3.1768 \(\int \frac{(a+b x)^3}{(c+d x) \sqrt{e+f x}} \, dx\)

Optimal. Leaf size=184 \[ \frac{2 b \sqrt{e+f x} \left (3 a^2 d^2 f^2-3 a b d f (c f+d e)+b^2 \left (c^2 f^2+c d e f+d^2 e^2\right )\right )}{d^3 f^3}-\frac{2 b^2 (e+f x)^{3/2} (-3 a d f+b c f+2 b d e)}{3 d^2 f^3}+\frac{2 (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{7/2} \sqrt{d e-c f}}+\frac{2 b^3 (e+f x)^{5/2}}{5 d f^3} \]

[Out]

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

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Rubi [A]  time = 0.347774, antiderivative size = 184, 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 \[ \frac{2 b \sqrt{e+f x} \left (3 a^2 d^2 f^2-3 a b d f (c f+d e)+b^2 \left (c^2 f^2+c d e f+d^2 e^2\right )\right )}{d^3 f^3}-\frac{2 b^2 (e+f x)^{3/2} (-3 a d f+b c f+2 b d e)}{3 d^2 f^3}+\frac{2 (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{7/2} \sqrt{d e-c f}}+\frac{2 b^3 (e+f x)^{5/2}}{5 d f^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 70.5545, size = 196, normalized size = 1.07 \[ \frac{2 b^{3} \left (e + f x\right )^{\frac{5}{2}}}{5 d f^{3}} + \frac{2 b^{2} \left (e + f x\right )^{\frac{3}{2}} \left (3 a d f - b c f - 2 b d e\right )}{3 d^{2} f^{3}} + \frac{2 b \sqrt{e + f x} \left (3 a^{2} d^{2} f^{2} - 3 a b c d f^{2} - 3 a b d^{2} e f + b^{2} c^{2} f^{2} + b^{2} c d e f + b^{2} d^{2} e^{2}\right )}{d^{3} f^{3}} + \frac{2 \left (a d - b c\right )^{3} \operatorname{atan}{\left (\frac{\sqrt{d} \sqrt{e + f x}}{\sqrt{c f - d e}} \right )}}{d^{\frac{7}{2}} \sqrt{c f - d e}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**3/(d*x+c)/(f*x+e)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.268279, size = 157, normalized size = 0.85 \[ \frac{2 b \sqrt{e+f x} \left (45 a^2 d^2 f^2+15 a b d f (-3 c f-2 d e+d f x)+b^2 \left (15 c^2 f^2-5 c d f (f x-2 e)+d^2 \left (8 e^2-4 e f x+3 f^2 x^2\right )\right )\right )}{15 d^3 f^3}+\frac{2 (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{7/2} \sqrt{d e-c f}} \]

Antiderivative was successfully verified.

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

[Out]

(2*b*Sqrt[e + f*x]*(45*a^2*d^2*f^2 + 15*a*b*d*f*(-2*d*e - 3*c*f + d*f*x) + b^2*(
15*c^2*f^2 - 5*c*d*f*(-2*e + f*x) + d^2*(8*e^2 - 4*e*f*x + 3*f^2*x^2))))/(15*d^3
*f^3) + (2*(b*c - a*d)^3*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f]])/(d^(7
/2)*Sqrt[d*e - c*f])

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Maple [B]  time = 0.016, size = 372, normalized size = 2. \[{\frac{2\,{b}^{3}}{5\,d{f}^{3}} \left ( fx+e \right ) ^{{\frac{5}{2}}}}+2\,{\frac{{b}^{2} \left ( fx+e \right ) ^{3/2}a}{d{f}^{2}}}-{\frac{2\,{b}^{3}c}{3\,{d}^{2}{f}^{2}} \left ( fx+e \right ) ^{{\frac{3}{2}}}}-{\frac{4\,{b}^{3}e}{3\,d{f}^{3}} \left ( fx+e \right ) ^{{\frac{3}{2}}}}+6\,{\frac{{a}^{2}b\sqrt{fx+e}}{df}}-6\,{\frac{a{b}^{2}c\sqrt{fx+e}}{{d}^{2}f}}-6\,{\frac{a{b}^{2}e\sqrt{fx+e}}{d{f}^{2}}}+2\,{\frac{{b}^{3}{c}^{2}\sqrt{fx+e}}{f{d}^{3}}}+2\,{\frac{ce{b}^{3}\sqrt{fx+e}}{{d}^{2}{f}^{2}}}+2\,{\frac{{b}^{3}{e}^{2}\sqrt{fx+e}}{d{f}^{3}}}+2\,{\frac{{a}^{3}}{\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }-6\,{\frac{{a}^{2}bc}{d\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }+6\,{\frac{a{b}^{2}{c}^{2}}{{d}^{2}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }-2\,{\frac{{b}^{3}{c}^{3}}{{d}^{3}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3/(d*x+c)/(f*x+e)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220908, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} f^{3} \log \left (\frac{\sqrt{d^{2} e - c d f}{\left (d f x + 2 \, d e - c f\right )} - 2 \,{\left (d^{2} e - c d f\right )} \sqrt{f x + e}}{d x + c}\right ) - 2 \,{\left (3 \, b^{3} d^{2} f^{2} x^{2} + 8 \, b^{3} d^{2} e^{2} + 10 \,{\left (b^{3} c d - 3 \, a b^{2} d^{2}\right )} e f + 15 \,{\left (b^{3} c^{2} - 3 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} f^{2} -{\left (4 \, b^{3} d^{2} e f + 5 \,{\left (b^{3} c d - 3 \, a b^{2} d^{2}\right )} f^{2}\right )} x\right )} \sqrt{d^{2} e - c d f} \sqrt{f x + e}}{15 \, \sqrt{d^{2} e - c d f} d^{3} f^{3}}, \frac{2 \,{\left (15 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} f^{3} \arctan \left (-\frac{d e - c f}{\sqrt{-d^{2} e + c d f} \sqrt{f x + e}}\right ) +{\left (3 \, b^{3} d^{2} f^{2} x^{2} + 8 \, b^{3} d^{2} e^{2} + 10 \,{\left (b^{3} c d - 3 \, a b^{2} d^{2}\right )} e f + 15 \,{\left (b^{3} c^{2} - 3 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} f^{2} -{\left (4 \, b^{3} d^{2} e f + 5 \,{\left (b^{3} c d - 3 \, a b^{2} d^{2}\right )} f^{2}\right )} x\right )} \sqrt{-d^{2} e + c d f} \sqrt{f x + e}\right )}}{15 \, \sqrt{-d^{2} e + c d f} d^{3} f^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/15*(15*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*f^3*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*(3*b^3*d^2*f^2*x^2 + 8*b^3*d^2*e^2 + 10*(b^3*c*d - 3*a*b^2*d^2)*e*f + 15*(b^
3*c^2 - 3*a*b^2*c*d + 3*a^2*b*d^2)*f^2 - (4*b^3*d^2*e*f + 5*(b^3*c*d - 3*a*b^2*d
^2)*f^2)*x)*sqrt(d^2*e - c*d*f)*sqrt(f*x + e))/(sqrt(d^2*e - c*d*f)*d^3*f^3), 2/
15*(15*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*f^3*arctan(-(d*e - c*
f)/(sqrt(-d^2*e + c*d*f)*sqrt(f*x + e))) + (3*b^3*d^2*f^2*x^2 + 8*b^3*d^2*e^2 +
10*(b^3*c*d - 3*a*b^2*d^2)*e*f + 15*(b^3*c^2 - 3*a*b^2*c*d + 3*a^2*b*d^2)*f^2 -
(4*b^3*d^2*e*f + 5*(b^3*c*d - 3*a*b^2*d^2)*f^2)*x)*sqrt(-d^2*e + c*d*f)*sqrt(f*x
+ e))/(sqrt(-d^2*e + c*d*f)*d^3*f^3)]

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Sympy [A]  time = 75.7796, size = 345, normalized size = 1.88 \[ \frac{2 b^{3} \left (e + f x\right )^{\frac{5}{2}}}{5 d f^{3}} + \frac{2 b^{2} \left (e + f x\right )^{\frac{3}{2}} \left (3 a d f - b c f - 2 b d e\right )}{3 d^{2} f^{3}} + \frac{2 b \sqrt{e + f x} \left (3 a^{2} d^{2} f^{2} - 3 a b c d f^{2} - 3 a b d^{2} e f + b^{2} c^{2} f^{2} + b^{2} c d e f + b^{2} d^{2} e^{2}\right )}{d^{3} f^{3}} - \frac{2 \left (a d - b c\right )^{3} \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{1}{\sqrt{\frac{d}{c f - d e}} \sqrt{e + f x}} \right )}}{\sqrt{\frac{d}{c f - d e}} \left (c f - d e\right )} & \text{for}\: \frac{d}{c f - d e} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{1}{\sqrt{- \frac{d}{c f - d e}} \sqrt{e + f x}} \right )}}{\sqrt{- \frac{d}{c f - d e}} \left (c f - d e\right )} & \text{for}\: \frac{1}{e + f x} > - \frac{d}{c f - d e} \wedge \frac{d}{c f - d e} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{1}{\sqrt{- \frac{d}{c f - d e}} \sqrt{e + f x}} \right )}}{\sqrt{- \frac{d}{c f - d e}} \left (c f - d e\right )} & \text{for}\: \frac{d}{c f - d e} < 0 \wedge \frac{1}{e + f x} < - \frac{d}{c f - d e} \end{cases}\right )}{d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**3/(d*x+c)/(f*x+e)**(1/2),x)

[Out]

2*b**3*(e + f*x)**(5/2)/(5*d*f**3) + 2*b**2*(e + f*x)**(3/2)*(3*a*d*f - b*c*f -
2*b*d*e)/(3*d**2*f**3) + 2*b*sqrt(e + f*x)*(3*a**2*d**2*f**2 - 3*a*b*c*d*f**2 -
3*a*b*d**2*e*f + b**2*c**2*f**2 + b**2*c*d*e*f + b**2*d**2*e**2)/(d**3*f**3) - 2
*(a*d - b*c)**3*Piecewise((atan(1/(sqrt(d/(c*f - d*e))*sqrt(e + f*x)))/(sqrt(d/(
c*f - d*e))*(c*f - d*e)), d/(c*f - d*e) > 0), (-acoth(1/(sqrt(-d/(c*f - d*e))*sq
rt(e + f*x)))/(sqrt(-d/(c*f - d*e))*(c*f - d*e)), (d/(c*f - d*e) < 0) & (1/(e +
f*x) > -d/(c*f - d*e))), (-atanh(1/(sqrt(-d/(c*f - d*e))*sqrt(e + f*x)))/(sqrt(-
d/(c*f - d*e))*(c*f - d*e)), (d/(c*f - d*e) < 0) & (1/(e + f*x) < -d/(c*f - d*e)
)))/d**3

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GIAC/XCAS [A]  time = 0.216274, size = 402, normalized size = 2.18 \[ -\frac{2 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{f x + e} d}{\sqrt{c d f - d^{2} e}}\right )}{\sqrt{c d f - d^{2} e} d^{3}} + \frac{2 \,{\left (3 \,{\left (f x + e\right )}^{\frac{5}{2}} b^{3} d^{4} f^{12} - 5 \,{\left (f x + e\right )}^{\frac{3}{2}} b^{3} c d^{3} f^{13} + 15 \,{\left (f x + e\right )}^{\frac{3}{2}} a b^{2} d^{4} f^{13} + 15 \, \sqrt{f x + e} b^{3} c^{2} d^{2} f^{14} - 45 \, \sqrt{f x + e} a b^{2} c d^{3} f^{14} + 45 \, \sqrt{f x + e} a^{2} b d^{4} f^{14} - 10 \,{\left (f x + e\right )}^{\frac{3}{2}} b^{3} d^{4} f^{12} e + 15 \, \sqrt{f x + e} b^{3} c d^{3} f^{13} e - 45 \, \sqrt{f x + e} a b^{2} d^{4} f^{13} e + 15 \, \sqrt{f x + e} b^{3} d^{4} f^{12} e^{2}\right )}}{15 \, d^{5} f^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3/((d*x + c)*sqrt(f*x + e)),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))/(sqrt(c*d*f - d^2*e)*d^3) + 2/15*(3*(f*x + e)^(5/2)*b^3*d^4*f
^12 - 5*(f*x + e)^(3/2)*b^3*c*d^3*f^13 + 15*(f*x + e)^(3/2)*a*b^2*d^4*f^13 + 15*
sqrt(f*x + e)*b^3*c^2*d^2*f^14 - 45*sqrt(f*x + e)*a*b^2*c*d^3*f^14 + 45*sqrt(f*x
+ e)*a^2*b*d^4*f^14 - 10*(f*x + e)^(3/2)*b^3*d^4*f^12*e + 15*sqrt(f*x + e)*b^3*
c*d^3*f^13*e - 45*sqrt(f*x + e)*a*b^2*d^4*f^13*e + 15*sqrt(f*x + e)*b^3*d^4*f^12
*e^2)/(d^5*f^15)