3.1767 \(\int \frac{(a+b x)^3 \sqrt{e+f x}}{c+d x} \, dx\)

Optimal. Leaf size=210 \[ \frac{2 b (e+f x)^{3/2} \left (3 a^2 d^2 f^2-3 a b d f (c f+d e)+b^2 \left (c^2 f^2+c d e f+d^2 e^2\right )\right )}{3 d^3 f^3}-\frac{2 b^2 (e+f x)^{5/2} (-3 a d f+b c f+2 b d e)}{5 d^2 f^3}+\frac{2 (b c-a d)^3 \sqrt{d e-c f} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{9/2}}-\frac{2 \sqrt{e+f x} (b c-a d)^3}{d^4}+\frac{2 b^3 (e+f x)^{7/2}}{7 d f^3} \]

[Out]

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

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Rubi [A]  time = 0.375833, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{2 b (e+f x)^{3/2} \left (3 a^2 d^2 f^2-3 a b d f (c f+d e)+b^2 \left (c^2 f^2+c d e f+d^2 e^2\right )\right )}{3 d^3 f^3}-\frac{2 b^2 (e+f x)^{5/2} (-3 a d f+b c f+2 b d e)}{5 d^2 f^3}+\frac{2 (b c-a d)^3 \sqrt{d e-c f} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{9/2}}-\frac{2 \sqrt{e+f x} (b c-a d)^3}{d^4}+\frac{2 b^3 (e+f x)^{7/2}}{7 d f^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 83.0947, size = 219, normalized size = 1.04 \[ \frac{2 b^{3} \left (e + f x\right )^{\frac{7}{2}}}{7 d f^{3}} + \frac{2 b^{2} \left (e + f x\right )^{\frac{5}{2}} \left (3 a d f - b c f - 2 b d e\right )}{5 d^{2} f^{3}} + \frac{2 b \left (e + f x\right )^{\frac{3}{2}} \left (3 a^{2} d^{2} f^{2} - 3 a b c d f^{2} - 3 a b d^{2} e f + b^{2} c^{2} f^{2} + b^{2} c d e f + b^{2} d^{2} e^{2}\right )}{3 d^{3} f^{3}} + \frac{2 \sqrt{e + f x} \left (a d - b c\right )^{3}}{d^{4}} - \frac{2 \left (a d - b c\right )^{3} \sqrt{c f - d e} \operatorname{atan}{\left (\frac{\sqrt{d} \sqrt{e + f x}}{\sqrt{c f - d e}} \right )}}{d^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.38069, size = 249, normalized size = 1.19 \[ \frac{2 \sqrt{e+f x} \left (105 a^3 d^3 f^3+105 a^2 b d^2 f^2 (d (e+f x)-3 c f)-21 a b^2 d f \left (-15 c^2 f^2+5 c d f (e+f x)+d^2 \left (2 e^2-e f x-3 f^2 x^2\right )\right )+b^3 \left (-105 c^3 f^3+35 c^2 d f^2 (e+f x)-7 c d^2 f \left (-2 e^2+e f x+3 f^2 x^2\right )+d^3 \left (8 e^3-4 e^2 f x+3 e f^2 x^2+15 f^3 x^3\right )\right )\right )}{105 d^4 f^3}+\frac{2 (b c-a d)^3 \sqrt{d e-c f} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[e + f*x]*(105*a^3*d^3*f^3 + 105*a^2*b*d^2*f^2*(-3*c*f + d*(e + f*x)) - 2
1*a*b^2*d*f*(-15*c^2*f^2 + 5*c*d*f*(e + f*x) + d^2*(2*e^2 - e*f*x - 3*f^2*x^2))
+ b^3*(-105*c^3*f^3 + 35*c^2*d*f^2*(e + f*x) - 7*c*d^2*f*(-2*e^2 + e*f*x + 3*f^2
*x^2) + d^3*(8*e^3 - 4*e^2*f*x + 3*e*f^2*x^2 + 15*f^3*x^3))))/(105*d^4*f^3) + (2
*(b*c - a*d)^3*Sqrt[d*e - c*f]*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f]])
/d^(9/2)

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Maple [B]  time = 0.019, size = 629, normalized size = 3. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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*sqrt(f*x + e)/(d*x + c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224183, size = 1, normalized size = 0. \[ \left [-\frac{105 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} f^{3} \sqrt{\frac{d e - c f}{d}} \log \left (\frac{d f x + 2 \, d e - c f - 2 \, \sqrt{f x + e} d \sqrt{\frac{d e - c f}{d}}}{d x + c}\right ) - 2 \,{\left (15 \, b^{3} d^{3} f^{3} x^{3} + 8 \, b^{3} d^{3} e^{3} + 14 \,{\left (b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} e^{2} f + 35 \,{\left (b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} e f^{2} - 105 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} f^{3} + 3 \,{\left (b^{3} d^{3} e f^{2} - 7 \,{\left (b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} f^{3}\right )} x^{2} -{\left (4 \, b^{3} d^{3} e^{2} f + 7 \,{\left (b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} e f^{2} - 35 \,{\left (b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} f^{3}\right )} x\right )} \sqrt{f x + e}}{105 \, d^{4} f^{3}}, \frac{2 \,{\left (105 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} f^{3} \sqrt{-\frac{d e - c f}{d}} \arctan \left (\frac{\sqrt{f x + e}}{\sqrt{-\frac{d e - c f}{d}}}\right ) +{\left (15 \, b^{3} d^{3} f^{3} x^{3} + 8 \, b^{3} d^{3} e^{3} + 14 \,{\left (b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} e^{2} f + 35 \,{\left (b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} e f^{2} - 105 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} f^{3} + 3 \,{\left (b^{3} d^{3} e f^{2} - 7 \,{\left (b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} f^{3}\right )} x^{2} -{\left (4 \, b^{3} d^{3} e^{2} f + 7 \,{\left (b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} e f^{2} - 35 \,{\left (b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} f^{3}\right )} x\right )} \sqrt{f x + e}\right )}}{105 \, d^{4} f^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/105*(105*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*f^3*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*(15*b^3*d^3*f^3*x^3 + 8*b^3*d^3*e^3 + 14*(b^3*c*d^2 - 3*a*b^2*d^3)*e^2
*f + 35*(b^3*c^2*d - 3*a*b^2*c*d^2 + 3*a^2*b*d^3)*e*f^2 - 105*(b^3*c^3 - 3*a*b^2
*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*f^3 + 3*(b^3*d^3*e*f^2 - 7*(b^3*c*d^2 - 3*a*b^
2*d^3)*f^3)*x^2 - (4*b^3*d^3*e^2*f + 7*(b^3*c*d^2 - 3*a*b^2*d^3)*e*f^2 - 35*(b^3
*c^2*d - 3*a*b^2*c*d^2 + 3*a^2*b*d^3)*f^3)*x)*sqrt(f*x + e))/(d^4*f^3), 2/105*(1
05*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*f^3*sqrt(-(d*e - c*f)/d)*
arctan(sqrt(f*x + e)/sqrt(-(d*e - c*f)/d)) + (15*b^3*d^3*f^3*x^3 + 8*b^3*d^3*e^3
+ 14*(b^3*c*d^2 - 3*a*b^2*d^3)*e^2*f + 35*(b^3*c^2*d - 3*a*b^2*c*d^2 + 3*a^2*b*
d^3)*e*f^2 - 105*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*f^3 + 3*(b^
3*d^3*e*f^2 - 7*(b^3*c*d^2 - 3*a*b^2*d^3)*f^3)*x^2 - (4*b^3*d^3*e^2*f + 7*(b^3*c
*d^2 - 3*a*b^2*d^3)*e*f^2 - 35*(b^3*c^2*d - 3*a*b^2*c*d^2 + 3*a^2*b*d^3)*f^3)*x)
*sqrt(f*x + e))/(d^4*f^3)]

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Sympy [A]  time = 25.3216, size = 388, normalized size = 1.85 \[ \frac{2 \left (\frac{b^{3} \left (e + f x\right )^{\frac{7}{2}}}{7 d f^{2}} + \frac{\left (e + f x\right )^{\frac{5}{2}} \left (3 a b^{2} d f - b^{3} c f - 2 b^{3} d e\right )}{5 d^{2} f^{2}} + \frac{\left (e + f x\right )^{\frac{3}{2}} \left (3 a^{2} b d^{2} f^{2} - 3 a b^{2} c d f^{2} - 3 a b^{2} d^{2} e f + b^{3} c^{2} f^{2} + b^{3} c d e f + b^{3} d^{2} e^{2}\right )}{3 d^{3} f^{2}} - \frac{f \left (a d - b c\right )^{3} \left (c f - d e\right ) \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{\sqrt{e + f x}}{\sqrt{\frac{c f - d e}{d}}} \right )}}{d \sqrt{\frac{c f - d e}{d}}} & \text{for}\: \frac{c f - d e}{d} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{\sqrt{e + f x}}{\sqrt{\frac{- c f + d e}{d}}} \right )}}{d \sqrt{\frac{- c f + d e}{d}}} & \text{for}\: e + f x > \frac{- c f + d e}{d} \wedge \frac{c f - d e}{d} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{\sqrt{e + f x}}{\sqrt{\frac{- c f + d e}{d}}} \right )}}{d \sqrt{\frac{- c f + d e}{d}}} & \text{for}\: \frac{c f - d e}{d} < 0 \wedge e + f x < \frac{- c f + d e}{d} \end{cases}\right )}{d^{4}} + \frac{\sqrt{e + f x} \left (a^{3} d^{3} f - 3 a^{2} b c d^{2} f + 3 a b^{2} c^{2} d f - b^{3} c^{3} f\right )}{d^{4}}\right )}{f} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.221507, size = 589, normalized size = 2.8 \[ \frac{2 \,{\left (b^{3} c^{4} f - 3 \, a b^{2} c^{3} d f + 3 \, a^{2} b c^{2} d^{2} f - a^{3} c d^{3} f - b^{3} c^{3} d e + 3 \, a b^{2} c^{2} d^{2} e - 3 \, a^{2} b c d^{3} e + a^{3} d^{4} e\right )} \arctan \left (\frac{\sqrt{f x + e} d}{\sqrt{c d f - d^{2} e}}\right )}{\sqrt{c d f - d^{2} e} d^{4}} + \frac{2 \,{\left (15 \,{\left (f x + e\right )}^{\frac{7}{2}} b^{3} d^{6} f^{18} - 21 \,{\left (f x + e\right )}^{\frac{5}{2}} b^{3} c d^{5} f^{19} + 63 \,{\left (f x + e\right )}^{\frac{5}{2}} a b^{2} d^{6} f^{19} + 35 \,{\left (f x + e\right )}^{\frac{3}{2}} b^{3} c^{2} d^{4} f^{20} - 105 \,{\left (f x + e\right )}^{\frac{3}{2}} a b^{2} c d^{5} f^{20} + 105 \,{\left (f x + e\right )}^{\frac{3}{2}} a^{2} b d^{6} f^{20} - 105 \, \sqrt{f x + e} b^{3} c^{3} d^{3} f^{21} + 315 \, \sqrt{f x + e} a b^{2} c^{2} d^{4} f^{21} - 315 \, \sqrt{f x + e} a^{2} b c d^{5} f^{21} + 105 \, \sqrt{f x + e} a^{3} d^{6} f^{21} - 42 \,{\left (f x + e\right )}^{\frac{5}{2}} b^{3} d^{6} f^{18} e + 35 \,{\left (f x + e\right )}^{\frac{3}{2}} b^{3} c d^{5} f^{19} e - 105 \,{\left (f x + e\right )}^{\frac{3}{2}} a b^{2} d^{6} f^{19} e + 35 \,{\left (f x + e\right )}^{\frac{3}{2}} b^{3} d^{6} f^{18} e^{2}\right )}}{105 \, d^{7} f^{21}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(b^3*c^4*f - 3*a*b^2*c^3*d*f + 3*a^2*b*c^2*d^2*f - a^3*c*d^3*f - b^3*c^3*d*e +
3*a*b^2*c^2*d^2*e - 3*a^2*b*c*d^3*e + a^3*d^4*e)*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^3*d^6*f^18
- 21*(f*x + e)^(5/2)*b^3*c*d^5*f^19 + 63*(f*x + e)^(5/2)*a*b^2*d^6*f^19 + 35*(f
*x + e)^(3/2)*b^3*c^2*d^4*f^20 - 105*(f*x + e)^(3/2)*a*b^2*c*d^5*f^20 + 105*(f*x
+ e)^(3/2)*a^2*b*d^6*f^20 - 105*sqrt(f*x + e)*b^3*c^3*d^3*f^21 + 315*sqrt(f*x +
e)*a*b^2*c^2*d^4*f^21 - 315*sqrt(f*x + e)*a^2*b*c*d^5*f^21 + 105*sqrt(f*x + e)*
a^3*d^6*f^21 - 42*(f*x + e)^(5/2)*b^3*d^6*f^18*e + 35*(f*x + e)^(3/2)*b^3*c*d^5*
f^19*e - 105*(f*x + e)^(3/2)*a*b^2*d^6*f^19*e + 35*(f*x + e)^(3/2)*b^3*d^6*f^18*
e^2)/(d^7*f^21)