### 3.2036 $$\int (d+e x)^{5/2} (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2} \, dx$$

Optimal. Leaf size=295 $\frac{256 \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{15015 c^5 d^5 (d+e x)^{5/2}}+\frac{128 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{3003 c^4 d^4 (d+e x)^{3/2}}+\frac{32 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{429 c^3 d^3 \sqrt{d+e x}}+\frac{16 \sqrt{d+e x} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{143 c^2 d^2}+\frac{2 (d+e x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{13 c d}$

[Out]

(256*(c*d^2 - a*e^2)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(15015*c^5*d^5*(d + e*x)^(5/2)) + (128*(
c*d^2 - a*e^2)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(3003*c^4*d^4*(d + e*x)^(3/2)) + (32*(c*d^2 -
a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(429*c^3*d^3*Sqrt[d + e*x]) + (16*(c*d^2 - a*e^2)*Sqrt
[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(143*c^2*d^2) + (2*(d + e*x)^(3/2)*(a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2)^(5/2))/(13*c*d)

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Rubi [A]  time = 0.296466, antiderivative size = 295, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 39, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.051, Rules used = {656, 648} $\frac{256 \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{15015 c^5 d^5 (d+e x)^{5/2}}+\frac{128 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{3003 c^4 d^4 (d+e x)^{3/2}}+\frac{32 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{429 c^3 d^3 \sqrt{d+e x}}+\frac{16 \sqrt{d+e x} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{143 c^2 d^2}+\frac{2 (d+e x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{13 c d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(256*(c*d^2 - a*e^2)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(15015*c^5*d^5*(d + e*x)^(5/2)) + (128*(
c*d^2 - a*e^2)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(3003*c^4*d^4*(d + e*x)^(3/2)) + (32*(c*d^2 -
a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(429*c^3*d^3*Sqrt[d + e*x]) + (16*(c*d^2 - a*e^2)*Sqrt
[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(143*c^2*d^2) + (2*(d + e*x)^(3/2)*(a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2)^(5/2))/(13*c*d)

Rule 656

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[(Simplify[m + p]*(2*c*d - b*e))/(c*(m + 2*p + 1)), In
t[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && E
qQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && IGtQ[Simplify[m + p], 0]

Rule 648

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*(p + 1)), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c
*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p, 0]

Rubi steps

\begin{align*} \int (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx &=\frac{2 (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{13 c d}+\frac{\left (8 \left (d^2-\frac{a e^2}{c}\right )\right ) \int (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{13 d}\\ &=\frac{16 \left (c d^2-a e^2\right ) \sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{143 c^2 d^2}+\frac{2 (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{13 c d}+\frac{\left (48 \left (d^2-\frac{a e^2}{c}\right )^2\right ) \int \sqrt{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}{143 d^2}\\ &=\frac{32 \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 )^{5/2}}{429 c^3 d^3 \sqrt{d+e x}}+\frac{16 \left (c d^2-a e^2\right ) \sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{143 c^2 d^2}+\frac{2 (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{13 c d}+\frac{\left (64 \left (d^2-\frac{a e^2}{c}\right )^3\right ) \int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{\sqrt{d+e x}} \, dx}{429 d^3}\\ &=\frac{128 \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}}{3003 c^4 d^4 (d+e x)^{3/2}}+\frac{32 \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 )^{5/2}}{429 c^3 d^3 \sqrt{d+e x}}+\frac{16 \left (c d^2-a e^2\right ) \sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{143 c^2 d^2}+\frac{2 (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{13 c d}+\frac{\left (128 \left (d^2-\frac{a e^2}{c}\right )^4\right ) \int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx}{3003 d^4}\\ &=\frac{256 \left (c d^2-a e^2\right )^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{15015 c^5 d^5 (d+e x)^{5/2}}+\frac{128 \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}}{3003 c^4 d^4 (d+e x)^{3/2}}+\frac{32 \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 )^{5/2}}{429 c^3 d^3 \sqrt{d+e x}}+\frac{16 \left (c d^2-a e^2\right ) \sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{143 c^2 d^2}+\frac{2 (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{13 c d}\\ \end{align*}

Mathematica [A]  time = 0.180637, size = 187, normalized size = 0.63 $\frac{2 ((d+e x) (a e+c d x))^{5/2} \left (16 a^2 c^2 d^2 e^4 \left (143 d^2+130 d e x+35 e^2 x^2\right )-64 a^3 c d e^6 (13 d+5 e x)+128 a^4 e^8-8 a c^3 d^3 e^2 \left (715 d^2 e x+429 d^3+455 d e^2 x^2+105 e^3 x^3\right )+c^4 d^4 \left (10010 d^2 e^2 x^2+8580 d^3 e x+3003 d^4+5460 d e^3 x^3+1155 e^4 x^4\right )\right )}{15015 c^5 d^5 (d+e x)^{5/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(128*a^4*e^8 - 64*a^3*c*d*e^6*(13*d + 5*e*x) + 16*a^2*c^2*d^2*e^4*(143*d^2
+ 130*d*e*x + 35*e^2*x^2) - 8*a*c^3*d^3*e^2*(429*d^3 + 715*d^2*e*x + 455*d*e^2*x^2 + 105*e^3*x^3) + c^4*d^4*(3
003*d^4 + 8580*d^3*e*x + 10010*d^2*e^2*x^2 + 5460*d*e^3*x^3 + 1155*e^4*x^4)))/(15015*c^5*d^5*(d + e*x)^(5/2))

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Maple [A]  time = 0.046, size = 243, normalized size = 0.8 \begin{align*}{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 1155\,{e}^{4}{x}^{4}{c}^{4}{d}^{4}-840\,a{c}^{3}{d}^{3}{e}^{5}{x}^{3}+5460\,{c}^{4}{d}^{5}{e}^{3}{x}^{3}+560\,{a}^{2}{c}^{2}{d}^{2}{e}^{6}{x}^{2}-3640\,a{c}^{3}{d}^{4}{e}^{4}{x}^{2}+10010\,{c}^{4}{d}^{6}{e}^{2}{x}^{2}-320\,{a}^{3}cd{e}^{7}x+2080\,{a}^{2}{c}^{2}{d}^{3}{e}^{5}x-5720\,a{c}^{3}{d}^{5}{e}^{3}x+8580\,{c}^{4}{d}^{7}ex+128\,{a}^{4}{e}^{8}-832\,{a}^{3}c{d}^{2}{e}^{6}+2288\,{a}^{2}{c}^{2}{d}^{4}{e}^{4}-3432\,a{c}^{3}{d}^{6}{e}^{2}+3003\,{c}^{4}{d}^{8} \right ) }{15015\,{c}^{5}{d}^{5}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15015*(c*d*x+a*e)*(1155*c^4*d^4*e^4*x^4-840*a*c^3*d^3*e^5*x^3+5460*c^4*d^5*e^3*x^3+560*a^2*c^2*d^2*e^6*x^2-3
640*a*c^3*d^4*e^4*x^2+10010*c^4*d^6*e^2*x^2-320*a^3*c*d*e^7*x+2080*a^2*c^2*d^3*e^5*x-5720*a*c^3*d^5*e^3*x+8580
*c^4*d^7*e*x+128*a^4*e^8-832*a^3*c*d^2*e^6+2288*a^2*c^2*d^4*e^4-3432*a*c^3*d^6*e^2+3003*c^4*d^8)*(c*d*e*x^2+a*
e^2*x+c*d^2*x+a*d*e)^(3/2)/c^5/d^5/(e*x+d)^(3/2)

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Maxima [A]  time = 1.1146, size = 504, normalized size = 1.71 \begin{align*} \frac{2 \,{\left (1155 \, c^{6} d^{6} e^{4} x^{6} + 3003 \, a^{2} c^{4} d^{8} e^{2} - 3432 \, a^{3} c^{3} d^{6} e^{4} + 2288 \, a^{4} c^{2} d^{4} e^{6} - 832 \, a^{5} c d^{2} e^{8} + 128 \, a^{6} e^{10} + 210 \,{\left (26 \, c^{6} d^{7} e^{3} + 7 \, a c^{5} d^{5} e^{5}\right )} x^{5} + 35 \,{\left (286 \, c^{6} d^{8} e^{2} + 208 \, a c^{5} d^{6} e^{4} + a^{2} c^{4} d^{4} e^{6}\right )} x^{4} + 20 \,{\left (429 \, c^{6} d^{9} e + 715 \, a c^{5} d^{7} e^{3} + 13 \, a^{2} c^{4} d^{5} e^{5} - 2 \, a^{3} c^{3} d^{3} e^{7}\right )} x^{3} + 3 \,{\left (1001 \, c^{6} d^{10} + 4576 \, a c^{5} d^{8} e^{2} + 286 \, a^{2} c^{4} d^{6} e^{4} - 104 \, a^{3} c^{3} d^{4} e^{6} + 16 \, a^{4} c^{2} d^{2} e^{8}\right )} x^{2} + 2 \,{\left (3003 \, a c^{5} d^{9} e + 858 \, a^{2} c^{4} d^{7} e^{3} - 572 \, a^{3} c^{3} d^{5} e^{5} + 208 \, a^{4} c^{2} d^{3} e^{7} - 32 \, a^{5} c d e^{9}\right )} x\right )} \sqrt{c d x + a e}{\left (e x + d\right )}}{15015 \,{\left (c^{5} d^{5} e x + c^{5} d^{6}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15015*(1155*c^6*d^6*e^4*x^6 + 3003*a^2*c^4*d^8*e^2 - 3432*a^3*c^3*d^6*e^4 + 2288*a^4*c^2*d^4*e^6 - 832*a^5*c
*d^2*e^8 + 128*a^6*e^10 + 210*(26*c^6*d^7*e^3 + 7*a*c^5*d^5*e^5)*x^5 + 35*(286*c^6*d^8*e^2 + 208*a*c^5*d^6*e^4
+ a^2*c^4*d^4*e^6)*x^4 + 20*(429*c^6*d^9*e + 715*a*c^5*d^7*e^3 + 13*a^2*c^4*d^5*e^5 - 2*a^3*c^3*d^3*e^7)*x^3
+ 3*(1001*c^6*d^10 + 4576*a*c^5*d^8*e^2 + 286*a^2*c^4*d^6*e^4 - 104*a^3*c^3*d^4*e^6 + 16*a^4*c^2*d^2*e^8)*x^2
+ 2*(3003*a*c^5*d^9*e + 858*a^2*c^4*d^7*e^3 - 572*a^3*c^3*d^5*e^5 + 208*a^4*c^2*d^3*e^7 - 32*a^5*c*d*e^9)*x)*s
qrt(c*d*x + a*e)*(e*x + d)/(c^5*d^5*e*x + c^5*d^6)

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Fricas [A]  time = 1.89956, size = 853, normalized size = 2.89 \begin{align*} \frac{2 \,{\left (1155 \, c^{6} d^{6} e^{4} x^{6} + 3003 \, a^{2} c^{4} d^{8} e^{2} - 3432 \, a^{3} c^{3} d^{6} e^{4} + 2288 \, a^{4} c^{2} d^{4} e^{6} - 832 \, a^{5} c d^{2} e^{8} + 128 \, a^{6} e^{10} + 210 \,{\left (26 \, c^{6} d^{7} e^{3} + 7 \, a c^{5} d^{5} e^{5}\right )} x^{5} + 35 \,{\left (286 \, c^{6} d^{8} e^{2} + 208 \, a c^{5} d^{6} e^{4} + a^{2} c^{4} d^{4} e^{6}\right )} x^{4} + 20 \,{\left (429 \, c^{6} d^{9} e + 715 \, a c^{5} d^{7} e^{3} + 13 \, a^{2} c^{4} d^{5} e^{5} - 2 \, a^{3} c^{3} d^{3} e^{7}\right )} x^{3} + 3 \,{\left (1001 \, c^{6} d^{10} + 4576 \, a c^{5} d^{8} e^{2} + 286 \, a^{2} c^{4} d^{6} e^{4} - 104 \, a^{3} c^{3} d^{4} e^{6} + 16 \, a^{4} c^{2} d^{2} e^{8}\right )} x^{2} + 2 \,{\left (3003 \, a c^{5} d^{9} e + 858 \, a^{2} c^{4} d^{7} e^{3} - 572 \, a^{3} c^{3} d^{5} e^{5} + 208 \, a^{4} c^{2} d^{3} e^{7} - 32 \, a^{5} c d e^{9}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d}}{15015 \,{\left (c^{5} d^{5} e x + c^{5} d^{6}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15015*(1155*c^6*d^6*e^4*x^6 + 3003*a^2*c^4*d^8*e^2 - 3432*a^3*c^3*d^6*e^4 + 2288*a^4*c^2*d^4*e^6 - 832*a^5*c
*d^2*e^8 + 128*a^6*e^10 + 210*(26*c^6*d^7*e^3 + 7*a*c^5*d^5*e^5)*x^5 + 35*(286*c^6*d^8*e^2 + 208*a*c^5*d^6*e^4
+ a^2*c^4*d^4*e^6)*x^4 + 20*(429*c^6*d^9*e + 715*a*c^5*d^7*e^3 + 13*a^2*c^4*d^5*e^5 - 2*a^3*c^3*d^3*e^7)*x^3
+ 3*(1001*c^6*d^10 + 4576*a*c^5*d^8*e^2 + 286*a^2*c^4*d^6*e^4 - 104*a^3*c^3*d^4*e^6 + 16*a^4*c^2*d^2*e^8)*x^2
+ 2*(3003*a*c^5*d^9*e + 858*a^2*c^4*d^7*e^3 - 572*a^3*c^3*d^5*e^5 + 208*a^4*c^2*d^3*e^7 - 32*a^5*c*d*e^9)*x)*s
qrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)/(c^5*d^5*e*x + c^5*d^6)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out