3.1917 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^6} \, dx\)

Optimal. Leaf size=111 \[ \frac{4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{35 (d+e x)^5 \left (c d^2-a e^2\right )^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 (d+e x)^6 \left (c d^2-a e^2\right )} \]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(7*(c*d^2 - a*e^2)*(d + e*x)^6
) + (4*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(35*(c*d^2 - a*e^2)^2*
(d + e*x)^5)

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Rubi [A]  time = 0.173744, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054 \[ \frac{4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{35 (d+e x)^5 \left (c d^2-a e^2\right )^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 (d+e x)^6 \left (c d^2-a e^2\right )} \]

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)^6,x]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(7*(c*d^2 - a*e^2)*(d + e*x)^6
) + (4*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(35*(c*d^2 - a*e^2)^2*
(d + e*x)^5)

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Rubi in Sympy [A]  time = 33.8966, size = 102, normalized size = 0.92 \[ \frac{4 c d \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{35 \left (d + e x\right )^{5} \left (a e^{2} - c d^{2}\right )^{2}} - \frac{2 \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{7 \left (d + e x\right )^{6} \left (a e^{2} - c d^{2}\right )} \]

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)**6,x)

[Out]

4*c*d*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(5/2)/(35*(d + e*x)**5*(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)/(7*(d + e*x
)**6*(a*e**2 - c*d**2))

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Mathematica [A]  time = 0.13751, size = 61, normalized size = 0.55 \[ \frac{2 ((d+e x) (a e+c d x))^{5/2} \left (c d (7 d+2 e x)-5 a e^2\right )}{35 (d+e x)^6 \left (c d^2-a e^2\right )^2} \]

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)^6,x]

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(-5*a*e^2 + c*d*(7*d + 2*e*x)))/(35*(c*d^2 -
a*e^2)^2*(d + e*x)^6)

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Maple [A]  time = 0.012, size = 90, normalized size = 0.8 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -2\,cdex+5\,a{e}^{2}-7\,c{d}^{2} \right ) }{35\, \left ( ex+d \right ) ^{5} \left ({a}^{2}{e}^{4}-2\,ac{d}^{2}{e}^{2}+{c}^{2}{d}^{4} \right ) } \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed \right ) ^{{\frac{3}{2}}}} \]

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)^6,x)

[Out]

-2/35*(c*d*x+a*e)*(-2*c*d*e*x+5*a*e^2-7*c*d^2)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)
^(3/2)/(e*x+d)^5/(a^2*e^4-2*a*c*d^2*e^2+c^2*d^4)

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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)^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.14487, size = 371, normalized size = 3.34 \[ \frac{2 \,{\left (2 \, c^{3} d^{3} e x^{3} + 7 \, a^{2} c d^{2} e^{2} - 5 \, a^{3} e^{4} +{\left (7 \, c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} x^{2} + 2 \,{\left (7 \, a c^{2} d^{3} e - 4 \, a^{2} c d e^{3}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{35 \,{\left (c^{2} d^{8} - 2 \, a c d^{6} e^{2} + a^{2} d^{4} e^{4} +{\left (c^{2} d^{4} e^{4} - 2 \, a c d^{2} e^{6} + a^{2} e^{8}\right )} x^{4} + 4 \,{\left (c^{2} d^{5} e^{3} - 2 \, a c d^{3} e^{5} + a^{2} d e^{7}\right )} x^{3} + 6 \,{\left (c^{2} d^{6} e^{2} - 2 \, a c d^{4} e^{4} + a^{2} d^{2} e^{6}\right )} x^{2} + 4 \,{\left (c^{2} d^{7} e - 2 \, a c d^{5} e^{3} + a^{2} d^{3} e^{5}\right )} x\right )}} \]

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)^6,x, algorithm="fricas")

[Out]

2/35*(2*c^3*d^3*e*x^3 + 7*a^2*c*d^2*e^2 - 5*a^3*e^4 + (7*c^3*d^4 - a*c^2*d^2*e^2
)*x^2 + 2*(7*a*c^2*d^3*e - 4*a^2*c*d*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a
*e^2)*x)/(c^2*d^8 - 2*a*c*d^6*e^2 + a^2*d^4*e^4 + (c^2*d^4*e^4 - 2*a*c*d^2*e^6 +
a^2*e^8)*x^4 + 4*(c^2*d^5*e^3 - 2*a*c*d^3*e^5 + a^2*d*e^7)*x^3 + 6*(c^2*d^6*e^2
- 2*a*c*d^4*e^4 + a^2*d^2*e^6)*x^2 + 4*(c^2*d^7*e - 2*a*c*d^5*e^3 + a^2*d^3*e^5
)*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)**6,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 29.2093, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

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)^6,x, algorithm="giac")

[Out]

Done