3.1918 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^7} \, dx\)

Optimal. Leaf size=171 \[ \frac{16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{315 (d+e x)^5 \left (c d^2-a e^2\right )^3}+\frac{8 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{63 (d+e x)^6 \left (c d^2-a e^2\right )^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{9 (d+e x)^7 \left (c d^2-a e^2\right )} \]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(9*(c*d^2 - a*e^2)*(d + e*x)^7
) + (8*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(63*(c*d^2 - a*e^2)^2*
(d + e*x)^6) + (16*c^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(315*(
c*d^2 - a*e^2)^3*(d + e*x)^5)

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Rubi [A]  time = 0.279351, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054 \[ \frac{16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{315 (d+e x)^5 \left (c d^2-a e^2\right )^3}+\frac{8 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{63 (d+e x)^6 \left (c d^2-a e^2\right )^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{9 (d+e x)^7 \left (c d^2-a e^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^7,x]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(9*(c*d^2 - a*e^2)*(d + e*x)^7
) + (8*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(63*(c*d^2 - a*e^2)^2*
(d + e*x)^6) + (16*c^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(315*(
c*d^2 - a*e^2)^3*(d + e*x)^5)

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Rubi in Sympy [A]  time = 53.7516, size = 160, normalized size = 0.94 \[ - \frac{16 c^{2} d^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{315 \left (d + e x\right )^{5} \left (a e^{2} - c d^{2}\right )^{3}} + \frac{8 c d \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{63 \left (d + e x\right )^{6} \left (a e^{2} - c d^{2}\right )^{2}} - \frac{2 \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{9 \left (d + e x\right )^{7} \left (a e^{2} - c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2)/(e*x+d)**7,x)

[Out]

-16*c**2*d**2*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(5/2)/(315*(d + e*x)**
5*(a*e**2 - c*d**2)**3) + 8*c*d*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(5/2
)/(63*(d + e*x)**6*(a*e**2 - c*d**2)**2) - 2*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c
*d**2))**(5/2)/(9*(d + e*x)**7*(a*e**2 - c*d**2))

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Mathematica [A]  time = 0.177947, size = 94, normalized size = 0.55 \[ \frac{2 ((d+e x) (a e+c d x))^{5/2} \left (35 a^2 e^4-10 a c d e^2 (9 d+2 e x)+c^2 d^2 \left (63 d^2+36 d e x+8 e^2 x^2\right )\right )}{315 (d+e x)^7 \left (c d^2-a e^2\right )^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^7,x]

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(35*a^2*e^4 - 10*a*c*d*e^2*(9*d + 2*e*x) + c^
2*d^2*(63*d^2 + 36*d*e*x + 8*e^2*x^2)))/(315*(c*d^2 - a*e^2)^3*(d + e*x)^7)

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Maple [A]  time = 0.015, size = 146, normalized size = 0.9 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 8\,{x}^{2}{c}^{2}{d}^{2}{e}^{2}-20\,xacd{e}^{3}+36\,x{c}^{2}{d}^{3}e+35\,{a}^{2}{e}^{4}-90\,ac{d}^{2}{e}^{2}+63\,{c}^{2}{d}^{4} \right ) }{315\, \left ( ex+d \right ) ^{6} \left ({a}^{3}{e}^{6}-3\,{a}^{2}c{d}^{2}{e}^{4}+3\,{c}^{2}{d}^{4}a{e}^{2}-{c}^{3}{d}^{6} \right ) } \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^7,x)

[Out]

-2/315*(c*d*x+a*e)*(8*c^2*d^2*e^2*x^2-20*a*c*d*e^3*x+36*c^2*d^3*e*x+35*a^2*e^4-9
0*a*c*d^2*e^2+63*c^2*d^4)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(3/2)/(e*x+d)^6/(a^3
*e^6-3*a^2*c*d^2*e^4+3*a*c^2*d^4*e^2-c^3*d^6)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)/(e*x + d)^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 4.60399, size = 637, normalized size = 3.73 \[ \frac{2 \,{\left (8 \, c^{4} d^{4} e^{2} x^{4} + 63 \, a^{2} c^{2} d^{4} e^{2} - 90 \, a^{3} c d^{2} e^{4} + 35 \, a^{4} e^{6} + 4 \,{\left (9 \, c^{4} d^{5} e - a c^{3} d^{3} e^{3}\right )} x^{3} + 3 \,{\left (21 \, c^{4} d^{6} - 6 \, a c^{3} d^{4} e^{2} + a^{2} c^{2} d^{2} e^{4}\right )} x^{2} + 2 \,{\left (63 \, a c^{3} d^{5} e - 72 \, a^{2} c^{2} d^{3} e^{3} + 25 \, a^{3} c d e^{5}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{315 \,{\left (c^{3} d^{11} - 3 \, a c^{2} d^{9} e^{2} + 3 \, a^{2} c d^{7} e^{4} - a^{3} d^{5} e^{6} +{\left (c^{3} d^{6} e^{5} - 3 \, a c^{2} d^{4} e^{7} + 3 \, a^{2} c d^{2} e^{9} - a^{3} e^{11}\right )} x^{5} + 5 \,{\left (c^{3} d^{7} e^{4} - 3 \, a c^{2} d^{5} e^{6} + 3 \, a^{2} c d^{3} e^{8} - a^{3} d e^{10}\right )} x^{4} + 10 \,{\left (c^{3} d^{8} e^{3} - 3 \, a c^{2} d^{6} e^{5} + 3 \, a^{2} c d^{4} e^{7} - a^{3} d^{2} e^{9}\right )} x^{3} + 10 \,{\left (c^{3} d^{9} e^{2} - 3 \, a c^{2} d^{7} e^{4} + 3 \, a^{2} c d^{5} e^{6} - a^{3} d^{3} e^{8}\right )} x^{2} + 5 \,{\left (c^{3} d^{10} e - 3 \, a c^{2} d^{8} e^{3} + 3 \, a^{2} c d^{6} e^{5} - a^{3} d^{4} e^{7}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)/(e*x + d)^7,x, algorithm="fricas")

[Out]

2/315*(8*c^4*d^4*e^2*x^4 + 63*a^2*c^2*d^4*e^2 - 90*a^3*c*d^2*e^4 + 35*a^4*e^6 +
4*(9*c^4*d^5*e - a*c^3*d^3*e^3)*x^3 + 3*(21*c^4*d^6 - 6*a*c^3*d^4*e^2 + a^2*c^2*
d^2*e^4)*x^2 + 2*(63*a*c^3*d^5*e - 72*a^2*c^2*d^3*e^3 + 25*a^3*c*d*e^5)*x)*sqrt(
c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(c^3*d^11 - 3*a*c^2*d^9*e^2 + 3*a^2*c*d^7
*e^4 - a^3*d^5*e^6 + (c^3*d^6*e^5 - 3*a*c^2*d^4*e^7 + 3*a^2*c*d^2*e^9 - a^3*e^11
)*x^5 + 5*(c^3*d^7*e^4 - 3*a*c^2*d^5*e^6 + 3*a^2*c*d^3*e^8 - a^3*d*e^10)*x^4 + 1
0*(c^3*d^8*e^3 - 3*a*c^2*d^6*e^5 + 3*a^2*c*d^4*e^7 - a^3*d^2*e^9)*x^3 + 10*(c^3*
d^9*e^2 - 3*a*c^2*d^7*e^4 + 3*a^2*c*d^5*e^6 - a^3*d^3*e^8)*x^2 + 5*(c^3*d^10*e -
3*a*c^2*d^8*e^3 + 3*a^2*c*d^6*e^5 - a^3*d^4*e^7)*x)

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

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)**(3/2)/(e*x+d)**7,x)

[Out]

Timed out

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)/(e*x + d)^7,x, algorithm="giac")

[Out]

Timed out