3.749 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{11/2}} \, dx\)

Optimal. Leaf size=198 \[ \frac{16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{315 (d+e x)^{5/2} (f+g x)^{5/2} (c d f-a e g)^3}+\frac{8 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{63 (d+e x)^{5/2} (f+g x)^{7/2} (c d f-a e g)^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{9 (d+e x)^{5/2} (f+g x)^{9/2} (c d f-a e g)} \]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(9*(c*d*f - a*e*g)*(d + e*x)^(
5/2)*(f + g*x)^(9/2)) + (8*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(6
3*(c*d*f - a*e*g)^2*(d + e*x)^(5/2)*(f + g*x)^(7/2)) + (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*f - a*e*g)^3*(d + e*x)^(5/2)*(f + g*
x)^(5/2))

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Rubi [A]  time = 0.814011, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{315 (d+e x)^{5/2} (f+g x)^{5/2} (c d f-a e g)^3}+\frac{8 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{63 (d+e x)^{5/2} (f+g x)^{7/2} (c d f-a e g)^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{9 (d+e x)^{5/2} (f+g x)^{9/2} (c d f-a e g)} \]

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)^(11/2)),x]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(9*(c*d*f - a*e*g)*(d + e*x)^(
5/2)*(f + g*x)^(9/2)) + (8*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(6
3*(c*d*f - a*e*g)^2*(d + e*x)^(5/2)*(f + g*x)^(7/2)) + (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*f - a*e*g)^3*(d + e*x)^(5/2)*(f + g*
x)^(5/2))

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Rubi in Sympy [A]  time = 68.8613, size = 190, normalized size = 0.96 \[ - \frac{16 c^{2} d^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{315 \left (d + e x\right )^{\frac{5}{2}} \left (f + g x\right )^{\frac{5}{2}} \left (a e g - c d f\right )^{3}} + \frac{8 c d \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{63 \left (d + e x\right )^{\frac{5}{2}} \left (f + g x\right )^{\frac{7}{2}} \left (a e g - c d f\right )^{2}} - \frac{2 \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{9 \left (d + e x\right )^{\frac{5}{2}} \left (f + g x\right )^{\frac{9}{2}} \left (a e g - c d f\right )} \]

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)**(11/2),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)**
(5/2)*(f + g*x)**(5/2)*(a*e*g - c*d*f)**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)**(5/2)*(f + g*x)**(7/2)*(a*e*g - c*d*f)**2) -
2*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(5/2)/(9*(d + e*x)**(5/2)*(f + g*
x)**(9/2)*(a*e*g - c*d*f))

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Mathematica [A]  time = 0.284453, size = 105, normalized size = 0.53 \[ \frac{2 ((d+e x) (a e+c d x))^{5/2} \left (35 a^2 e^2 g^2-10 a c d e g (9 f+2 g x)+c^2 d^2 \left (63 f^2+36 f g x+8 g^2 x^2\right )\right )}{315 (d+e x)^{5/2} (f+g x)^{9/2} (c d f-a e g)^3} \]

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)^(3/2)*(f + g*x)^(11/2)),x]

[Out]

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

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Maple [A]  time = 0.014, size = 169, normalized size = 0.9 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 8\,{c}^{2}{d}^{2}{g}^{2}{x}^{2}-20\,acde{g}^{2}x+36\,{c}^{2}{d}^{2}fgx+35\,{a}^{2}{e}^{2}{g}^{2}-90\,acdefg+63\,{c}^{2}{d}^{2}{f}^{2} \right ) }{315\,{a}^{3}{e}^{3}{g}^{3}-945\,{a}^{2}cd{e}^{2}f{g}^{2}+945\,a{c}^{2}{d}^{2}e{f}^{2}g-315\,{c}^{3}{d}^{3}{f}^{3}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{{\frac{3}{2}}} \left ( gx+f \right ) ^{-{\frac{9}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]

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)^(11/2),x)

[Out]

-2/315*(c*d*x+a*e)*(8*c^2*d^2*g^2*x^2-20*a*c*d*e*g^2*x+36*c^2*d^2*f*g*x+35*a^2*e
^2*g^2-90*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)/(g
*x+f)^(9/2)/(a^3*e^3*g^3-3*a^2*c*d*e^2*f*g^2+3*a*c^2*d^2*e*f^2*g-c^3*d^3*f^3)/(e
*x+d)^(3/2)

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Maxima [A]  time = 1.40244, size = 1245, normalized size = 6.29 \[ \text{result too large to display} \]

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)^(11/2)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.306683, size = 1239, normalized size = 6.26 \[ \text{result too large to display} \]

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)^(11/2)),x, algorithm="fricas")

[Out]

2/315*(8*c^4*d^4*g^2*x^4 + 63*a^2*c^2*d^2*e^2*f^2 - 90*a^3*c*d*e^3*f*g + 35*a^4*
e^4*g^2 + 4*(9*c^4*d^4*f*g - a*c^3*d^3*e*g^2)*x^3 + 3*(21*c^4*d^4*f^2 - 6*a*c^3*
d^3*e*f*g + a^2*c^2*d^2*e^2*g^2)*x^2 + 2*(63*a*c^3*d^3*e*f^2 - 72*a^2*c^2*d^2*e^
2*f*g + 25*a^3*c*d*e^3*g^2)*x)*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^3*d^4*f^8 - 3*a*c^2*d^3*e*f^7*g + 3*a^2*c*d^2*e^2*f^6*
g^2 - a^3*d*e^3*f^5*g^3 + (c^3*d^3*e*f^3*g^5 - 3*a*c^2*d^2*e^2*f^2*g^6 + 3*a^2*c
*d*e^3*f*g^7 - a^3*e^4*g^8)*x^6 + (5*c^3*d^3*e*f^4*g^4 - a^3*d*e^3*g^8 + (c^3*d^
4 - 15*a*c^2*d^2*e^2)*f^3*g^5 - 3*(a*c^2*d^3*e - 5*a^2*c*d*e^3)*f^2*g^6 + (3*a^2
*c*d^2*e^2 - 5*a^3*e^4)*f*g^7)*x^5 + 5*(2*c^3*d^3*e*f^5*g^3 - a^3*d*e^3*f*g^7 +
(c^3*d^4 - 6*a*c^2*d^2*e^2)*f^4*g^4 - 3*(a*c^2*d^3*e - 2*a^2*c*d*e^3)*f^3*g^5 +
(3*a^2*c*d^2*e^2 - 2*a^3*e^4)*f^2*g^6)*x^4 + 10*(c^3*d^3*e*f^6*g^2 - a^3*d*e^3*f
^2*g^6 + (c^3*d^4 - 3*a*c^2*d^2*e^2)*f^5*g^3 - 3*(a*c^2*d^3*e - a^2*c*d*e^3)*f^4
*g^4 + (3*a^2*c*d^2*e^2 - a^3*e^4)*f^3*g^5)*x^3 + 5*(c^3*d^3*e*f^7*g - 2*a^3*d*e
^3*f^3*g^5 + (2*c^3*d^4 - 3*a*c^2*d^2*e^2)*f^6*g^2 - 3*(2*a*c^2*d^3*e - a^2*c*d*
e^3)*f^5*g^3 + (6*a^2*c*d^2*e^2 - a^3*e^4)*f^4*g^4)*x^2 + (c^3*d^3*e*f^8 - 5*a^3
*d*e^3*f^4*g^4 + (5*c^3*d^4 - 3*a*c^2*d^2*e^2)*f^7*g - 3*(5*a*c^2*d^3*e - a^2*c*
d*e^3)*f^6*g^2 + (15*a^2*c*d^2*e^2 - a^3*e^4)*f^5*g^3)*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)**(11/2),x)

[Out]

Timed out

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

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)^(11/2)),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError