3.702 \(\int \frac{(f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=200 \[ -\frac{8 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g) \left (2 a e^2 g-c d (9 e f-7 d g)\right )}{693 c^3 d^3 e (d+e x)^{7/2}}+\frac{8 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g)}{99 c^2 d^2 e (d+e x)^{5/2}}+\frac{2 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{11 c d (d+e x)^{7/2}} \]

[Out]

(-8*(c*d*f - a*e*g)*(2*a*e^2*g - c*d*(9*e*f - 7*d*g))*(a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2)^(7/2))/(693*c^3*d^3*e*(d + e*x)^(7/2)) + (8*g*(c*d*f - a*e*g)*(a*d
*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(99*c^2*d^2*e*(d + e*x)^(5/2)) + (2*(
f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(11*c*d*(d + e*x)^(7/2
))

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Rubi [A]  time = 0.709366, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ -\frac{8 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g) \left (2 a e^2 g-c d (9 e f-7 d g)\right )}{693 c^3 d^3 e (d+e x)^{7/2}}+\frac{8 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g)}{99 c^2 d^2 e (d+e x)^{5/2}}+\frac{2 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{11 c d (d+e x)^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-8*(c*d*f - a*e*g)*(2*a*e^2*g - c*d*(9*e*f - 7*d*g))*(a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2)^(7/2))/(693*c^3*d^3*e*(d + e*x)^(7/2)) + (8*g*(c*d*f - a*e*g)*(a*d
*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(99*c^2*d^2*e*(d + e*x)^(5/2)) + (2*(
f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(11*c*d*(d + e*x)^(7/2
))

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Rubi in Sympy [A]  time = 60.5693, size = 196, normalized size = 0.98 \[ \frac{2 \left (f + g x\right )^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{7}{2}}}{11 c d \left (d + e x\right )^{\frac{7}{2}}} - \frac{8 g \left (a e g - c d f\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{7}{2}}}{99 c^{2} d^{2} e \left (d + e x\right )^{\frac{5}{2}}} + \frac{8 \left (a e g - c d f\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{7}{2}} \left (2 a e^{2} g + 7 c d^{2} g - 9 c d e f\right )}{693 c^{3} d^{3} e \left (d + e x\right )^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x+f)**2*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/(e*x+d)**(5/2),x)

[Out]

2*(f + g*x)**2*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(7/2)/(11*c*d*(d + e*
x)**(7/2)) - 8*g*(a*e*g - c*d*f)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(7/
2)/(99*c**2*d**2*e*(d + e*x)**(5/2)) + 8*(a*e*g - c*d*f)*(a*d*e + c*d*e*x**2 + x
*(a*e**2 + c*d**2))**(7/2)*(2*a*e**2*g + 7*c*d**2*g - 9*c*d*e*f)/(693*c**3*d**3*
e*(d + e*x)**(7/2))

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Mathematica [A]  time = 0.208595, size = 100, normalized size = 0.5 \[ \frac{2 (a e+c d x)^3 \sqrt{(d+e x) (a e+c d x)} \left (8 a^2 e^2 g^2-4 a c d e g (11 f+7 g x)+c^2 d^2 \left (99 f^2+154 f g x+63 g^2 x^2\right )\right )}{693 c^3 d^3 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(8*a^2*e^2*g^2 - 4*a*c*d*e*g*(1
1*f + 7*g*x) + c^2*d^2*(99*f^2 + 154*f*g*x + 63*g^2*x^2)))/(693*c^3*d^3*Sqrt[d +
e*x])

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Maple [A]  time = 0.011, size = 116, normalized size = 0.6 \[{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 63\,{g}^{2}{x}^{2}{c}^{2}{d}^{2}-28\,acde{g}^{2}x+154\,{c}^{2}{d}^{2}fgx+8\,{a}^{2}{e}^{2}{g}^{2}-44\,acdefg+99\,{f}^{2}{c}^{2}{d}^{2} \right ) }{693\,{c}^{3}{d}^{3}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((g*x+f)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2),x)

[Out]

2/693*(c*d*x+a*e)*(63*c^2*d^2*g^2*x^2-28*a*c*d*e*g^2*x+154*c^2*d^2*f*g*x+8*a^2*e
^2*g^2-44*a*c*d*e*f*g+99*c^2*d^2*f^2)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(5/2)/c^
3/d^3/(e*x+d)^(5/2)

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Maxima [A]  time = 0.735522, size = 328, normalized size = 1.64 \[ \frac{2 \,{\left (c^{3} d^{3} x^{3} + 3 \, a c^{2} d^{2} e x^{2} + 3 \, a^{2} c d e^{2} x + a^{3} e^{3}\right )} \sqrt{c d x + a e} f^{2}}{7 \, c d} + \frac{4 \,{\left (7 \, c^{4} d^{4} x^{4} + 19 \, a c^{3} d^{3} e x^{3} + 15 \, a^{2} c^{2} d^{2} e^{2} x^{2} + a^{3} c d e^{3} x - 2 \, a^{4} e^{4}\right )} \sqrt{c d x + a e} f g}{63 \, c^{2} d^{2}} + \frac{2 \,{\left (63 \, c^{5} d^{5} x^{5} + 161 \, a c^{4} d^{4} e x^{4} + 113 \, a^{2} c^{3} d^{3} e^{2} x^{3} + 3 \, a^{3} c^{2} d^{2} e^{3} x^{2} - 4 \, a^{4} c d e^{4} x + 8 \, a^{5} e^{5}\right )} \sqrt{c d x + a e} g^{2}}{693 \, c^{3} d^{3}} \]

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)^(5/2)*(g*x + f)^2/(e*x + d)^(5/2),x, algorithm="maxima")

[Out]

2/7*(c^3*d^3*x^3 + 3*a*c^2*d^2*e*x^2 + 3*a^2*c*d*e^2*x + a^3*e^3)*sqrt(c*d*x + a
*e)*f^2/(c*d) + 4/63*(7*c^4*d^4*x^4 + 19*a*c^3*d^3*e*x^3 + 15*a^2*c^2*d^2*e^2*x^
2 + a^3*c*d*e^3*x - 2*a^4*e^4)*sqrt(c*d*x + a*e)*f*g/(c^2*d^2) + 2/693*(63*c^5*d
^5*x^5 + 161*a*c^4*d^4*e*x^4 + 113*a^2*c^3*d^3*e^2*x^3 + 3*a^3*c^2*d^2*e^3*x^2 -
4*a^4*c*d*e^4*x + 8*a^5*e^5)*sqrt(c*d*x + a*e)*g^2/(c^3*d^3)

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Fricas [A]  time = 0.278832, size = 809, normalized size = 4.04 \[ \frac{2 \,{\left (63 \, c^{6} d^{6} e g^{2} x^{7} + 99 \, a^{4} c^{2} d^{3} e^{4} f^{2} - 44 \, a^{5} c d^{2} e^{5} f g + 8 \, a^{6} d e^{6} g^{2} + 7 \,{\left (22 \, c^{6} d^{6} e f g +{\left (9 \, c^{6} d^{7} + 32 \, a c^{5} d^{5} e^{2}\right )} g^{2}\right )} x^{6} +{\left (99 \, c^{6} d^{6} e f^{2} + 22 \,{\left (7 \, c^{6} d^{7} + 26 \, a c^{5} d^{5} e^{2}\right )} f g + 2 \,{\left (112 \, a c^{5} d^{6} e + 137 \, a^{2} c^{4} d^{4} e^{3}\right )} g^{2}\right )} x^{5} +{\left (99 \,{\left (c^{6} d^{7} + 4 \, a c^{5} d^{5} e^{2}\right )} f^{2} + 44 \,{\left (13 \, a c^{5} d^{6} e + 17 \, a^{2} c^{4} d^{4} e^{3}\right )} f g + 2 \,{\left (137 \, a^{2} c^{4} d^{5} e^{2} + 58 \, a^{3} c^{3} d^{3} e^{4}\right )} g^{2}\right )} x^{4} +{\left (198 \,{\left (2 \, a c^{5} d^{6} e + 3 \, a^{2} c^{4} d^{4} e^{3}\right )} f^{2} + 44 \,{\left (17 \, a^{2} c^{4} d^{5} e^{2} + 8 \, a^{3} c^{3} d^{3} e^{4}\right )} f g +{\left (116 \, a^{3} c^{3} d^{4} e^{3} - a^{4} c^{2} d^{2} e^{5}\right )} g^{2}\right )} x^{3} +{\left (198 \,{\left (3 \, a^{2} c^{4} d^{5} e^{2} + 2 \, a^{3} c^{3} d^{3} e^{4}\right )} f^{2} + 22 \,{\left (16 \, a^{3} c^{3} d^{4} e^{3} - a^{4} c^{2} d^{2} e^{5}\right )} f g -{\left (a^{4} c^{2} d^{3} e^{4} - 4 \, a^{5} c d e^{6}\right )} g^{2}\right )} x^{2} +{\left (99 \,{\left (4 \, a^{3} c^{3} d^{4} e^{3} + a^{4} c^{2} d^{2} e^{5}\right )} f^{2} - 22 \,{\left (a^{4} c^{2} d^{3} e^{4} + 2 \, a^{5} c d e^{6}\right )} f g + 4 \,{\left (a^{5} c d^{2} e^{5} + 2 \, a^{6} e^{7}\right )} g^{2}\right )} x\right )}}{693 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} c^{3} d^{3}} \]

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)^(5/2)*(g*x + f)^2/(e*x + d)^(5/2),x, algorithm="fricas")

[Out]

2/693*(63*c^6*d^6*e*g^2*x^7 + 99*a^4*c^2*d^3*e^4*f^2 - 44*a^5*c*d^2*e^5*f*g + 8*
a^6*d*e^6*g^2 + 7*(22*c^6*d^6*e*f*g + (9*c^6*d^7 + 32*a*c^5*d^5*e^2)*g^2)*x^6 +
(99*c^6*d^6*e*f^2 + 22*(7*c^6*d^7 + 26*a*c^5*d^5*e^2)*f*g + 2*(112*a*c^5*d^6*e +
137*a^2*c^4*d^4*e^3)*g^2)*x^5 + (99*(c^6*d^7 + 4*a*c^5*d^5*e^2)*f^2 + 44*(13*a*
c^5*d^6*e + 17*a^2*c^4*d^4*e^3)*f*g + 2*(137*a^2*c^4*d^5*e^2 + 58*a^3*c^3*d^3*e^
4)*g^2)*x^4 + (198*(2*a*c^5*d^6*e + 3*a^2*c^4*d^4*e^3)*f^2 + 44*(17*a^2*c^4*d^5*
e^2 + 8*a^3*c^3*d^3*e^4)*f*g + (116*a^3*c^3*d^4*e^3 - a^4*c^2*d^2*e^5)*g^2)*x^3
+ (198*(3*a^2*c^4*d^5*e^2 + 2*a^3*c^3*d^3*e^4)*f^2 + 22*(16*a^3*c^3*d^4*e^3 - a^
4*c^2*d^2*e^5)*f*g - (a^4*c^2*d^3*e^4 - 4*a^5*c*d*e^6)*g^2)*x^2 + (99*(4*a^3*c^3
*d^4*e^3 + a^4*c^2*d^2*e^5)*f^2 - 22*(a^4*c^2*d^3*e^4 + 2*a^5*c*d*e^6)*f*g + 4*(
a^5*c*d^2*e^5 + 2*a^6*e^7)*g^2)*x)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*
sqrt(e*x + d)*c^3*d^3)

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

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: AttributeError} \]

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)^(5/2)*(g*x + f)^2/(e*x + d)^(5/2),x, algorithm="giac")

[Out]

Exception raised: AttributeError