3.1749 \(\int \frac{(a+b x) (e+f x)^{5/2}}{c+d x} \, dx\)

Optimal. Leaf size=164 \[ \frac{2 (b c-a d) (d e-c f)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{9/2}}-\frac{2 \sqrt{e+f x} (b c-a d) (d e-c f)^2}{d^4}-\frac{2 (e+f x)^{3/2} (b c-a d) (d e-c f)}{3 d^3}-\frac{2 (e+f x)^{5/2} (b c-a d)}{5 d^2}+\frac{2 b (e+f x)^{7/2}}{7 d f} \]

[Out]

(-2*(b*c - a*d)*(d*e - c*f)^2*Sqrt[e + f*x])/d^4 - (2*(b*c - a*d)*(d*e - c*f)*(e
+ f*x)^(3/2))/(3*d^3) - (2*(b*c - a*d)*(e + f*x)^(5/2))/(5*d^2) + (2*b*(e + f*x
)^(7/2))/(7*d*f) + (2*(b*c - a*d)*(d*e - c*f)^(5/2)*ArcTanh[(Sqrt[d]*Sqrt[e + f*
x])/Sqrt[d*e - c*f]])/d^(9/2)

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Rubi [A]  time = 0.408381, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{2 (b c-a d) (d e-c f)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{9/2}}-\frac{2 \sqrt{e+f x} (b c-a d) (d e-c f)^2}{d^4}-\frac{2 (e+f x)^{3/2} (b c-a d) (d e-c f)}{3 d^3}-\frac{2 (e+f x)^{5/2} (b c-a d)}{5 d^2}+\frac{2 b (e+f x)^{7/2}}{7 d f} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(b*c - a*d)*(d*e - c*f)^2*Sqrt[e + f*x])/d^4 - (2*(b*c - a*d)*(d*e - c*f)*(e
+ f*x)^(3/2))/(3*d^3) - (2*(b*c - a*d)*(e + f*x)^(5/2))/(5*d^2) + (2*b*(e + f*x
)^(7/2))/(7*d*f) + (2*(b*c - a*d)*(d*e - c*f)^(5/2)*ArcTanh[(Sqrt[d]*Sqrt[e + f*
x])/Sqrt[d*e - c*f]])/d^(9/2)

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Rubi in Sympy [A]  time = 33.8015, size = 144, normalized size = 0.88 \[ \frac{2 b \left (e + f x\right )^{\frac{7}{2}}}{7 d f} + \frac{2 \left (e + f x\right )^{\frac{5}{2}} \left (a d - b c\right )}{5 d^{2}} - \frac{2 \left (e + f x\right )^{\frac{3}{2}} \left (a d - b c\right ) \left (c f - d e\right )}{3 d^{3}} + \frac{2 \sqrt{e + f x} \left (a d - b c\right ) \left (c f - d e\right )^{2}}{d^{4}} - \frac{2 \left (a d - b c\right ) \left (c f - d e\right )^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{d} \sqrt{e + f x}}{\sqrt{c f - d e}} \right )}}{d^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b*(e + f*x)**(7/2)/(7*d*f) + 2*(e + f*x)**(5/2)*(a*d - b*c)/(5*d**2) - 2*(e +
f*x)**(3/2)*(a*d - b*c)*(c*f - d*e)/(3*d**3) + 2*sqrt(e + f*x)*(a*d - b*c)*(c*f
- d*e)**2/d**4 - 2*(a*d - b*c)*(c*f - d*e)**(5/2)*atan(sqrt(d)*sqrt(e + f*x)/sqr
t(c*f - d*e))/d**(9/2)

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Mathematica [A]  time = 0.337608, size = 190, normalized size = 1.16 \[ \frac{2 \sqrt{e+f x} \left (7 a d f \left (15 c^2 f^2-5 c d f (7 e+f x)+d^2 \left (23 e^2+11 e f x+3 f^2 x^2\right )\right )+b \left (-105 c^3 f^3+35 c^2 d f^2 (7 e+f x)-7 c d^2 f \left (23 e^2+11 e f x+3 f^2 x^2\right )+15 d^3 (e+f x)^3\right )\right )}{105 d^4 f}+\frac{2 (b c-a d) (d e-c f)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[e + f*x]*(7*a*d*f*(15*c^2*f^2 - 5*c*d*f*(7*e + f*x) + d^2*(23*e^2 + 11*e
*f*x + 3*f^2*x^2)) + b*(-105*c^3*f^3 + 15*d^3*(e + f*x)^3 + 35*c^2*d*f^2*(7*e +
f*x) - 7*c*d^2*f*(23*e^2 + 11*e*f*x + 3*f^2*x^2))))/(105*d^4*f) + (2*(b*c - a*d)
*(d*e - c*f)^(5/2)*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f]])/d^(9/2)

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Maple [B]  time = 0.023, size = 573, normalized size = 3.5 \[{\frac{2\,b}{7\,df} \left ( fx+e \right ) ^{{\frac{7}{2}}}}+{\frac{2\,a}{5\,d} \left ( fx+e \right ) ^{{\frac{5}{2}}}}-{\frac{2\,bc}{5\,{d}^{2}} \left ( fx+e \right ) ^{{\frac{5}{2}}}}-{\frac{2\,acf}{3\,{d}^{2}} \left ( fx+e \right ) ^{{\frac{3}{2}}}}+{\frac{2\,ae}{3\,d} \left ( fx+e \right ) ^{{\frac{3}{2}}}}+{\frac{2\,bf{c}^{2}}{3\,{d}^{3}} \left ( fx+e \right ) ^{{\frac{3}{2}}}}-{\frac{2\,bce}{3\,{d}^{2}} \left ( fx+e \right ) ^{{\frac{3}{2}}}}+2\,{\frac{{f}^{2}a{c}^{2}\sqrt{fx+e}}{{d}^{3}}}-4\,{\frac{acfe\sqrt{fx+e}}{{d}^{2}}}+2\,{\frac{a{e}^{2}\sqrt{fx+e}}{d}}-2\,{\frac{b{f}^{2}{c}^{3}\sqrt{fx+e}}{{d}^{4}}}+4\,{\frac{bf{c}^{2}e\sqrt{fx+e}}{{d}^{3}}}-2\,{\frac{bc{e}^{2}\sqrt{fx+e}}{{d}^{2}}}-2\,{\frac{{f}^{3}a{c}^{3}}{{d}^{3}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }+6\,{\frac{{f}^{2}a{c}^{2}e}{{d}^{2}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }-6\,{\frac{acf{e}^{2}}{d\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }+2\,{\frac{a{e}^{3}}{\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }+2\,{\frac{b{c}^{4}{f}^{3}}{{d}^{4}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }-6\,{\frac{b{f}^{2}{c}^{3}e}{{d}^{3}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }+6\,{\frac{bf{c}^{2}{e}^{2}}{{d}^{2}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }-2\,{\frac{bc{e}^{3}}{d\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/7*b*(f*x+e)^(7/2)/d/f+2/5/d*(f*x+e)^(5/2)*a-2/5/d^2*(f*x+e)^(5/2)*b*c-2/3*f/d^
2*(f*x+e)^(3/2)*a*c+2/3/d*(f*x+e)^(3/2)*a*e+2/3*f/d^3*(f*x+e)^(3/2)*b*c^2-2/3/d^
2*(f*x+e)^(3/2)*b*c*e+2*f^2/d^3*a*c^2*(f*x+e)^(1/2)-4*f/d^2*a*c*e*(f*x+e)^(1/2)+
2/d*a*e^2*(f*x+e)^(1/2)-2*f^2/d^4*b*c^3*(f*x+e)^(1/2)+4*f/d^3*b*c^2*e*(f*x+e)^(1
/2)-2/d^2*b*c*e^2*(f*x+e)^(1/2)-2*f^3/d^3/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/
2)*d/((c*f-d*e)*d)^(1/2))*a*c^3+6*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*e-6*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+2/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c
*f-d*e)*d)^(1/2))*a*e^3+2*f^3/d^4/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c
*f-d*e)*d)^(1/2))*b*c^4-6*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*e+6*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^2-2/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^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)*(f*x + e)^(5/2)/(d*x + c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224025, size = 1, normalized size = 0.01 \[ \left [\frac{105 \,{\left ({\left (b c d^{2} - a d^{3}\right )} e^{2} f - 2 \,{\left (b c^{2} d - a c d^{2}\right )} e f^{2} +{\left (b c^{3} - a c^{2} d\right )} f^{3}\right )} \sqrt{\frac{d e - c f}{d}} \log \left (\frac{d f x + 2 \, d e - c f + 2 \, \sqrt{f x + e} d \sqrt{\frac{d e - c f}{d}}}{d x + c}\right ) + 2 \,{\left (15 \, b d^{3} f^{3} x^{3} + 15 \, b d^{3} e^{3} - 161 \,{\left (b c d^{2} - a d^{3}\right )} e^{2} f + 245 \,{\left (b c^{2} d - a c d^{2}\right )} e f^{2} - 105 \,{\left (b c^{3} - a c^{2} d\right )} f^{3} + 3 \,{\left (15 \, b d^{3} e f^{2} - 7 \,{\left (b c d^{2} - a d^{3}\right )} f^{3}\right )} x^{2} +{\left (45 \, b d^{3} e^{2} f - 77 \,{\left (b c d^{2} - a d^{3}\right )} e f^{2} + 35 \,{\left (b c^{2} d - a c d^{2}\right )} f^{3}\right )} x\right )} \sqrt{f x + e}}{105 \, d^{4} f}, \frac{2 \,{\left (105 \,{\left ({\left (b c d^{2} - a d^{3}\right )} e^{2} f - 2 \,{\left (b c^{2} d - a c d^{2}\right )} e f^{2} +{\left (b c^{3} - a c^{2} d\right )} f^{3}\right )} \sqrt{-\frac{d e - c f}{d}} \arctan \left (\frac{\sqrt{f x + e}}{\sqrt{-\frac{d e - c f}{d}}}\right ) +{\left (15 \, b d^{3} f^{3} x^{3} + 15 \, b d^{3} e^{3} - 161 \,{\left (b c d^{2} - a d^{3}\right )} e^{2} f + 245 \,{\left (b c^{2} d - a c d^{2}\right )} e f^{2} - 105 \,{\left (b c^{3} - a c^{2} d\right )} f^{3} + 3 \,{\left (15 \, b d^{3} e f^{2} - 7 \,{\left (b c d^{2} - a d^{3}\right )} f^{3}\right )} x^{2} +{\left (45 \, b d^{3} e^{2} f - 77 \,{\left (b c d^{2} - a d^{3}\right )} e f^{2} + 35 \,{\left (b c^{2} d - a c d^{2}\right )} f^{3}\right )} x\right )} \sqrt{f x + e}\right )}}{105 \, d^{4} f}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/105*(105*((b*c*d^2 - a*d^3)*e^2*f - 2*(b*c^2*d - a*c*d^2)*e*f^2 + (b*c^3 - a*
c^2*d)*f^3)*sqrt((d*e - c*f)/d)*log((d*f*x + 2*d*e - c*f + 2*sqrt(f*x + e)*d*sqr
t((d*e - c*f)/d))/(d*x + c)) + 2*(15*b*d^3*f^3*x^3 + 15*b*d^3*e^3 - 161*(b*c*d^2
- a*d^3)*e^2*f + 245*(b*c^2*d - a*c*d^2)*e*f^2 - 105*(b*c^3 - a*c^2*d)*f^3 + 3*
(15*b*d^3*e*f^2 - 7*(b*c*d^2 - a*d^3)*f^3)*x^2 + (45*b*d^3*e^2*f - 77*(b*c*d^2 -
a*d^3)*e*f^2 + 35*(b*c^2*d - a*c*d^2)*f^3)*x)*sqrt(f*x + e))/(d^4*f), 2/105*(10
5*((b*c*d^2 - a*d^3)*e^2*f - 2*(b*c^2*d - a*c*d^2)*e*f^2 + (b*c^3 - a*c^2*d)*f^3
)*sqrt(-(d*e - c*f)/d)*arctan(sqrt(f*x + e)/sqrt(-(d*e - c*f)/d)) + (15*b*d^3*f^
3*x^3 + 15*b*d^3*e^3 - 161*(b*c*d^2 - a*d^3)*e^2*f + 245*(b*c^2*d - a*c*d^2)*e*f
^2 - 105*(b*c^3 - a*c^2*d)*f^3 + 3*(15*b*d^3*e*f^2 - 7*(b*c*d^2 - a*d^3)*f^3)*x^
2 + (45*b*d^3*e^2*f - 77*(b*c*d^2 - a*d^3)*e*f^2 + 35*(b*c^2*d - a*c*d^2)*f^3)*x
)*sqrt(f*x + e))/(d^4*f)]

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Sympy [A]  time = 78.5803, size = 340, normalized size = 2.07 \[ \frac{2 b \left (e + f x\right )^{\frac{7}{2}}}{7 d f} + \frac{\left (e + f x\right )^{\frac{5}{2}} \left (2 a d - 2 b c\right )}{5 d^{2}} + \frac{\left (e + f x\right )^{\frac{3}{2}} \left (- 2 a c d f + 2 a d^{2} e + 2 b c^{2} f - 2 b c d e\right )}{3 d^{3}} + \frac{\sqrt{e + f x} \left (2 a c^{2} d f^{2} - 4 a c d^{2} e f + 2 a d^{3} e^{2} - 2 b c^{3} f^{2} + 4 b c^{2} d e f - 2 b c d^{2} e^{2}\right )}{d^{4}} - \frac{2 \left (a d - b c\right ) \left (c f - d e\right )^{3} \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{\sqrt{e + f x}}{\sqrt{\frac{c f - d e}{d}}} \right )}}{d \sqrt{\frac{c f - d e}{d}}} & \text{for}\: \frac{c f - d e}{d} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{\sqrt{e + f x}}{\sqrt{\frac{- c f + d e}{d}}} \right )}}{d \sqrt{\frac{- c f + d e}{d}}} & \text{for}\: e + f x > \frac{- c f + d e}{d} \wedge \frac{c f - d e}{d} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{\sqrt{e + f x}}{\sqrt{\frac{- c f + d e}{d}}} \right )}}{d \sqrt{\frac{- c f + d e}{d}}} & \text{for}\: \frac{c f - d e}{d} < 0 \wedge e + f x < \frac{- c f + d e}{d} \end{cases}\right )}{d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b*(e + f*x)**(7/2)/(7*d*f) + (e + f*x)**(5/2)*(2*a*d - 2*b*c)/(5*d**2) + (e +
f*x)**(3/2)*(-2*a*c*d*f + 2*a*d**2*e + 2*b*c**2*f - 2*b*c*d*e)/(3*d**3) + sqrt(e
+ f*x)*(2*a*c**2*d*f**2 - 4*a*c*d**2*e*f + 2*a*d**3*e**2 - 2*b*c**3*f**2 + 4*b*
c**2*d*e*f - 2*b*c*d**2*e**2)/d**4 - 2*(a*d - b*c)*(c*f - d*e)**3*Piecewise((ata
n(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*sqrt((-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**4

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GIAC/XCAS [A]  time = 0.220656, size = 522, normalized size = 3.18 \[ \frac{2 \,{\left (b c^{4} f^{3} - a c^{3} d f^{3} - 3 \, b c^{3} d f^{2} e + 3 \, a c^{2} d^{2} f^{2} e + 3 \, b c^{2} d^{2} f e^{2} - 3 \, a c d^{3} f e^{2} - b c d^{3} e^{3} + a d^{4} e^{3}\right )} \arctan \left (\frac{\sqrt{f x + e} d}{\sqrt{c d f - d^{2} e}}\right )}{\sqrt{c d f - d^{2} e} d^{4}} + \frac{2 \,{\left (15 \,{\left (f x + e\right )}^{\frac{7}{2}} b d^{6} f^{6} - 21 \,{\left (f x + e\right )}^{\frac{5}{2}} b c d^{5} f^{7} + 21 \,{\left (f x + e\right )}^{\frac{5}{2}} a d^{6} f^{7} + 35 \,{\left (f x + e\right )}^{\frac{3}{2}} b c^{2} d^{4} f^{8} - 35 \,{\left (f x + e\right )}^{\frac{3}{2}} a c d^{5} f^{8} - 105 \, \sqrt{f x + e} b c^{3} d^{3} f^{9} + 105 \, \sqrt{f x + e} a c^{2} d^{4} f^{9} - 35 \,{\left (f x + e\right )}^{\frac{3}{2}} b c d^{5} f^{7} e + 35 \,{\left (f x + e\right )}^{\frac{3}{2}} a d^{6} f^{7} e + 210 \, \sqrt{f x + e} b c^{2} d^{4} f^{8} e - 210 \, \sqrt{f x + e} a c d^{5} f^{8} e - 105 \, \sqrt{f x + e} b c d^{5} f^{7} e^{2} + 105 \, \sqrt{f x + e} a d^{6} f^{7} e^{2}\right )}}{105 \, d^{7} f^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(b*c^4*f^3 - a*c^3*d*f^3 - 3*b*c^3*d*f^2*e + 3*a*c^2*d^2*f^2*e + 3*b*c^2*d^2*f
*e^2 - 3*a*c*d^3*f*e^2 - b*c*d^3*e^3 + a*d^4*e^3)*arctan(sqrt(f*x + e)*d/sqrt(c*
d*f - d^2*e))/(sqrt(c*d*f - d^2*e)*d^4) + 2/105*(15*(f*x + e)^(7/2)*b*d^6*f^6 -
21*(f*x + e)^(5/2)*b*c*d^5*f^7 + 21*(f*x + e)^(5/2)*a*d^6*f^7 + 35*(f*x + e)^(3/
2)*b*c^2*d^4*f^8 - 35*(f*x + e)^(3/2)*a*c*d^5*f^8 - 105*sqrt(f*x + e)*b*c^3*d^3*
f^9 + 105*sqrt(f*x + e)*a*c^2*d^4*f^9 - 35*(f*x + e)^(3/2)*b*c*d^5*f^7*e + 35*(f
*x + e)^(3/2)*a*d^6*f^7*e + 210*sqrt(f*x + e)*b*c^2*d^4*f^8*e - 210*sqrt(f*x + e
)*a*c*d^5*f^8*e - 105*sqrt(f*x + e)*b*c*d^5*f^7*e^2 + 105*sqrt(f*x + e)*a*d^6*f^
7*e^2)/(d^7*f^7)