3.1766 \(\int \frac{(a+b x)^3 (e+f x)^{3/2}}{c+d x} \, dx\)

Optimal. Leaf size=244 \[ \frac{2 b (e+f x)^{5/2} \left (3 a^2 d^2 f^2-3 a b d f (c f+d e)+b^2 \left (c^2 f^2+c d e f+d^2 e^2\right )\right )}{5 d^3 f^3}-\frac{2 b^2 (e+f x)^{7/2} (-3 a d f+b c f+2 b d e)}{7 d^2 f^3}+\frac{2 (b c-a d)^3 (d e-c f)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{11/2}}-\frac{2 \sqrt{e+f x} (b c-a d)^3 (d e-c f)}{d^5}-\frac{2 (e+f x)^{3/2} (b c-a d)^3}{3 d^4}+\frac{2 b^3 (e+f x)^{9/2}}{9 d f^3} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.481759, antiderivative size = 244, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{2 b (e+f x)^{5/2} \left (3 a^2 d^2 f^2-3 a b d f (c f+d e)+b^2 \left (c^2 f^2+c d e f+d^2 e^2\right )\right )}{5 d^3 f^3}-\frac{2 b^2 (e+f x)^{7/2} (-3 a d f+b c f+2 b d e)}{7 d^2 f^3}+\frac{2 (b c-a d)^3 (d e-c f)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{11/2}}-\frac{2 \sqrt{e+f x} (b c-a d)^3 (d e-c f)}{d^5}-\frac{2 (e+f x)^{3/2} (b c-a d)^3}{3 d^4}+\frac{2 b^3 (e+f x)^{9/2}}{9 d f^3} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 85.6216, size = 250, normalized size = 1.02 \[ \frac{2 b^{3} \left (e + f x\right )^{\frac{9}{2}}}{9 d f^{3}} + \frac{2 b^{2} \left (e + f x\right )^{\frac{7}{2}} \left (3 a d f - b c f - 2 b d e\right )}{7 d^{2} f^{3}} + \frac{2 b \left (e + f x\right )^{\frac{5}{2}} \left (3 a^{2} d^{2} f^{2} - 3 a b c d f^{2} - 3 a b d^{2} e f + b^{2} c^{2} f^{2} + b^{2} c d e f + b^{2} d^{2} e^{2}\right )}{5 d^{3} f^{3}} + \frac{2 \left (e + f x\right )^{\frac{3}{2}} \left (a d - b c\right )^{3}}{3 d^{4}} - \frac{2 \sqrt{e + f x} \left (a d - b c\right )^{3} \left (c f - d e\right )}{d^{5}} + \frac{2 \left (a d - b c\right )^{3} \left (c f - d e\right )^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{d} \sqrt{e + f x}}{\sqrt{c f - d e}} \right )}}{d^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b**3*(e + f*x)**(9/2)/(9*d*f**3) + 2*b**2*(e + f*x)**(7/2)*(3*a*d*f - b*c*f -
2*b*d*e)/(7*d**2*f**3) + 2*b*(e + f*x)**(5/2)*(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)/(5*d**3*f**3
) + 2*(e + f*x)**(3/2)*(a*d - b*c)**3/(3*d**4) - 2*sqrt(e + f*x)*(a*d - b*c)**3*
(c*f - d*e)/d**5 + 2*(a*d - b*c)**3*(c*f - d*e)**(3/2)*atan(sqrt(d)*sqrt(e + f*x
)/sqrt(c*f - d*e))/d**(11/2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.606743, size = 313, normalized size = 1.28 \[ \frac{2 \sqrt{e+f x} \left (105 a^3 d^3 f^3 (-3 c f+4 d e+d f x)+63 a^2 b d^2 f^2 \left (15 c^2 f^2-5 c d f (4 e+f x)+3 d^2 (e+f x)^2\right )-9 a b^2 d f \left (105 c^3 f^3-35 c^2 d f^2 (4 e+f x)+21 c d^2 f (e+f x)^2+3 d^3 (2 e-5 f x) (e+f x)^2\right )+b^3 \left (315 c^4 f^4-105 c^3 d f^3 (4 e+f x)+63 c^2 d^2 f^2 (e+f x)^2+9 c d^3 f (2 e-5 f x) (e+f x)^2+d^4 (e+f x)^2 \left (8 e^2-20 e f x+35 f^2 x^2\right )\right )\right )}{315 d^5 f^3}+\frac{2 (b c-a d)^3 (d e-c f)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{11/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[e + f*x]*(105*a^3*d^3*f^3*(4*d*e - 3*c*f + d*f*x) + 63*a^2*b*d^2*f^2*(15
*c^2*f^2 + 3*d^2*(e + f*x)^2 - 5*c*d*f*(4*e + f*x)) - 9*a*b^2*d*f*(105*c^3*f^3 +
21*c*d^2*f*(e + f*x)^2 + 3*d^3*(2*e - 5*f*x)*(e + f*x)^2 - 35*c^2*d*f^2*(4*e +
f*x)) + b^3*(315*c^4*f^4 + 63*c^2*d^2*f^2*(e + f*x)^2 + 9*c*d^3*f*(2*e - 5*f*x)*
(e + f*x)^2 - 105*c^3*d*f^3*(4*e + f*x) + d^4*(e + f*x)^2*(8*e^2 - 20*e*f*x + 35
*f^2*x^2))))/(315*d^5*f^3) + (2*(b*c - a*d)^3*(d*e - c*f)^(3/2)*ArcTanh[(Sqrt[d]
*Sqrt[e + f*x])/Sqrt[d*e - c*f]])/d^(11/2)

_______________________________________________________________________________________

Maple [B]  time = 0.022, size = 984, normalized size = 4. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-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
^2-2*f^2/d^5/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)^(1/2))*b^3
*c^5-6/5/f/d^2*(f*x+e)^(5/2)*a*b^2*c-6/5/f^2/d*(f*x+e)^(5/2)*a*b^2*e+2/5/f^2/d^2
*(f*x+e)^(5/2)*b^3*c*e+6*f/d^3*a^2*b*c^2*(f*x+e)^(1/2)+2/3/d*(f*x+e)^(3/2)*a^3-2
*f/d^2*a^3*c*(f*x+e)^(1/2)-2/3/d^4*(f*x+e)^(3/2)*b^3*c^3+2/d*a^3*e*(f*x+e)^(1/2)
+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^2+6/5/f
/d*(f*x+e)^(5/2)*a^2*b+2/5/f/d^3*(f*x+e)^(5/2)*b^3*c^2+2/9*b^3*(f*x+e)^(9/2)/d/f
^3+12*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*e-12*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*e+4*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*e+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^2-6/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^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^3*c*e-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))*a^2*b*c^3+6*f^2/d^4/((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^4-6*f/d^4*a*b^2*c^3*(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^3*c^2-6/d^2*a^2*b*c*e*(f*x+
e)^(1/2)+6/d^3*a*b^2*c^2*e*(f*x+e)^(1/2)+6/7/f^2/d*(f*x+e)^(7/2)*a*b^2-2/7/f^2/d
^2*(f*x+e)^(7/2)*b^3*c-2/d^2*(f*x+e)^(3/2)*a^2*b*c+2/d^3*(f*x+e)^(3/2)*a*b^2*c^2
-2/d^4*b^3*c^3*e*(f*x+e)^(1/2)-4/7/f^3/d*(f*x+e)^(7/2)*b^3*e+2/5/f^3/d*(f*x+e)^(
5/2)*b^3*e^2+2*f/d^5*b^3*c^4*(f*x+e)^(1/2)

_______________________________________________________________________________________

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*(f*x + e)^(3/2)/(d*x + c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.231866, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/315*(315*((b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*e*f^3 - (b^
3*c^4 - 3*a*b^2*c^3*d + 3*a^2*b*c^2*d^2 - a^3*c*d^3)*f^4)*sqrt((d*e - c*f)/d)*lo
g((d*f*x + 2*d*e - c*f + 2*sqrt(f*x + e)*d*sqrt((d*e - c*f)/d))/(d*x + c)) + 2*(
35*b^3*d^4*f^4*x^4 + 8*b^3*d^4*e^4 + 18*(b^3*c*d^3 - 3*a*b^2*d^4)*e^3*f + 63*(b^
3*c^2*d^2 - 3*a*b^2*c*d^3 + 3*a^2*b*d^4)*e^2*f^2 - 420*(b^3*c^3*d - 3*a*b^2*c^2*
d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*e*f^3 + 315*(b^3*c^4 - 3*a*b^2*c^3*d + 3*a^2*b*c^
2*d^2 - a^3*c*d^3)*f^4 + 5*(10*b^3*d^4*e*f^3 - 9*(b^3*c*d^3 - 3*a*b^2*d^4)*f^4)*
x^3 + 3*(b^3*d^4*e^2*f^2 - 24*(b^3*c*d^3 - 3*a*b^2*d^4)*e*f^3 + 21*(b^3*c^2*d^2
- 3*a*b^2*c*d^3 + 3*a^2*b*d^4)*f^4)*x^2 - (4*b^3*d^4*e^3*f + 9*(b^3*c*d^3 - 3*a*
b^2*d^4)*e^2*f^2 - 126*(b^3*c^2*d^2 - 3*a*b^2*c*d^3 + 3*a^2*b*d^4)*e*f^3 + 105*(
b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*f^4)*x)*sqrt(f*x + e))/(d
^5*f^3), 2/315*(315*((b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*e*f
^3 - (b^3*c^4 - 3*a*b^2*c^3*d + 3*a^2*b*c^2*d^2 - a^3*c*d^3)*f^4)*sqrt(-(d*e - c
*f)/d)*arctan(sqrt(f*x + e)/sqrt(-(d*e - c*f)/d)) + (35*b^3*d^4*f^4*x^4 + 8*b^3*
d^4*e^4 + 18*(b^3*c*d^3 - 3*a*b^2*d^4)*e^3*f + 63*(b^3*c^2*d^2 - 3*a*b^2*c*d^3 +
3*a^2*b*d^4)*e^2*f^2 - 420*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d
^4)*e*f^3 + 315*(b^3*c^4 - 3*a*b^2*c^3*d + 3*a^2*b*c^2*d^2 - a^3*c*d^3)*f^4 + 5*
(10*b^3*d^4*e*f^3 - 9*(b^3*c*d^3 - 3*a*b^2*d^4)*f^4)*x^3 + 3*(b^3*d^4*e^2*f^2 -
24*(b^3*c*d^3 - 3*a*b^2*d^4)*e*f^3 + 21*(b^3*c^2*d^2 - 3*a*b^2*c*d^3 + 3*a^2*b*d
^4)*f^4)*x^2 - (4*b^3*d^4*e^3*f + 9*(b^3*c*d^3 - 3*a*b^2*d^4)*e^2*f^2 - 126*(b^3
*c^2*d^2 - 3*a*b^2*c*d^3 + 3*a^2*b*d^4)*e*f^3 + 105*(b^3*c^3*d - 3*a*b^2*c^2*d^2
+ 3*a^2*b*c*d^3 - a^3*d^4)*f^4)*x)*sqrt(f*x + e))/(d^5*f^3)]

_______________________________________________________________________________________

Sympy [A]  time = 136.518, size = 500, normalized size = 2.05 \[ \frac{2 b^{3} \left (e + f x\right )^{\frac{9}{2}}}{9 d f^{3}} + \frac{\left (e + f x\right )^{\frac{7}{2}} \left (6 a b^{2} d f - 2 b^{3} c f - 4 b^{3} d e\right )}{7 d^{2} f^{3}} + \frac{\left (e + f x\right )^{\frac{5}{2}} \left (6 a^{2} b d^{2} f^{2} - 6 a b^{2} c d f^{2} - 6 a b^{2} d^{2} e f + 2 b^{3} c^{2} f^{2} + 2 b^{3} c d e f + 2 b^{3} d^{2} e^{2}\right )}{5 d^{3} f^{3}} + \frac{\left (e + f x\right )^{\frac{3}{2}} \left (2 a^{3} d^{3} - 6 a^{2} b c d^{2} + 6 a b^{2} c^{2} d - 2 b^{3} c^{3}\right )}{3 d^{4}} + \frac{\sqrt{e + f x} \left (- 2 a^{3} c d^{3} f + 2 a^{3} d^{4} e + 6 a^{2} b c^{2} d^{2} f - 6 a^{2} b c d^{3} e - 6 a b^{2} c^{3} d f + 6 a b^{2} c^{2} d^{2} e + 2 b^{3} c^{4} f - 2 b^{3} c^{3} d e\right )}{d^{5}} + \frac{2 \left (a d - b c\right )^{3} \left (c f - d e\right )^{2} \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{\sqrt{e + f x}}{\sqrt{\frac{c f - d e}{d}}} \right )}}{d \sqrt{\frac{c f - d e}{d}}} & \text{for}\: \frac{c f - d e}{d} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{\sqrt{e + f x}}{\sqrt{\frac{- c f + d e}{d}}} \right )}}{d \sqrt{\frac{- c f + d e}{d}}} & \text{for}\: e + f x > \frac{- c f + d e}{d} \wedge \frac{c f - d e}{d} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{\sqrt{e + f x}}{\sqrt{\frac{- c f + d e}{d}}} \right )}}{d \sqrt{\frac{- c f + d e}{d}}} & \text{for}\: \frac{c f - d e}{d} < 0 \wedge e + f x < \frac{- c f + d e}{d} \end{cases}\right )}{d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b**3*(e + f*x)**(9/2)/(9*d*f**3) + (e + f*x)**(7/2)*(6*a*b**2*d*f - 2*b**3*c*f
- 4*b**3*d*e)/(7*d**2*f**3) + (e + f*x)**(5/2)*(6*a**2*b*d**2*f**2 - 6*a*b**2*c
*d*f**2 - 6*a*b**2*d**2*e*f + 2*b**3*c**2*f**2 + 2*b**3*c*d*e*f + 2*b**3*d**2*e*
*2)/(5*d**3*f**3) + (e + f*x)**(3/2)*(2*a**3*d**3 - 6*a**2*b*c*d**2 + 6*a*b**2*c
**2*d - 2*b**3*c**3)/(3*d**4) + sqrt(e + f*x)*(-2*a**3*c*d**3*f + 2*a**3*d**4*e
+ 6*a**2*b*c**2*d**2*f - 6*a**2*b*c*d**3*e - 6*a*b**2*c**3*d*f + 6*a*b**2*c**2*d
**2*e + 2*b**3*c**4*f - 2*b**3*c**3*d*e)/d**5 + 2*(a*d - b*c)**3*(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*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)/s
qrt((-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**5

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.234999, size = 922, normalized size = 3.78 \[ -\frac{2 \,{\left (b^{3} c^{5} f^{2} - 3 \, a b^{2} c^{4} d f^{2} + 3 \, a^{2} b c^{3} d^{2} f^{2} - a^{3} c^{2} d^{3} f^{2} - 2 \, b^{3} c^{4} d f e + 6 \, a b^{2} c^{3} d^{2} f e - 6 \, a^{2} b c^{2} d^{3} f e + 2 \, a^{3} c d^{4} f e + b^{3} c^{3} d^{2} e^{2} - 3 \, a b^{2} c^{2} d^{3} e^{2} + 3 \, a^{2} b c d^{4} e^{2} - a^{3} d^{5} e^{2}\right )} \arctan \left (\frac{\sqrt{f x + e} d}{\sqrt{c d f - d^{2} e}}\right )}{\sqrt{c d f - d^{2} e} d^{5}} + \frac{2 \,{\left (35 \,{\left (f x + e\right )}^{\frac{9}{2}} b^{3} d^{8} f^{24} - 45 \,{\left (f x + e\right )}^{\frac{7}{2}} b^{3} c d^{7} f^{25} + 135 \,{\left (f x + e\right )}^{\frac{7}{2}} a b^{2} d^{8} f^{25} + 63 \,{\left (f x + e\right )}^{\frac{5}{2}} b^{3} c^{2} d^{6} f^{26} - 189 \,{\left (f x + e\right )}^{\frac{5}{2}} a b^{2} c d^{7} f^{26} + 189 \,{\left (f x + e\right )}^{\frac{5}{2}} a^{2} b d^{8} f^{26} - 105 \,{\left (f x + e\right )}^{\frac{3}{2}} b^{3} c^{3} d^{5} f^{27} + 315 \,{\left (f x + e\right )}^{\frac{3}{2}} a b^{2} c^{2} d^{6} f^{27} - 315 \,{\left (f x + e\right )}^{\frac{3}{2}} a^{2} b c d^{7} f^{27} + 105 \,{\left (f x + e\right )}^{\frac{3}{2}} a^{3} d^{8} f^{27} + 315 \, \sqrt{f x + e} b^{3} c^{4} d^{4} f^{28} - 945 \, \sqrt{f x + e} a b^{2} c^{3} d^{5} f^{28} + 945 \, \sqrt{f x + e} a^{2} b c^{2} d^{6} f^{28} - 315 \, \sqrt{f x + e} a^{3} c d^{7} f^{28} - 90 \,{\left (f x + e\right )}^{\frac{7}{2}} b^{3} d^{8} f^{24} e + 63 \,{\left (f x + e\right )}^{\frac{5}{2}} b^{3} c d^{7} f^{25} e - 189 \,{\left (f x + e\right )}^{\frac{5}{2}} a b^{2} d^{8} f^{25} e - 315 \, \sqrt{f x + e} b^{3} c^{3} d^{5} f^{27} e + 945 \, \sqrt{f x + e} a b^{2} c^{2} d^{6} f^{27} e - 945 \, \sqrt{f x + e} a^{2} b c d^{7} f^{27} e + 315 \, \sqrt{f x + e} a^{3} d^{8} f^{27} e + 63 \,{\left (f x + e\right )}^{\frac{5}{2}} b^{3} d^{8} f^{24} e^{2}\right )}}{315 \, d^{9} f^{27}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(b^3*c^5*f^2 - 3*a*b^2*c^4*d*f^2 + 3*a^2*b*c^3*d^2*f^2 - a^3*c^2*d^3*f^2 - 2*
b^3*c^4*d*f*e + 6*a*b^2*c^3*d^2*f*e - 6*a^2*b*c^2*d^3*f*e + 2*a^3*c*d^4*f*e + b^
3*c^3*d^2*e^2 - 3*a*b^2*c^2*d^3*e^2 + 3*a^2*b*c*d^4*e^2 - a^3*d^5*e^2)*arctan(sq
rt(f*x + e)*d/sqrt(c*d*f - d^2*e))/(sqrt(c*d*f - d^2*e)*d^5) + 2/315*(35*(f*x +
e)^(9/2)*b^3*d^8*f^24 - 45*(f*x + e)^(7/2)*b^3*c*d^7*f^25 + 135*(f*x + e)^(7/2)*
a*b^2*d^8*f^25 + 63*(f*x + e)^(5/2)*b^3*c^2*d^6*f^26 - 189*(f*x + e)^(5/2)*a*b^2
*c*d^7*f^26 + 189*(f*x + e)^(5/2)*a^2*b*d^8*f^26 - 105*(f*x + e)^(3/2)*b^3*c^3*d
^5*f^27 + 315*(f*x + e)^(3/2)*a*b^2*c^2*d^6*f^27 - 315*(f*x + e)^(3/2)*a^2*b*c*d
^7*f^27 + 105*(f*x + e)^(3/2)*a^3*d^8*f^27 + 315*sqrt(f*x + e)*b^3*c^4*d^4*f^28
- 945*sqrt(f*x + e)*a*b^2*c^3*d^5*f^28 + 945*sqrt(f*x + e)*a^2*b*c^2*d^6*f^28 -
315*sqrt(f*x + e)*a^3*c*d^7*f^28 - 90*(f*x + e)^(7/2)*b^3*d^8*f^24*e + 63*(f*x +
e)^(5/2)*b^3*c*d^7*f^25*e - 189*(f*x + e)^(5/2)*a*b^2*d^8*f^25*e - 315*sqrt(f*x
+ e)*b^3*c^3*d^5*f^27*e + 945*sqrt(f*x + e)*a*b^2*c^2*d^6*f^27*e - 945*sqrt(f*x
+ e)*a^2*b*c*d^7*f^27*e + 315*sqrt(f*x + e)*a^3*d^8*f^27*e + 63*(f*x + e)^(5/2)
*b^3*d^8*f^24*e^2)/(d^9*f^27)