3.1770 \(\int \frac{(a+b x)^3}{(c+d x) (e+f x)^{5/2}} \, dx\)

Optimal. Leaf size=163 \[ \frac{2 (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{3/2} (d e-c f)^{5/2}}+\frac{2 (b e-a f)^2 (a d f-3 b c f+2 b d e)}{f^3 \sqrt{e+f x} (d e-c f)^2}-\frac{2 (b e-a f)^3}{3 f^3 (e+f x)^{3/2} (d e-c f)}+\frac{2 b^3 \sqrt{e+f x}}{d f^3} \]

[Out]

(-2*(b*e - a*f)^3)/(3*f^3*(d*e - c*f)*(e + f*x)^(3/2)) + (2*(b*e - a*f)^2*(2*b*d
*e - 3*b*c*f + a*d*f))/(f^3*(d*e - c*f)^2*Sqrt[e + f*x]) + (2*b^3*Sqrt[e + f*x])
/(d*f^3) + (2*(b*c - a*d)^3*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f]])/(d
^(3/2)*(d*e - c*f)^(5/2))

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Rubi [A]  time = 0.456902, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{2 (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{3/2} (d e-c f)^{5/2}}+\frac{2 (b e-a f)^2 (a d f-3 b c f+2 b d e)}{f^3 \sqrt{e+f x} (d e-c f)^2}-\frac{2 (b e-a f)^3}{3 f^3 (e+f x)^{3/2} (d e-c f)}+\frac{2 b^3 \sqrt{e+f x}}{d f^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^3/((c + d*x)*(e + f*x)^(5/2)),x]

[Out]

(-2*(b*e - a*f)^3)/(3*f^3*(d*e - c*f)*(e + f*x)^(3/2)) + (2*(b*e - a*f)^2*(2*b*d
*e - 3*b*c*f + a*d*f))/(f^3*(d*e - c*f)^2*Sqrt[e + f*x]) + (2*b^3*Sqrt[e + f*x])
/(d*f^3) + (2*(b*c - a*d)^3*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f]])/(d
^(3/2)*(d*e - c*f)^(5/2))

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Rubi in Sympy [A]  time = 140.259, size = 253, normalized size = 1.55 \[ \frac{2 \sqrt{e + f x} \left (a f - b e\right ) \left (a^{2} d^{2} f^{2} - 3 a b c d f^{2} + a b d^{2} e f + 3 b^{2} c^{2} f^{2} - 3 b^{2} c d e f + b^{2} d^{2} e^{2}\right )}{f^{3} \left (c f - d e\right )^{3}} + \frac{2 \left (a f - b e\right )^{2} \left (a d f - 3 b c f + 2 b d e\right )}{f^{3} \sqrt{e + f x} \left (c f - d e\right )^{2}} - \frac{2 \left (a f - b e\right )^{3}}{3 f^{3} \left (e + f x\right )^{\frac{3}{2}} \left (c f - d e\right )} - \frac{2 \sqrt{e + f x} \left (a d - b c\right )^{3}}{d \left (c f - d e\right )^{3}} + \frac{2 \left (a d - b c\right )^{3} \operatorname{atan}{\left (\frac{\sqrt{d} \sqrt{e + f x}}{\sqrt{c f - d e}} \right )}}{d^{\frac{3}{2}} \left (c f - d e\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**3/(d*x+c)/(f*x+e)**(5/2),x)

[Out]

2*sqrt(e + f*x)*(a*f - b*e)*(a**2*d**2*f**2 - 3*a*b*c*d*f**2 + a*b*d**2*e*f + 3*
b**2*c**2*f**2 - 3*b**2*c*d*e*f + b**2*d**2*e**2)/(f**3*(c*f - d*e)**3) + 2*(a*f
- b*e)**2*(a*d*f - 3*b*c*f + 2*b*d*e)/(f**3*sqrt(e + f*x)*(c*f - d*e)**2) - 2*(
a*f - b*e)**3/(3*f**3*(e + f*x)**(3/2)*(c*f - d*e)) - 2*sqrt(e + f*x)*(a*d - b*c
)**3/(d*(c*f - d*e)**3) + 2*(a*d - b*c)**3*atan(sqrt(d)*sqrt(e + f*x)/sqrt(c*f -
d*e))/(d**(3/2)*(c*f - d*e)**(5/2))

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Mathematica [A]  time = 0.684428, size = 156, normalized size = 0.96 \[ \frac{2 \sqrt{e+f x} \left (\frac{3 (b e-a f)^2 (a d f-3 b c f+2 b d e)}{(e+f x) (d e-c f)^2}-\frac{(b e-a f)^3}{(e+f x)^2 (d e-c f)}+\frac{3 b^3}{d}\right )}{3 f^3}+\frac{2 (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{3/2} (d e-c f)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^3/((c + d*x)*(e + f*x)^(5/2)),x]

[Out]

(2*Sqrt[e + f*x]*((3*b^3)/d - (b*e - a*f)^3/((d*e - c*f)*(e + f*x)^2) + (3*(b*e
- a*f)^2*(2*b*d*e - 3*b*c*f + a*d*f))/((d*e - c*f)^2*(e + f*x))))/(3*f^3) + (2*(
b*c - a*d)^3*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f]])/(d^(3/2)*(d*e - c
*f)^(5/2))

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Maple [B]  time = 0.026, size = 501, normalized size = 3.1 \[ 2\,{\frac{{b}^{3}\sqrt{fx+e}}{d{f}^{3}}}-{\frac{2\,{a}^{3}}{3\,cf-3\,de} \left ( fx+e \right ) ^{-{\frac{3}{2}}}}+2\,{\frac{{a}^{2}be}{ \left ( cf-de \right ) f \left ( fx+e \right ) ^{3/2}}}-2\,{\frac{a{b}^{2}{e}^{2}}{{f}^{2} \left ( cf-de \right ) \left ( fx+e \right ) ^{3/2}}}+{\frac{2\,{b}^{3}{e}^{3}}{3\,{f}^{3} \left ( cf-de \right ) } \left ( fx+e \right ) ^{-{\frac{3}{2}}}}+2\,{\frac{{a}^{3}d}{ \left ( cf-de \right ) ^{2}\sqrt{fx+e}}}-6\,{\frac{{a}^{2}bc}{ \left ( cf-de \right ) ^{2}\sqrt{fx+e}}}+12\,{\frac{a{b}^{2}ce}{f \left ( cf-de \right ) ^{2}\sqrt{fx+e}}}-6\,{\frac{a{b}^{2}d{e}^{2}}{{f}^{2} \left ( cf-de \right ) ^{2}\sqrt{fx+e}}}-6\,{\frac{{b}^{3}c{e}^{2}}{{f}^{2} \left ( cf-de \right ) ^{2}\sqrt{fx+e}}}+4\,{\frac{{b}^{3}d{e}^{3}}{{f}^{3} \left ( cf-de \right ) ^{2}\sqrt{fx+e}}}+2\,{\frac{{d}^{2}{a}^{3}}{ \left ( cf-de \right ) ^{2}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }-6\,{\frac{d{a}^{2}cb}{ \left ( cf-de \right ) ^{2}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }+6\,{\frac{a{b}^{2}{c}^{2}}{ \left ( cf-de \right ) ^{2}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }-2\,{\frac{{b}^{3}{c}^{3}}{d \left ( cf-de \right ) ^{2}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3/(d*x+c)/(f*x+e)^(5/2),x)

[Out]

2*b^3*(f*x+e)^(1/2)/d/f^3-2/3/(c*f-d*e)/(f*x+e)^(3/2)*a^3+2/f/(c*f-d*e)/(f*x+e)^
(3/2)*a^2*b*e-2/f^2/(c*f-d*e)/(f*x+e)^(3/2)*a*b^2*e^2+2/3/f^3/(c*f-d*e)/(f*x+e)^
(3/2)*b^3*e^3+2/(c*f-d*e)^2/(f*x+e)^(1/2)*a^3*d-6/(c*f-d*e)^2/(f*x+e)^(1/2)*a^2*
b*c+12/f/(c*f-d*e)^2/(f*x+e)^(1/2)*a*b^2*c*e-6/f^2/(c*f-d*e)^2/(f*x+e)^(1/2)*a*b
^2*d*e^2-6/f^2/(c*f-d*e)^2/(f*x+e)^(1/2)*b^3*c*e^2+4/f^3/(c*f-d*e)^2/(f*x+e)^(1/
2)*b^3*d*e^3+2*d^2/(c*f-d*e)^2/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-
d*e)*d)^(1/2))*a^3-6*d/(c*f-d*e)^2/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((
c*f-d*e)*d)^(1/2))*a^2*c*b+6/(c*f-d*e)^2/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2
)*d/((c*f-d*e)*d)^(1/2))*a*b^2*c^2-2/d/(c*f-d*e)^2/((c*f-d*e)*d)^(1/2)*arctan((f
*x+e)^(1/2)*d/((c*f-d*e)*d)^(1/2))*b^3*c^3

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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((b*x + a)^3/((d*x + c)*(f*x + e)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23141, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3/((d*x + c)*(f*x + e)^(5/2)),x, algorithm="fricas")

[Out]

[-1/3*(3*((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*f^4*x + (b^3*c^3 -
3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*e*f^3)*sqrt(f*x + e)*log((sqrt(d^2*e -
c*d*f)*(d*f*x + 2*d*e - c*f) - 2*(d^2*e - c*d*f)*sqrt(f*x + e))/(d*x + c)) - 2*
(8*b^3*d^2*e^4 - a^3*c*d*f^4 - 2*(7*b^3*c*d + 3*a*b^2*d^2)*e^3*f + 3*(b^3*c^2 +
5*a*b^2*c*d - a^2*b*d^2)*e^2*f^2 - 2*(3*a^2*b*c*d - 2*a^3*d^2)*e*f^3 + 3*(b^3*d^
2*e^2*f^2 - 2*b^3*c*d*e*f^3 + b^3*c^2*f^4)*x^2 + 3*(4*b^3*d^2*e^3*f - (7*b^3*c*d
+ 3*a*b^2*d^2)*e^2*f^2 + 2*(b^3*c^2 + 3*a*b^2*c*d)*e*f^3 - (3*a^2*b*c*d - a^3*d
^2)*f^4)*x)*sqrt(d^2*e - c*d*f))/((d^3*e^3*f^3 - 2*c*d^2*e^2*f^4 + c^2*d*e*f^5 +
(d^3*e^2*f^4 - 2*c*d^2*e*f^5 + c^2*d*f^6)*x)*sqrt(d^2*e - c*d*f)*sqrt(f*x + e))
, 2/3*(3*((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*f^4*x + (b^3*c^3 -
3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*e*f^3)*sqrt(f*x + e)*arctan(-(d*e - c*
f)/(sqrt(-d^2*e + c*d*f)*sqrt(f*x + e))) + (8*b^3*d^2*e^4 - a^3*c*d*f^4 - 2*(7*b
^3*c*d + 3*a*b^2*d^2)*e^3*f + 3*(b^3*c^2 + 5*a*b^2*c*d - a^2*b*d^2)*e^2*f^2 - 2*
(3*a^2*b*c*d - 2*a^3*d^2)*e*f^3 + 3*(b^3*d^2*e^2*f^2 - 2*b^3*c*d*e*f^3 + b^3*c^2
*f^4)*x^2 + 3*(4*b^3*d^2*e^3*f - (7*b^3*c*d + 3*a*b^2*d^2)*e^2*f^2 + 2*(b^3*c^2
+ 3*a*b^2*c*d)*e*f^3 - (3*a^2*b*c*d - a^3*d^2)*f^4)*x)*sqrt(-d^2*e + c*d*f))/((d
^3*e^3*f^3 - 2*c*d^2*e^2*f^4 + c^2*d*e*f^5 + (d^3*e^2*f^4 - 2*c*d^2*e*f^5 + c^2*
d*f^6)*x)*sqrt(-d^2*e + c*d*f)*sqrt(f*x + e))]

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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((b*x+a)**3/(d*x+c)/(f*x+e)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.221206, size = 455, normalized size = 2.79 \[ -\frac{2 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{f x + e} d}{\sqrt{c d f - d^{2} e}}\right )}{{\left (c^{2} d f^{2} - 2 \, c d^{2} f e + d^{3} e^{2}\right )} \sqrt{c d f - d^{2} e}} + \frac{2 \, \sqrt{f x + e} b^{3}}{d f^{3}} - \frac{2 \,{\left (9 \,{\left (f x + e\right )} a^{2} b c f^{3} - 3 \,{\left (f x + e\right )} a^{3} d f^{3} + a^{3} c f^{4} - 18 \,{\left (f x + e\right )} a b^{2} c f^{2} e - 3 \, a^{2} b c f^{3} e - a^{3} d f^{3} e + 9 \,{\left (f x + e\right )} b^{3} c f e^{2} + 9 \,{\left (f x + e\right )} a b^{2} d f e^{2} + 3 \, a b^{2} c f^{2} e^{2} + 3 \, a^{2} b d f^{2} e^{2} - 6 \,{\left (f x + e\right )} b^{3} d e^{3} - b^{3} c f e^{3} - 3 \, a b^{2} d f e^{3} + b^{3} d e^{4}\right )}}{3 \,{\left (c^{2} f^{5} - 2 \, c d f^{4} e + d^{2} f^{3} e^{2}\right )}{\left (f x + e\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3/((d*x + c)*(f*x + e)^(5/2)),x, algorithm="giac")

[Out]

-2*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*arctan(sqrt(f*x + e)*d/sq
rt(c*d*f - d^2*e))/((c^2*d*f^2 - 2*c*d^2*f*e + d^3*e^2)*sqrt(c*d*f - d^2*e)) + 2
*sqrt(f*x + e)*b^3/(d*f^3) - 2/3*(9*(f*x + e)*a^2*b*c*f^3 - 3*(f*x + e)*a^3*d*f^
3 + a^3*c*f^4 - 18*(f*x + e)*a*b^2*c*f^2*e - 3*a^2*b*c*f^3*e - a^3*d*f^3*e + 9*(
f*x + e)*b^3*c*f*e^2 + 9*(f*x + e)*a*b^2*d*f*e^2 + 3*a*b^2*c*f^2*e^2 + 3*a^2*b*d
*f^2*e^2 - 6*(f*x + e)*b^3*d*e^3 - b^3*c*f*e^3 - 3*a*b^2*d*f*e^3 + b^3*d*e^4)/((
c^2*f^5 - 2*c*d*f^4*e + d^2*f^3*e^2)*(f*x + e)^(3/2))