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

Optimal. Leaf size=63 \[ \frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 (d+e x)^{5/2} (f+g x)^{5/2} (c d f-a e g)} \]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(5*(c*d*f - a*e*g)*(d + e*x)^(
5/2)*(f + g*x)^(5/2))

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Rubi [A]  time = 0.254732, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.021 \[ \frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 (d+e x)^{5/2} (f+g x)^{5/2} (c d f-a e g)} \]

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)^(3/2)*(f + g*x)^(7/2)),x]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(5*(c*d*f - a*e*g)*(d + e*x)^(
5/2)*(f + g*x)^(5/2))

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Rubi in Sympy [A]  time = 21.6618, size = 60, normalized size = 0.95 \[ - \frac{2 \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{5 \left (d + e x\right )^{\frac{5}{2}} \left (f + g x\right )^{\frac{5}{2}} \left (a e g - c d f\right )} \]

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)**(3/2)/(g*x+f)**(7/2),x)

[Out]

-2*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(5/2)/(5*(d + e*x)**(5/2)*(f + g*
x)**(5/2)*(a*e*g - c*d*f))

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Mathematica [A]  time = 0.169936, size = 52, normalized size = 0.83 \[ \frac{2 ((d+e x) (a e+c d x))^{5/2}}{5 (d+e x)^{5/2} (f+g x)^{5/2} (c d f-a e g)} \]

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)^(3/2)*(f + g*x)^(7/2)),x]

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(5/2))/(5*(c*d*f - a*e*g)*(d + e*x)^(5/2)*(f + g*x)
^(5/2))

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Maple [A]  time = 0.01, size = 63, normalized size = 1. \[ -{\frac{2\,cdx+2\,ae}{5\,aeg-5\,cdf} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{{\frac{3}{2}}} \left ( gx+f \right ) ^{-{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/5/(g*x+f)^(5/2)*(c*d*x+a*e)/(a*e*g-c*d*f)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(
3/2)/(e*x+d)^(3/2)

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Maxima [A]  time = 1.06745, size = 308, normalized size = 4.89 \[ \frac{2 \,{\left (c^{2} d^{2} x^{2} + 2 \, a c d e x + a^{2} e^{2}\right )} \sqrt{c d x + a e}{\left (e x + d\right )} \sqrt{g x + f}}{5 \,{\left (c d^{2} f^{4} - a d e f^{3} g +{\left (c d e f g^{3} - a e^{2} g^{4}\right )} x^{4} -{\left ({\left (3 \, e^{2} f g^{3} + d e g^{4}\right )} a -{\left (3 \, d e f^{2} g^{2} + d^{2} f g^{3}\right )} c\right )} x^{3} - 3 \,{\left ({\left (e^{2} f^{2} g^{2} + d e f g^{3}\right )} a -{\left (d e f^{3} g + d^{2} f^{2} g^{2}\right )} c\right )} x^{2} -{\left ({\left (e^{2} f^{3} g + 3 \, d e f^{2} g^{2}\right )} a -{\left (d e f^{4} + 3 \, d^{2} f^{3} g\right )} c\right )} x\right )}} \]

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)^(3/2)*(g*x + f)^(7/2)),x, algorithm="maxima")

[Out]

2/5*(c^2*d^2*x^2 + 2*a*c*d*e*x + a^2*e^2)*sqrt(c*d*x + a*e)*(e*x + d)*sqrt(g*x +
f)/(c*d^2*f^4 - a*d*e*f^3*g + (c*d*e*f*g^3 - a*e^2*g^4)*x^4 - ((3*e^2*f*g^3 + d
*e*g^4)*a - (3*d*e*f^2*g^2 + d^2*f*g^3)*c)*x^3 - 3*((e^2*f^2*g^2 + d*e*f*g^3)*a
- (d*e*f^3*g + d^2*f^2*g^2)*c)*x^2 - ((e^2*f^3*g + 3*d*e*f^2*g^2)*a - (d*e*f^4 +
3*d^2*f^3*g)*c)*x)

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Fricas [A]  time = 0.289347, size = 313, normalized size = 4.97 \[ \frac{2 \,{\left (c^{2} d^{2} x^{2} + 2 \, a c d e x + a^{2} e^{2}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} \sqrt{g x + f}}{5 \,{\left (c d^{2} f^{4} - a d e f^{3} g +{\left (c d e f g^{3} - a e^{2} g^{4}\right )} x^{4} +{\left (3 \, c d e f^{2} g^{2} - a d e g^{4} +{\left (c d^{2} - 3 \, a e^{2}\right )} f g^{3}\right )} x^{3} + 3 \,{\left (c d e f^{3} g - a d e f g^{3} +{\left (c d^{2} - a e^{2}\right )} f^{2} g^{2}\right )} x^{2} +{\left (c d e f^{4} - 3 \, a d e f^{2} g^{2} +{\left (3 \, c d^{2} - a e^{2}\right )} f^{3} g\right )} x\right )}} \]

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)^(3/2)*(g*x + f)^(7/2)),x, algorithm="fricas")

[Out]

2/5*(c^2*d^2*x^2 + 2*a*c*d*e*x + a^2*e^2)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^
2)*x)*sqrt(e*x + d)*sqrt(g*x + f)/(c*d^2*f^4 - a*d*e*f^3*g + (c*d*e*f*g^3 - a*e^
2*g^4)*x^4 + (3*c*d*e*f^2*g^2 - a*d*e*g^4 + (c*d^2 - 3*a*e^2)*f*g^3)*x^3 + 3*(c*
d*e*f^3*g - a*d*e*f*g^3 + (c*d^2 - a*e^2)*f^2*g^2)*x^2 + (c*d*e*f^4 - 3*a*d*e*f^
2*g^2 + (3*c*d^2 - a*e^2)*f^3*g)*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)**(3/2)/(g*x+f)**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{3}{2}}}{{\left (e x + d\right )}^{\frac{3}{2}}{\left (g x + f\right )}^{\frac{7}{2}}}\,{d x} \]

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)^(3/2)*(g*x + f)^(7/2)),x, algorithm="giac")

[Out]

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)/((e*x + d)^(3/2)*(g*x +
f)^(7/2)), x)