6144 \, c^{4} d^{4} e^{4}}, \frac {15 \, {\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}\right ) + 2 \, {\left (256 \, c^{6} d^{6} e^{6} x^{5} + 15 \, c^{6} d^{11} e - 85 \, a c^{5} d^{9} e^{3} + 198 \, a^{2} c^{4} d^{7} e^{5} + 198 \, a^{3} c^{3} d^{5} e^{7} - 85 \, a^{4} c^{2} d^{3} e^{9} + 15 \, a^{5} c d e^{11} + 640 \, {\left (c^{6} d^{7} e^{5} + a c^{5} d^{5} e^{7}\right )} x^{4} + 16 \, {\left (27 \, c^{6} d^{8} e^{4} + 106 \, a c^{5} d^{6} e^{6} + 27 \, a^{2} c^{4} d^{4} e^{8}\right )} x^{3} + 8 \, {\left (c^{6} d^{9} e^{3} + 159 \, a c^{5} d^{7} e^{5} + 159 \, a^{2} c^{4} d^{5} e^{7} + a^{3} c^{3} d^{3} e^{9}\right )} x^{2} - 2 \, {\left (5 \, c^{6} d^{10} e^{2} - 28 \, a c^{5} d^{8} e^{4} - 594 \, a^{2} c^{4} d^{6} e^{6} - 28 \, a^{3} c^{3} d^{4} e^{8} + 5 \, a^{4} c^{2} d^{2} e^{10}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{3072 \, c^{4} d^{4} e^{4}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6144*(15*(c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 - 20*a^3*c^3*d^6*e^6 + 15*a^4*c^2*d^4*e^8 - 6*a^
5*c*d^2*e^10 + a^6*e^12)*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*(2
56*c^6*d^6*e^6*x^5 + 15*c^6*d^11*e - 85*a*c^5*d^9*e^3 + 198*a^2*c^4*d^7*e^5 + 198*a^3*c^3*d^5*e^7 - 85*a^4*c^2
*d^3*e^9 + 15*a^5*c*d*e^11 + 640*(c^6*d^7*e^5 + a*c^5*d^5*e^7)*x^4 + 16*(27*c^6*d^8*e^4 + 106*a*c^5*d^6*e^6 +
27*a^2*c^4*d^4*e^8)*x^3 + 8*(c^6*d^9*e^3 + 159*a*c^5*d^7*e^5 + 159*a^2*c^4*d^5*e^7 + a^3*c^3*d^3*e^9)*x^2 - 2*
(5*c^6*d^10*e^2 - 28*a*c^5*d^8*e^4 - 594*a^2*c^4*d^6*e^6 - 28*a^3*c^3*d^4*e^8 + 5*a^4*c^2*d^2*e^10)*x)*sqrt(c*
d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^4*d^4*e^4), 1/3072*(15*(c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*
e^4 - 20*a^3*c^3*d^6*e^6 + 15*a^4*c^2*d^4*e^8 - 6*a^5*c*d^2*e^10 + a^6*e^12)*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*(256*c^6*d^6*e^6*x^5 + 15*c^6*d^11*e - 85*a*c^5*d^9*e^3 + 198*a^2*c^4*d^7*e^5 +
198*a^3*c^3*d^5*e^7 - 85*a^4*c^2*d^3*e^9 + 15*a^5*c*d*e^11 + 640*(c^6*d^7*e^5 + a*c^5*d^5*e^7)*x^4 + 16*(27*c
^6*d^8*e^4 + 106*a*c^5*d^6*e^6 + 27*a^2*c^4*d^4*e^8)*x^3 + 8*(c^6*d^9*e^3 + 159*a*c^5*d^7*e^5 + 159*a^2*c^4*d^
5*e^7 + a^3*c^3*d^3*e^9)*x^2 - 2*(5*c^6*d^10*e^2 - 28*a*c^5*d^8*e^4 - 594*a^2*c^4*d^6*e^6 - 28*a^3*c^3*d^4*e^8
+ 5*a^4*c^2*d^2*e^10)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^4*d^4*e^4)]

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giac [A]  time = 0.55, size = 493, normalized size = 1.62 \begin {gather*} \frac {1}{1536} \, \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 \, c^{2} d^{2} x e^{2} + \frac {5 \, {\left (c^{7} d^{8} e^{6} + a c^{6} d^{6} e^{8}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac {{\left (27 \, c^{7} d^{9} e^{5} + 106 \, a c^{6} d^{7} e^{7} + 27 \, a^{2} c^{5} d^{5} e^{9}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac {{\left (c^{7} d^{10} e^{4} + 159 \, a c^{6} d^{8} e^{6} + 159 \, a^{2} c^{5} d^{6} e^{8} + a^{3} c^{4} d^{4} e^{10}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x - \frac {{\left (5 \, c^{7} d^{11} e^{3} - 28 \, a c^{6} d^{9} e^{5} - 594 \, a^{2} c^{5} d^{7} e^{7} - 28 \, a^{3} c^{4} d^{5} e^{9} + 5 \, a^{4} c^{3} d^{3} e^{11}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac {{\left (15 \, c^{7} d^{12} e^{2} - 85 \, a c^{6} d^{10} e^{4} + 198 \, a^{2} c^{5} d^{8} e^{6} + 198 \, a^{3} c^{4} d^{6} e^{8} - 85 \, a^{4} c^{3} d^{4} e^{10} + 15 \, a^{5} c^{2} d^{2} e^{12}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} + \frac {5 \, {\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\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 )}{1024 \, \sqrt {c d} c^{3} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1536*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*(2*(4*(2*(8*(2*c^2*d^2*x*e^2 + 5*(c^7*d^8*e^6 + a*c^6*d^6*e
^8)*e^(-5)/(c^5*d^5))*x + (27*c^7*d^9*e^5 + 106*a*c^6*d^7*e^7 + 27*a^2*c^5*d^5*e^9)*e^(-5)/(c^5*d^5))*x + (c^7
*d^10*e^4 + 159*a*c^6*d^8*e^6 + 159*a^2*c^5*d^6*e^8 + a^3*c^4*d^4*e^10)*e^(-5)/(c^5*d^5))*x - (5*c^7*d^11*e^3
- 28*a*c^6*d^9*e^5 - 594*a^2*c^5*d^7*e^7 - 28*a^3*c^4*d^5*e^9 + 5*a^4*c^3*d^3*e^11)*e^(-5)/(c^5*d^5))*x + (15*
c^7*d^12*e^2 - 85*a*c^6*d^10*e^4 + 198*a^2*c^5*d^8*e^6 + 198*a^3*c^4*d^6*e^8 - 85*a^4*c^3*d^4*e^10 + 15*a^5*c^
2*d^2*e^12)*e^(-5)/(c^5*d^5)) + 5/1024*(c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 - 20*a^3*c^3*d^6*e^6
+ 15*a^4*c^2*d^4*e^8 - 6*a^5*c*d^2*e^10 + a^6*e^12)*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^3*d^3)

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maple [B]  time = 0.06, size = 1247, normalized size = 4.09

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

15/128*d^2*e^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*a^2-15/512/c^2/d*e^5*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x
)^(1/2)*a^4-5/64/c*e^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*a^3+5/256/c^2/d^2*e^6*(c*d*e*x^2+a*d*e+(a*e^2
+c*d^2)*x)^(1/2)*x*a^4-5/96/c/d*e^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x*a^2-5/1024/c^3/d^3*e^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^6-75/1024/c*d*e
^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+1
5/512*c^2*d^7/e*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-75/1024*c*d^5*e*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+15/512/c^2/d*e^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^5-5/192*c*d^4/e^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)+5/512*c^2*d^7/e^3*(
c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)+5/192/c*e^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a^2+5/256*d^3*e*(c*
d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^2+5/192*d^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a+5/48*d*e*(c*d*e*x
^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x*a-5/192/c^2/d^2*e^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a^3-15/512*c*d^5
/e*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a-5/96*c*d^3/e*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x-5/1024*c^3
*d^9/e^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)
+5/256/c*d*e^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^3+5/512/c^3/d^3*e^7*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)
^(1/2)*a^5+5/256*c^2*d^6/e^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x+25/256*d^3*e^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^3-5/64*c*d^4*(c*d*e*x^2+a*d*e+
(a*e^2+c*d^2)*x)^(1/2)*x*a+1/12*(2*c*d*e*x+a*e^2+c*d^2)*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)/c/d/e

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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((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 [B]  time = 0.79, size = 319, normalized size = 1.05 \begin {gather*} \frac {\left (\frac {c\,d^2}{2}+c\,x\,d\,e+\frac {a\,e^2}{2}\right )\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{6\,c\,d\,e}-\frac {\left (\frac {5\,{\left (c\,d^2+a\,e^2\right )}^2}{4}-5\,a\,c\,d^2\,e^2\right )\,\left (\frac {\left (\frac {c\,d^2}{2}+c\,x\,d\,e+\frac {a\,e^2}{2}\right )\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}}{4\,c\,d\,e}-\frac {\left (\frac {3\,{\left (c\,d^2+a\,e^2\right )}^2}{4}-3\,a\,c\,d^2\,e^2\right )\,\left (\left (\frac {x}{2}+\frac {c\,d^2+a\,e^2}{4\,c\,d\,e}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}-\frac {\ln \left (2\,\sqrt {\left (a\,e+c\,d\,x\right )\,\left (d+e\,x\right )}\,\sqrt {c\,d\,e}+a\,e^2+c\,d^2+2\,c\,d\,e\,x\right )\,\left (\frac {{\left (c\,d^2+a\,e^2\right )}^2}{4}-a\,c\,d^2\,e^2\right )}{2\,{\left (c\,d\,e\right )}^{3/2}}\right )}{4\,c\,d\,e}\right )}{6\,c\,d\,e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(((a*e^2)/2 + (c*d^2)/2 + c*d*e*x)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2))/(6*c*d*e) - (((5*(a*e^2 + c*
d^2)^2)/4 - 5*a*c*d^2*e^2)*((((a*e^2)/2 + (c*d^2)/2 + c*d*e*x)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2))/
(4*c*d*e) - (((3*(a*e^2 + c*d^2)^2)/4 - 3*a*c*d^2*e^2)*((x/2 + (a*e^2 + c*d^2)/(4*c*d*e))*(x*(a*e^2 + c*d^2) +
a*d*e + c*d*e*x^2)^(1/2) - (log(2*((a*e + c*d*x)*(d + e*x))^(1/2)*(c*d*e)^(1/2) + a*e^2 + c*d^2 + 2*c*d*e*x)*
((a*e^2 + c*d^2)^2/4 - a*c*d^2*e^2))/(2*(c*d*e)^(3/2))))/(4*c*d*e)))/(6*c*d*e)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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