3.1769 \(\int \frac{(a+b x)^3}{(c+d x) (e+f x)^{3/2}} \, dx\)

Optimal. Leaf size=149 \[ -\frac{2 b^2 \sqrt{e+f x} (-3 a d f+b c f+2 b d e)}{d^2 f^3}+\frac{2 (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{5/2} (d e-c f)^{3/2}}-\frac{2 (b e-a f)^3}{f^3 \sqrt{e+f x} (d e-c f)}+\frac{2 b^3 (e+f x)^{3/2}}{3 d f^3} \]

[Out]

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

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Rubi [A]  time = 0.44494, antiderivative size = 169, normalized size of antiderivative = 1.13, number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{2 b^2 \sqrt{e+f x} (-3 a d f+b c f+b d e)}{d^2 f^3}+\frac{2 (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{5/2} (d e-c f)^{3/2}}-\frac{2 (b e-a f)^3}{f^3 \sqrt{e+f x} (d e-c f)}+\frac{2 b^3 (e+f x)^{3/2}}{3 d f^3}-\frac{2 b^3 e \sqrt{e+f x}}{d f^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 73.6328, size = 178, normalized size = 1.19 \[ \frac{2 b^{3} \left (e + f x\right )^{\frac{3}{2}}}{3 d f^{3}} - \frac{2 \sqrt{e + f x} \left (a f - b e\right )^{2} \left (a d f - 3 b c f + 2 b d e\right )}{f^{3} \left (c f - d e\right )^{2}} - \frac{2 \left (a f - b e\right )^{3}}{f^{3} \sqrt{e + f x} \left (c f - d e\right )} + \frac{2 \sqrt{e + f x} \left (a d - b c\right )^{3}}{d^{2} \left (c f - d e\right )^{2}} - \frac{2 \left (a d - b c\right )^{3} \operatorname{atan}{\left (\frac{\sqrt{d} \sqrt{e + f x}}{\sqrt{c f - d e}} \right )}}{d^{\frac{5}{2}} \left (c f - d e\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.479248, size = 137, normalized size = 0.92 \[ \frac{2 \sqrt{e+f x} \left (-\frac{b^2 (-9 a d f+3 b c f+5 b d e)}{d^2}-\frac{3 (b e-a f)^3}{(e+f x) (d e-c f)}+\frac{b^3 f x}{d}\right )}{3 f^3}+\frac{2 (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{5/2} (d e-c f)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[e + f*x]*(-((b^2*(5*b*d*e + 3*b*c*f - 9*a*d*f))/d^2) + (b^3*f*x)/d - (3*
(b*e - a*f)^3)/((d*e - c*f)*(e + f*x))))/(3*f^3) + (2*(b*c - a*d)^3*ArcTanh[(Sqr
t[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f]])/(d^(5/2)*(d*e - c*f)^(3/2))

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Maple [B]  time = 0.021, size = 395, normalized size = 2.7 \[{\frac{2\,{b}^{3}}{3\,d{f}^{3}} \left ( fx+e \right ) ^{{\frac{3}{2}}}}+6\,{\frac{a{b}^{2}\sqrt{fx+e}}{d{f}^{2}}}-2\,{\frac{{b}^{3}c\sqrt{fx+e}}{{d}^{2}{f}^{2}}}-4\,{\frac{{b}^{3}e\sqrt{fx+e}}{d{f}^{3}}}-2\,{\frac{d{a}^{3}}{ \left ( cf-de \right ) \sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }+6\,{\frac{{a}^{2}bc}{ \left ( cf-de \right ) \sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }-6\,{\frac{a{b}^{2}{c}^{2}}{ \left ( cf-de \right ) d\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }+2\,{\frac{{b}^{3}{c}^{3}}{ \left ( cf-de \right ){d}^{2}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }-2\,{\frac{{a}^{3}}{ \left ( cf-de \right ) \sqrt{fx+e}}}+6\,{\frac{{a}^{2}be}{ \left ( cf-de \right ) f\sqrt{fx+e}}}-6\,{\frac{a{b}^{2}{e}^{2}}{{f}^{2} \left ( cf-de \right ) \sqrt{fx+e}}}+2\,{\frac{{b}^{3}{e}^{3}}{{f}^{3} \left ( cf-de \right ) \sqrt{fx+e}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*b^3*(f*x+e)^(3/2)/d/f^3+6/f^2*b^2/d*a*(f*x+e)^(1/2)-2/f^2*b^3/d^2*c*(f*x+e)^
(1/2)-4*b^3*e*(f*x+e)^(1/2)/d/f^3-2/(c*f-d*e)*d/((c*f-d*e)*d)^(1/2)*arctan((f*x+
e)^(1/2)*d/((c*f-d*e)*d)^(1/2))*a^3+6/(c*f-d*e)/((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)/d/((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/(c*f-d*e)/d^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^3-2/(c*f-d*e)/(f*x+e)^(1/2
)*a^3+6/f/(c*f-d*e)/(f*x+e)^(1/2)*a^2*b*e-6/f^2/(c*f-d*e)/(f*x+e)^(1/2)*a*b^2*e^
2+2/f^3/(c*f-d*e)/(f*x+e)^(1/2)*b^3*e^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)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227149, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt{f x + e} f^{3} \log \left (\frac{\sqrt{d^{2} e - c d f}{\left (d f x + 2 \, d e - c f\right )} + 2 \,{\left (d^{2} e - c d f\right )} \sqrt{f x + e}}{d x + c}\right ) - 2 \,{\left (8 \, b^{3} d^{2} e^{3} - 3 \, a^{3} d^{2} f^{3} - 2 \,{\left (b^{3} c d + 9 \, a b^{2} d^{2}\right )} e^{2} f - 3 \,{\left (b^{3} c^{2} - 3 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} e f^{2} -{\left (b^{3} d^{2} e f^{2} - b^{3} c d f^{3}\right )} x^{2} +{\left (4 \, b^{3} d^{2} e^{2} f -{\left (b^{3} c d + 9 \, a b^{2} d^{2}\right )} e f^{2} - 3 \,{\left (b^{3} c^{2} - 3 \, a b^{2} c d\right )} f^{3}\right )} x\right )} \sqrt{d^{2} e - c d f}}{3 \,{\left (d^{3} e f^{3} - c d^{2} f^{4}\right )} \sqrt{d^{2} e - c d f} \sqrt{f x + e}}, \frac{2 \,{\left (3 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt{f x + e} f^{3} \arctan \left (-\frac{d e - c f}{\sqrt{-d^{2} e + c d f} \sqrt{f x + e}}\right ) -{\left (8 \, b^{3} d^{2} e^{3} - 3 \, a^{3} d^{2} f^{3} - 2 \,{\left (b^{3} c d + 9 \, a b^{2} d^{2}\right )} e^{2} f - 3 \,{\left (b^{3} c^{2} - 3 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} e f^{2} -{\left (b^{3} d^{2} e f^{2} - b^{3} c d f^{3}\right )} x^{2} +{\left (4 \, b^{3} d^{2} e^{2} f -{\left (b^{3} c d + 9 \, a b^{2} d^{2}\right )} e f^{2} - 3 \,{\left (b^{3} c^{2} - 3 \, a b^{2} c d\right )} f^{3}\right )} x\right )} \sqrt{-d^{2} e + c d f}\right )}}{3 \,{\left (d^{3} e f^{3} - c d^{2} f^{4}\right )} \sqrt{-d^{2} e + c d f} \sqrt{f x + e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3/((d*x + c)*(f*x + e)^(3/2)),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)*sqrt(f*x + e)*f^3*lo
g((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^3 - 3*a^3*d^2*f^3 - 2*(b^3*c*d + 9*a*b^2*d^2)*e^2*f
- 3*(b^3*c^2 - 3*a*b^2*c*d - 3*a^2*b*d^2)*e*f^2 - (b^3*d^2*e*f^2 - b^3*c*d*f^3)*
x^2 + (4*b^3*d^2*e^2*f - (b^3*c*d + 9*a*b^2*d^2)*e*f^2 - 3*(b^3*c^2 - 3*a*b^2*c*
d)*f^3)*x)*sqrt(d^2*e - c*d*f))/((d^3*e*f^3 - c*d^2*f^4)*sqrt(d^2*e - c*d*f)*sqr
t(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)*sqrt(f*x
+ e)*f^3*arctan(-(d*e - c*f)/(sqrt(-d^2*e + c*d*f)*sqrt(f*x + e))) - (8*b^3*d^2
*e^3 - 3*a^3*d^2*f^3 - 2*(b^3*c*d + 9*a*b^2*d^2)*e^2*f - 3*(b^3*c^2 - 3*a*b^2*c*
d - 3*a^2*b*d^2)*e*f^2 - (b^3*d^2*e*f^2 - b^3*c*d*f^3)*x^2 + (4*b^3*d^2*e^2*f -
(b^3*c*d + 9*a*b^2*d^2)*e*f^2 - 3*(b^3*c^2 - 3*a*b^2*c*d)*f^3)*x)*sqrt(-d^2*e +
c*d*f))/((d^3*e*f^3 - c*d^2*f^4)*sqrt(-d^2*e + c*d*f)*sqrt(f*x + e))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x\right )^{3}}{\left (c + d x\right ) \left (e + f x\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.217885, size = 325, normalized size = 2.18 \[ \frac{2 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{f x + e} d}{\sqrt{c d f - d^{2} e}}\right )}{{\left (c d^{2} f - d^{3} e\right )} \sqrt{c d f - d^{2} e}} - \frac{2 \,{\left (a^{3} f^{3} - 3 \, a^{2} b f^{2} e + 3 \, a b^{2} f e^{2} - b^{3} e^{3}\right )}}{{\left (c f^{4} - d f^{3} e\right )} \sqrt{f x + e}} + \frac{2 \,{\left ({\left (f x + e\right )}^{\frac{3}{2}} b^{3} d^{2} f^{6} - 3 \, \sqrt{f x + e} b^{3} c d f^{7} + 9 \, \sqrt{f x + e} a b^{2} d^{2} f^{7} - 6 \, \sqrt{f x + e} b^{3} d^{2} f^{6} e\right )}}{3 \, d^{3} f^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3/((d*x + c)*(f*x + e)^(3/2)),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/sqr
t(c*d*f - d^2*e))/((c*d^2*f - d^3*e)*sqrt(c*d*f - d^2*e)) - 2*(a^3*f^3 - 3*a^2*b
*f^2*e + 3*a*b^2*f*e^2 - b^3*e^3)/((c*f^4 - d*f^3*e)*sqrt(f*x + e)) + 2/3*((f*x
+ e)^(3/2)*b^3*d^2*f^6 - 3*sqrt(f*x + e)*b^3*c*d*f^7 + 9*sqrt(f*x + e)*a*b^2*d^2
*f^7 - 6*sqrt(f*x + e)*b^3*d^2*f^6*e)/(d^3*f^9)