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

Optimal. Leaf size=200 \[ -\frac{8 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g) \left (2 a e^2 g-c d (7 e f-5 d g)\right )}{315 c^3 d^3 e (d+e x)^{5/2}}+\frac{8 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g)}{63 c^2 d^2 e (d+e x)^{3/2}}+\frac{2 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{9 c d (d+e x)^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.709869, 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 \[ -\frac{8 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g) \left (2 a e^2 g-c d (7 e f-5 d g)\right )}{315 c^3 d^3 e (d+e x)^{5/2}}+\frac{8 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g)}{63 c^2 d^2 e (d+e x)^{3/2}}+\frac{2 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{9 c d (d+e x)^{5/2}} \]

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)^(3/2))/(d + e*x)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 60.3884, size = 196, normalized size = 0.98 \[ \frac{2 \left (f + g x\right )^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{9 c d \left (d + e x\right )^{\frac{5}{2}}} - \frac{8 g \left (a e g - c d f\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{63 c^{2} d^{2} e \left (d + e x\right )^{\frac{3}{2}}} + \frac{8 \left (a e g - c d f\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}} \left (2 a e^{2} g + 5 c d^{2} g - 7 c d e f\right )}{315 c^{3} d^{3} e \left (d + e x\right )^{\frac{5}{2}}} \]

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

[Out]

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

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Mathematica [A]  time = 0.18955, size = 90, normalized size = 0.45 \[ \frac{2 ((d+e x) (a e+c d x))^{5/2} \left (8 a^2 e^2 g^2-4 a c d e g (9 f+5 g x)+c^2 d^2 \left (63 f^2+90 f g x+35 g^2 x^2\right )\right )}{315 c^3 d^3 (d+e x)^{5/2}} \]

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)^(3/2))/(d + e*x)^(3/2),x]

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(8*a^2*e^2*g^2 - 4*a*c*d*e*g*(9*f + 5*g*x) +
c^2*d^2*(63*f^2 + 90*f*g*x + 35*g^2*x^2)))/(315*c^3*d^3*(d + e*x)^(5/2))

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Maple [A]  time = 0.011, size = 116, normalized size = 0.6 \[{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 35\,{g}^{2}{x}^{2}{c}^{2}{d}^{2}-20\,acde{g}^{2}x+90\,{c}^{2}{d}^{2}fgx+8\,{a}^{2}{e}^{2}{g}^{2}-36\,acdefg+63\,{f}^{2}{c}^{2}{d}^{2} \right ) }{315\,{c}^{3}{d}^{3}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]

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

[Out]

2/315*(c*d*x+a*e)*(35*c^2*d^2*g^2*x^2-20*a*c*d*e*g^2*x+90*c^2*d^2*f*g*x+8*a^2*e^
2*g^2-36*a*c*d*e*f*g+63*c^2*d^2*f^2)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(3/2)/c^3
/d^3/(e*x+d)^(3/2)

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Maxima [A]  time = 0.729543, size = 259, normalized size = 1.3 \[ \frac{2 \,{\left (c^{2} d^{2} x^{2} + 2 \, a c d e x + a^{2} e^{2}\right )} \sqrt{c d x + a e} f^{2}}{5 \, c d} + \frac{4 \,{\left (5 \, c^{3} d^{3} x^{3} + 8 \, a c^{2} d^{2} e x^{2} + a^{2} c d e^{2} x - 2 \, a^{3} e^{3}\right )} \sqrt{c d x + a e} f g}{35 \, c^{2} d^{2}} + \frac{2 \,{\left (35 \, c^{4} d^{4} x^{4} + 50 \, a c^{3} d^{3} e x^{3} + 3 \, a^{2} c^{2} d^{2} e^{2} x^{2} - 4 \, a^{3} c d e^{3} x + 8 \, a^{4} e^{4}\right )} \sqrt{c d x + a e} g^{2}}{315 \, c^{3} d^{3}} \]

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)^(3/2)*(g*x + f)^2/(e*x + d)^(3/2),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)*f^2/(c*d) + 4/35*(5*
c^3*d^3*x^3 + 8*a*c^2*d^2*e*x^2 + a^2*c*d*e^2*x - 2*a^3*e^3)*sqrt(c*d*x + a*e)*f
*g/(c^2*d^2) + 2/315*(35*c^4*d^4*x^4 + 50*a*c^3*d^3*e*x^3 + 3*a^2*c^2*d^2*e^2*x^
2 - 4*a^3*c*d*e^3*x + 8*a^4*e^4)*sqrt(c*d*x + a*e)*g^2/(c^3*d^3)

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Fricas [A]  time = 0.271004, size = 662, normalized size = 3.31 \[ \frac{2 \,{\left (35 \, c^{5} d^{5} e g^{2} x^{6} + 63 \, a^{3} c^{2} d^{3} e^{3} f^{2} - 36 \, a^{4} c d^{2} e^{4} f g + 8 \, a^{5} d e^{5} g^{2} + 5 \,{\left (18 \, c^{5} d^{5} e f g +{\left (7 \, c^{5} d^{6} + 17 \, a c^{4} d^{4} e^{2}\right )} g^{2}\right )} x^{5} +{\left (63 \, c^{5} d^{5} e f^{2} + 18 \,{\left (5 \, c^{5} d^{6} + 13 \, a c^{4} d^{4} e^{2}\right )} f g +{\left (85 \, a c^{4} d^{5} e + 53 \, a^{2} c^{3} d^{3} e^{3}\right )} g^{2}\right )} x^{4} +{\left (63 \,{\left (c^{5} d^{6} + 3 \, a c^{4} d^{4} e^{2}\right )} f^{2} + 18 \,{\left (13 \, a c^{4} d^{5} e + 9 \, a^{2} c^{3} d^{3} e^{3}\right )} f g +{\left (53 \, a^{2} c^{3} d^{4} e^{2} - a^{3} c^{2} d^{2} e^{4}\right )} g^{2}\right )} x^{3} +{\left (189 \,{\left (a c^{4} d^{5} e + a^{2} c^{3} d^{3} e^{3}\right )} f^{2} + 18 \,{\left (9 \, a^{2} c^{3} d^{4} e^{2} - a^{3} c^{2} d^{2} e^{4}\right )} f g -{\left (a^{3} c^{2} d^{3} e^{3} - 4 \, a^{4} c d e^{5}\right )} g^{2}\right )} x^{2} +{\left (63 \,{\left (3 \, a^{2} c^{3} d^{4} e^{2} + a^{3} c^{2} d^{2} e^{4}\right )} f^{2} - 18 \,{\left (a^{3} c^{2} d^{3} e^{3} + 2 \, a^{4} c d e^{5}\right )} f g + 4 \,{\left (a^{4} c d^{2} e^{4} + 2 \, a^{5} e^{6}\right )} g^{2}\right )} x\right )}}{315 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} c^{3} d^{3}} \]

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

[Out]

2/315*(35*c^5*d^5*e*g^2*x^6 + 63*a^3*c^2*d^3*e^3*f^2 - 36*a^4*c*d^2*e^4*f*g + 8*
a^5*d*e^5*g^2 + 5*(18*c^5*d^5*e*f*g + (7*c^5*d^6 + 17*a*c^4*d^4*e^2)*g^2)*x^5 +
(63*c^5*d^5*e*f^2 + 18*(5*c^5*d^6 + 13*a*c^4*d^4*e^2)*f*g + (85*a*c^4*d^5*e + 53
*a^2*c^3*d^3*e^3)*g^2)*x^4 + (63*(c^5*d^6 + 3*a*c^4*d^4*e^2)*f^2 + 18*(13*a*c^4*
d^5*e + 9*a^2*c^3*d^3*e^3)*f*g + (53*a^2*c^3*d^4*e^2 - a^3*c^2*d^2*e^4)*g^2)*x^3
+ (189*(a*c^4*d^5*e + a^2*c^3*d^3*e^3)*f^2 + 18*(9*a^2*c^3*d^4*e^2 - a^3*c^2*d^
2*e^4)*f*g - (a^3*c^2*d^3*e^3 - 4*a^4*c*d*e^5)*g^2)*x^2 + (63*(3*a^2*c^3*d^4*e^2
+ a^3*c^2*d^2*e^4)*f^2 - 18*(a^3*c^2*d^3*e^3 + 2*a^4*c*d*e^5)*f*g + 4*(a^4*c*d^
2*e^4 + 2*a^5*e^6)*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)**(3/2)/(e*x+d)**(3/2),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)*(g*x + f)^2/(e*x + d)^(3/2),x, algorithm="giac")

[Out]

Timed out