3.1218 \(\int \frac{\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^8} \, dx\)

Optimal. Leaf size=39 \[ \frac{2 \left (a+b x+c x^2\right )^{7/2}}{7 d^8 \left (b^2-4 a c\right ) (b+2 c x)^7} \]

[Out]

(2*(a + b*x + c*x^2)^(7/2))/(7*(b^2 - 4*a*c)*d^8*(b + 2*c*x)^7)

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Rubi [A]  time = 0.055534, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ \frac{2 \left (a+b x+c x^2\right )^{7/2}}{7 d^8 \left (b^2-4 a c\right ) (b+2 c x)^7} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^8,x]

[Out]

(2*(a + b*x + c*x^2)^(7/2))/(7*(b^2 - 4*a*c)*d^8*(b + 2*c*x)^7)

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Rubi in Sympy [A]  time = 13.7624, size = 36, normalized size = 0.92 \[ \frac{2 \left (a + b x + c x^{2}\right )^{\frac{7}{2}}}{7 d^{8} \left (b + 2 c x\right )^{7} \left (- 4 a c + b^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)**(5/2)/(2*c*d*x+b*d)**8,x)

[Out]

2*(a + b*x + c*x**2)**(7/2)/(7*d**8*(b + 2*c*x)**7*(-4*a*c + b**2))

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Mathematica [A]  time = 0.200414, size = 38, normalized size = 0.97 \[ \frac{2 (a+x (b+c x))^{7/2}}{7 d^8 \left (b^2-4 a c\right ) (b+2 c x)^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^8,x]

[Out]

(2*(a + x*(b + c*x))^(7/2))/(7*(b^2 - 4*a*c)*d^8*(b + 2*c*x)^7)

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Maple [A]  time = 0.008, size = 38, normalized size = 1. \[ -{\frac{2}{7\, \left ( 2\,cx+b \right ) ^{7}{d}^{8} \left ( 4\,ac-{b}^{2} \right ) } \left ( c{x}^{2}+bx+a \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^8,x)

[Out]

-2/7*(c*x^2+b*x+a)^(7/2)/(2*c*x+b)^7/d^8/(4*a*c-b^2)

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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*x^2 + b*x + a)^(5/2)/(2*c*d*x + b*d)^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.10928, size = 365, normalized size = 9.36 \[ \frac{2 \,{\left (c^{3} x^{6} + 3 \, b c^{2} x^{5} + 3 \,{\left (b^{2} c + a c^{2}\right )} x^{4} + 3 \, a^{2} b x +{\left (b^{3} + 6 \, a b c\right )} x^{3} + a^{3} + 3 \,{\left (a b^{2} + a^{2} c\right )} x^{2}\right )} \sqrt{c x^{2} + b x + a}}{7 \,{\left (128 \,{\left (b^{2} c^{7} - 4 \, a c^{8}\right )} d^{8} x^{7} + 448 \,{\left (b^{3} c^{6} - 4 \, a b c^{7}\right )} d^{8} x^{6} + 672 \,{\left (b^{4} c^{5} - 4 \, a b^{2} c^{6}\right )} d^{8} x^{5} + 560 \,{\left (b^{5} c^{4} - 4 \, a b^{3} c^{5}\right )} d^{8} x^{4} + 280 \,{\left (b^{6} c^{3} - 4 \, a b^{4} c^{4}\right )} d^{8} x^{3} + 84 \,{\left (b^{7} c^{2} - 4 \, a b^{5} c^{3}\right )} d^{8} x^{2} + 14 \,{\left (b^{8} c - 4 \, a b^{6} c^{2}\right )} d^{8} x +{\left (b^{9} - 4 \, a b^{7} c\right )} d^{8}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(5/2)/(2*c*d*x + b*d)^8,x, algorithm="fricas")

[Out]

2/7*(c^3*x^6 + 3*b*c^2*x^5 + 3*(b^2*c + a*c^2)*x^4 + 3*a^2*b*x + (b^3 + 6*a*b*c)
*x^3 + a^3 + 3*(a*b^2 + a^2*c)*x^2)*sqrt(c*x^2 + b*x + a)/(128*(b^2*c^7 - 4*a*c^
8)*d^8*x^7 + 448*(b^3*c^6 - 4*a*b*c^7)*d^8*x^6 + 672*(b^4*c^5 - 4*a*b^2*c^6)*d^8
*x^5 + 560*(b^5*c^4 - 4*a*b^3*c^5)*d^8*x^4 + 280*(b^6*c^3 - 4*a*b^4*c^4)*d^8*x^3
+ 84*(b^7*c^2 - 4*a*b^5*c^3)*d^8*x^2 + 14*(b^8*c - 4*a*b^6*c^2)*d^8*x + (b^9 -
4*a*b^7*c)*d^8)

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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((c*x**2+b*x+a)**(5/2)/(2*c*d*x+b*d)**8,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 1.14012, size = 4, normalized size = 0.1 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(5/2)/(2*c*d*x + b*d)^8,x, algorithm="giac")

[Out]

sage0*x