3.1750 \(\int \frac{(a+b x) (e+f x)^{3/2}}{c+d x} \, dx\)

Optimal. Leaf size=130 \[ \frac{2 (b c-a d) (d e-c f)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{7/2}}-\frac{2 \sqrt{e+f x} (b c-a d) (d e-c f)}{d^3}-\frac{2 (e+f x)^{3/2} (b c-a d)}{3 d^2}+\frac{2 b (e+f x)^{5/2}}{5 d f} \]

[Out]

(-2*(b*c - a*d)*(d*e - c*f)*Sqrt[e + f*x])/d^3 - (2*(b*c - a*d)*(e + f*x)^(3/2))
/(3*d^2) + (2*b*(e + f*x)^(5/2))/(5*d*f) + (2*(b*c - a*d)*(d*e - c*f)^(3/2)*ArcT
anh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f]])/d^(7/2)

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Rubi [A]  time = 0.233401, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{2 (b c-a d) (d e-c f)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{7/2}}-\frac{2 \sqrt{e+f x} (b c-a d) (d e-c f)}{d^3}-\frac{2 (e+f x)^{3/2} (b c-a d)}{3 d^2}+\frac{2 b (e+f x)^{5/2}}{5 d f} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(b*c - a*d)*(d*e - c*f)*Sqrt[e + f*x])/d^3 - (2*(b*c - a*d)*(e + f*x)^(3/2))
/(3*d^2) + (2*b*(e + f*x)^(5/2))/(5*d*f) + (2*(b*c - a*d)*(d*e - c*f)^(3/2)*ArcT
anh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f]])/d^(7/2)

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Rubi in Sympy [A]  time = 24.8086, size = 114, normalized size = 0.88 \[ \frac{2 b \left (e + f x\right )^{\frac{5}{2}}}{5 d f} + \frac{2 \left (e + f x\right )^{\frac{3}{2}} \left (a d - b c\right )}{3 d^{2}} - \frac{2 \sqrt{e + f x} \left (a d - b c\right ) \left (c f - d e\right )}{d^{3}} + \frac{2 \left (a d - b c\right ) \left (c f - d e\right )^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{d} \sqrt{e + f x}}{\sqrt{c f - d e}} \right )}}{d^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b*(e + f*x)**(5/2)/(5*d*f) + 2*(e + f*x)**(3/2)*(a*d - b*c)/(3*d**2) - 2*sqrt(
e + f*x)*(a*d - b*c)*(c*f - d*e)/d**3 + 2*(a*d - b*c)*(c*f - d*e)**(3/2)*atan(sq
rt(d)*sqrt(e + f*x)/sqrt(c*f - d*e))/d**(7/2)

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Mathematica [A]  time = 0.203358, size = 129, normalized size = 0.99 \[ \frac{2 \sqrt{e+f x} \left (5 a d f (-3 c f+4 d e+d f x)+b \left (15 c^2 f^2-5 c d f (4 e+f x)+3 d^2 (e+f x)^2\right )\right )}{15 d^3 f}+\frac{2 (b c-a d) (d e-c f)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[e + f*x]*(5*a*d*f*(4*d*e - 3*c*f + d*f*x) + b*(15*c^2*f^2 + 3*d^2*(e + f
*x)^2 - 5*c*d*f*(4*e + f*x))))/(15*d^3*f) + (2*(b*c - a*d)*(d*e - c*f)^(3/2)*Arc
Tanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f]])/d^(7/2)

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Maple [B]  time = 0.014, size = 370, normalized size = 2.9 \[{\frac{2\,b}{5\,df} \left ( fx+e \right ) ^{{\frac{5}{2}}}}+{\frac{2\,a}{3\,d} \left ( fx+e \right ) ^{{\frac{3}{2}}}}-{\frac{2\,bc}{3\,{d}^{2}} \left ( fx+e \right ) ^{{\frac{3}{2}}}}-2\,{\frac{acf\sqrt{fx+e}}{{d}^{2}}}+2\,{\frac{ae\sqrt{fx+e}}{d}}+2\,{\frac{bf{c}^{2}\sqrt{fx+e}}{{d}^{3}}}-2\,{\frac{bce\sqrt{fx+e}}{{d}^{2}}}+2\,{\frac{{f}^{2}a{c}^{2}}{{d}^{2}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }-4\,{\frac{acfe}{d\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }+2\,{\frac{a{e}^{2}}{\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }-2\,{\frac{b{c}^{3}{f}^{2}}{{d}^{3}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }+4\,{\frac{bf{c}^{2}e}{{d}^{2}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }-2\,{\frac{bc{e}^{2}}{d\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*b*(f*x+e)^(5/2)/d/f+2/3/d*(f*x+e)^(3/2)*a-2/3/d^2*(f*x+e)^(3/2)*b*c-2*f/d^2*
a*c*(f*x+e)^(1/2)+2/d*a*e*(f*x+e)^(1/2)+2*f/d^3*b*c^2*(f*x+e)^(1/2)-2/d^2*b*c*e*
(f*x+e)^(1/2)+2*f^2/d^2/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)
^(1/2))*a*c^2-4*f/d/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)^(1/
2))*a*c*e+2/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)^(1/2))*a*e^
2-2*f^2/d^3/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)^(1/2))*b*c^
3+4*f/d^2/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)^(1/2))*b*c^2*
e-2/d/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)^(1/2))*b*c*e^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((b*x + a)*(f*x + e)^(3/2)/(d*x + c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221155, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left ({\left (b c d - a d^{2}\right )} e f -{\left (b c^{2} - a c d\right )} f^{2}\right )} \sqrt{\frac{d e - c f}{d}} \log \left (\frac{d f x + 2 \, d e - c f - 2 \, \sqrt{f x + e} d \sqrt{\frac{d e - c f}{d}}}{d x + c}\right ) - 2 \,{\left (3 \, b d^{2} f^{2} x^{2} + 3 \, b d^{2} e^{2} - 20 \,{\left (b c d - a d^{2}\right )} e f + 15 \,{\left (b c^{2} - a c d\right )} f^{2} +{\left (6 \, b d^{2} e f - 5 \,{\left (b c d - a d^{2}\right )} f^{2}\right )} x\right )} \sqrt{f x + e}}{15 \, d^{3} f}, \frac{2 \,{\left (15 \,{\left ({\left (b c d - a d^{2}\right )} e f -{\left (b c^{2} - a c d\right )} f^{2}\right )} \sqrt{-\frac{d e - c f}{d}} \arctan \left (\frac{\sqrt{f x + e}}{\sqrt{-\frac{d e - c f}{d}}}\right ) +{\left (3 \, b d^{2} f^{2} x^{2} + 3 \, b d^{2} e^{2} - 20 \,{\left (b c d - a d^{2}\right )} e f + 15 \,{\left (b c^{2} - a c d\right )} f^{2} +{\left (6 \, b d^{2} e f - 5 \,{\left (b c d - a d^{2}\right )} f^{2}\right )} x\right )} \sqrt{f x + e}\right )}}{15 \, d^{3} f}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/15*(15*((b*c*d - a*d^2)*e*f - (b*c^2 - a*c*d)*f^2)*sqrt((d*e - c*f)/d)*log((
d*f*x + 2*d*e - c*f - 2*sqrt(f*x + e)*d*sqrt((d*e - c*f)/d))/(d*x + c)) - 2*(3*b
*d^2*f^2*x^2 + 3*b*d^2*e^2 - 20*(b*c*d - a*d^2)*e*f + 15*(b*c^2 - a*c*d)*f^2 + (
6*b*d^2*e*f - 5*(b*c*d - a*d^2)*f^2)*x)*sqrt(f*x + e))/(d^3*f), 2/15*(15*((b*c*d
- a*d^2)*e*f - (b*c^2 - a*c*d)*f^2)*sqrt(-(d*e - c*f)/d)*arctan(sqrt(f*x + e)/s
qrt(-(d*e - c*f)/d)) + (3*b*d^2*f^2*x^2 + 3*b*d^2*e^2 - 20*(b*c*d - a*d^2)*e*f +
15*(b*c^2 - a*c*d)*f^2 + (6*b*d^2*e*f - 5*(b*c*d - a*d^2)*f^2)*x)*sqrt(f*x + e)
)/(d^3*f)]

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Sympy [A]  time = 46.4003, size = 258, normalized size = 1.98 \[ \frac{2 b \left (e + f x\right )^{\frac{5}{2}}}{5 d f} + \frac{\left (e + f x\right )^{\frac{3}{2}} \left (2 a d - 2 b c\right )}{3 d^{2}} + \frac{\sqrt{e + f x} \left (- 2 a c d f + 2 a d^{2} e + 2 b c^{2} f - 2 b c d e\right )}{d^{3}} + \frac{2 \left (a d - b c\right ) \left (c f - d e\right )^{2} \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{\sqrt{e + f x}}{\sqrt{\frac{c f - d e}{d}}} \right )}}{d \sqrt{\frac{c f - d e}{d}}} & \text{for}\: \frac{c f - d e}{d} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{\sqrt{e + f x}}{\sqrt{\frac{- c f + d e}{d}}} \right )}}{d \sqrt{\frac{- c f + d e}{d}}} & \text{for}\: e + f x > \frac{- c f + d e}{d} \wedge \frac{c f - d e}{d} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{\sqrt{e + f x}}{\sqrt{\frac{- c f + d e}{d}}} \right )}}{d \sqrt{\frac{- c f + d e}{d}}} & \text{for}\: \frac{c f - d e}{d} < 0 \wedge e + f x < \frac{- c f + d e}{d} \end{cases}\right )}{d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b*(e + f*x)**(5/2)/(5*d*f) + (e + f*x)**(3/2)*(2*a*d - 2*b*c)/(3*d**2) + sqrt(
e + f*x)*(-2*a*c*d*f + 2*a*d**2*e + 2*b*c**2*f - 2*b*c*d*e)/d**3 + 2*(a*d - b*c)
*(c*f - d*e)**2*Piecewise((atan(sqrt(e + f*x)/sqrt((c*f - d*e)/d))/(d*sqrt((c*f
- d*e)/d)), (c*f - d*e)/d > 0), (-acoth(sqrt(e + f*x)/sqrt((-c*f + d*e)/d))/(d*s
qrt((-c*f + d*e)/d)), ((c*f - d*e)/d < 0) & (e + f*x > (-c*f + d*e)/d)), (-atanh
(sqrt(e + f*x)/sqrt((-c*f + d*e)/d))/(d*sqrt((-c*f + d*e)/d)), ((c*f - d*e)/d <
0) & (e + f*x < (-c*f + d*e)/d)))/d**3

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GIAC/XCAS [A]  time = 0.216712, size = 321, normalized size = 2.47 \[ -\frac{2 \,{\left (b c^{3} f^{2} - a c^{2} d f^{2} - 2 \, b c^{2} d f e + 2 \, a c d^{2} f e + b c d^{2} e^{2} - a d^{3} e^{2}\right )} \arctan \left (\frac{\sqrt{f x + e} d}{\sqrt{c d f - d^{2} e}}\right )}{\sqrt{c d f - d^{2} e} d^{3}} + \frac{2 \,{\left (3 \,{\left (f x + e\right )}^{\frac{5}{2}} b d^{4} f^{4} - 5 \,{\left (f x + e\right )}^{\frac{3}{2}} b c d^{3} f^{5} + 5 \,{\left (f x + e\right )}^{\frac{3}{2}} a d^{4} f^{5} + 15 \, \sqrt{f x + e} b c^{2} d^{2} f^{6} - 15 \, \sqrt{f x + e} a c d^{3} f^{6} - 15 \, \sqrt{f x + e} b c d^{3} f^{5} e + 15 \, \sqrt{f x + e} a d^{4} f^{5} e\right )}}{15 \, d^{5} f^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(b*c^3*f^2 - a*c^2*d*f^2 - 2*b*c^2*d*f*e + 2*a*c*d^2*f*e + b*c*d^2*e^2 - a*d^
3*e^2)*arctan(sqrt(f*x + e)*d/sqrt(c*d*f - d^2*e))/(sqrt(c*d*f - d^2*e)*d^3) + 2
/15*(3*(f*x + e)^(5/2)*b*d^4*f^4 - 5*(f*x + e)^(3/2)*b*c*d^3*f^5 + 5*(f*x + e)^(
3/2)*a*d^4*f^5 + 15*sqrt(f*x + e)*b*c^2*d^2*f^6 - 15*sqrt(f*x + e)*a*c*d^3*f^6 -
15*sqrt(f*x + e)*b*c*d^3*f^5*e + 15*sqrt(f*x + e)*a*d^4*f^5*e)/(d^5*f^5)