3.1758 \(\int \frac{(a+b x)^2 (e+f x)^{3/2}}{c+d x} \, dx\)

Optimal. Leaf size=172 \[ -\frac{2 (b c-a d)^2 (d e-c f)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{9/2}}+\frac{2 \sqrt{e+f x} (b c-a d)^2 (d e-c f)}{d^4}+\frac{2 (e+f x)^{3/2} (b c-a d)^2}{3 d^3}-\frac{2 b (e+f x)^{5/2} (-2 a d f+b c f+b d e)}{5 d^2 f^2}+\frac{2 b^2 (e+f x)^{7/2}}{7 d f^2} \]

[Out]

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

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Rubi [A]  time = 0.333745, antiderivative size = 172, 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 \[ -\frac{2 (b c-a d)^2 (d e-c f)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{9/2}}+\frac{2 \sqrt{e+f x} (b c-a d)^2 (d e-c f)}{d^4}+\frac{2 (e+f x)^{3/2} (b c-a d)^2}{3 d^3}-\frac{2 b (e+f x)^{5/2} (-2 a d f+b c f+b d e)}{5 d^2 f^2}+\frac{2 b^2 (e+f x)^{7/2}}{7 d f^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 45.6134, size = 160, normalized size = 0.93 \[ \frac{2 b^{2} \left (e + f x\right )^{\frac{7}{2}}}{7 d f^{2}} + \frac{2 b \left (e + f x\right )^{\frac{5}{2}} \left (2 a d f - b c f - b d e\right )}{5 d^{2} f^{2}} + \frac{2 \left (e + f x\right )^{\frac{3}{2}} \left (a d - b c\right )^{2}}{3 d^{3}} - \frac{2 \sqrt{e + f x} \left (a d - b c\right )^{2} \left (c f - d e\right )}{d^{4}} + \frac{2 \left (a d - b c\right )^{2} \left (c f - d e\right )^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{d} \sqrt{e + f x}}{\sqrt{c f - d e}} \right )}}{d^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.51647, size = 204, normalized size = 1.19 \[ \frac{2 \sqrt{e+f x} \left (35 a^2 d^2 f^2 (-3 c f+4 d e+d f x)+14 a b d f \left (15 c^2 f^2-5 c d f (4 e+f x)+3 d^2 (e+f x)^2\right )+b^2 \left (-105 c^3 f^3+35 c^2 d f^2 (4 e+f x)-21 c d^2 f (e+f x)^2-3 d^3 (2 e-5 f x) (e+f x)^2\right )\right )}{105 d^4 f^2}-\frac{2 (b c-a d)^2 (d e-c f)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[e + f*x]*(35*a^2*d^2*f^2*(4*d*e - 3*c*f + d*f*x) + 14*a*b*d*f*(15*c^2*f^
2 + 3*d^2*(e + f*x)^2 - 5*c*d*f*(4*e + f*x)) + b^2*(-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))))/(105*d
^4*f^2) - (2*(b*c - a*d)^2*(d*e - c*f)^(3/2)*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqr
t[d*e - c*f]])/d^(9/2)

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Maple [B]  time = 0.02, size = 644, normalized size = 3.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/7*b^2*(f*x+e)^(7/2)/d/f^2+4/5/f/d*(f*x+e)^(5/2)*a*b-2/5/f/d^2*(f*x+e)^(5/2)*b^
2*c-2/5/f^2/d*(f*x+e)^(5/2)*b^2*e+2/3/d*(f*x+e)^(3/2)*a^2-4/3/d^2*(f*x+e)^(3/2)*
a*b*c+2/3/d^3*(f*x+e)^(3/2)*b^2*c^2-2*f/d^2*a^2*c*(f*x+e)^(1/2)+2/d*a^2*e*(f*x+e
)^(1/2)+4*f/d^3*a*b*c^2*(f*x+e)^(1/2)-4/d^2*a*b*c*e*(f*x+e)^(1/2)-2*f/d^4*b^2*c^
3*(f*x+e)^(1/2)+2/d^3*b^2*c^2*e*(f*x+e)^(1/2)+2*f^2/d^2/((c*f-d*e)*d)^(1/2)*arct
an((f*x+e)^(1/2)*d/((c*f-d*e)*d)^(1/2))*a^2*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^2*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^2*e^2-4*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*b*c^3+8*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*b*c^2*e-4/d/((c*f-d*e)*d)^(1/2)*arctan((f*x+
e)^(1/2)*d/((c*f-d*e)*d)^(1/2))*a*b*c*e^2+2*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))*b^2*c^4-4*f/d^3/((c*f-d*e)*d)^(1/2)*arctan((
f*x+e)^(1/2)*d/((c*f-d*e)*d)^(1/2))*b^2*c^3*e+2/d^2/((c*f-d*e)*d)^(1/2)*arctan((
f*x+e)^(1/2)*d/((c*f-d*e)*d)^(1/2))*b^2*c^2*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)^2*(f*x + e)^(3/2)/(d*x + c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.228018, size = 1, normalized size = 0.01 \[ \left [-\frac{105 \,{\left ({\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} e f^{2} -{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} f^{3}\right )} \sqrt{\frac{d e - c f}{d}} \log \left (\frac{d f x + 2 \, d e - c f + 2 \, \sqrt{f x + e} d \sqrt{\frac{d e - c f}{d}}}{d x + c}\right ) - 2 \,{\left (15 \, b^{2} d^{3} f^{3} x^{3} - 6 \, b^{2} d^{3} e^{3} - 21 \,{\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )} e^{2} f + 140 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} e f^{2} - 105 \,{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} f^{3} + 3 \,{\left (8 \, b^{2} d^{3} e f^{2} - 7 \,{\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )} f^{3}\right )} x^{2} +{\left (3 \, b^{2} d^{3} e^{2} f - 42 \,{\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )} e f^{2} + 35 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} f^{3}\right )} x\right )} \sqrt{f x + e}}{105 \, d^{4} f^{2}}, -\frac{2 \,{\left (105 \,{\left ({\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} e f^{2} -{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} f^{3}\right )} \sqrt{-\frac{d e - c f}{d}} \arctan \left (\frac{\sqrt{f x + e}}{\sqrt{-\frac{d e - c f}{d}}}\right ) -{\left (15 \, b^{2} d^{3} f^{3} x^{3} - 6 \, b^{2} d^{3} e^{3} - 21 \,{\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )} e^{2} f + 140 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} e f^{2} - 105 \,{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} f^{3} + 3 \,{\left (8 \, b^{2} d^{3} e f^{2} - 7 \,{\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )} f^{3}\right )} x^{2} +{\left (3 \, b^{2} d^{3} e^{2} f - 42 \,{\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )} e f^{2} + 35 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} f^{3}\right )} x\right )} \sqrt{f x + e}\right )}}{105 \, d^{4} f^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/105*(105*((b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*e*f^2 - (b^2*c^3 - 2*a*b*c^2*d
+ a^2*c*d^2)*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^2*d^3*f^3*x^3 - 6*b^2*d^3*e^3 - 21
*(b^2*c*d^2 - 2*a*b*d^3)*e^2*f + 140*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*e*f^2 -
105*(b^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2)*f^3 + 3*(8*b^2*d^3*e*f^2 - 7*(b^2*c*d^2
- 2*a*b*d^3)*f^3)*x^2 + (3*b^2*d^3*e^2*f - 42*(b^2*c*d^2 - 2*a*b*d^3)*e*f^2 + 3
5*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*f^3)*x)*sqrt(f*x + e))/(d^4*f^2), -2/105*(
105*((b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*e*f^2 - (b^2*c^3 - 2*a*b*c^2*d + a^2*c*
d^2)*f^3)*sqrt(-(d*e - c*f)/d)*arctan(sqrt(f*x + e)/sqrt(-(d*e - c*f)/d)) - (15*
b^2*d^3*f^3*x^3 - 6*b^2*d^3*e^3 - 21*(b^2*c*d^2 - 2*a*b*d^3)*e^2*f + 140*(b^2*c^
2*d - 2*a*b*c*d^2 + a^2*d^3)*e*f^2 - 105*(b^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2)*f^3
+ 3*(8*b^2*d^3*e*f^2 - 7*(b^2*c*d^2 - 2*a*b*d^3)*f^3)*x^2 + (3*b^2*d^3*e^2*f -
42*(b^2*c*d^2 - 2*a*b*d^3)*e*f^2 + 35*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*f^3)*x
)*sqrt(f*x + e))/(d^4*f^2)]

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Sympy [A]  time = 76.7458, size = 355, normalized size = 2.06 \[ \frac{2 b^{2} \left (e + f x\right )^{\frac{7}{2}}}{7 d f^{2}} + \frac{\left (e + f x\right )^{\frac{5}{2}} \left (4 a b d f - 2 b^{2} c f - 2 b^{2} d e\right )}{5 d^{2} f^{2}} + \frac{\left (e + f x\right )^{\frac{3}{2}} \left (2 a^{2} d^{2} - 4 a b c d + 2 b^{2} c^{2}\right )}{3 d^{3}} + \frac{\sqrt{e + f x} \left (- 2 a^{2} c d^{2} f + 2 a^{2} d^{3} e + 4 a b c^{2} d f - 4 a b c d^{2} e - 2 b^{2} c^{3} f + 2 b^{2} c^{2} d e\right )}{d^{4}} + \frac{2 \left (a d - b c\right )^{2} \left (c f - d e\right )^{2} \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{\sqrt{e + f x}}{\sqrt{\frac{c f - d e}{d}}} \right )}}{d \sqrt{\frac{c f - d e}{d}}} & \text{for}\: \frac{c f - d e}{d} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{\sqrt{e + f x}}{\sqrt{\frac{- c f + d e}{d}}} \right )}}{d \sqrt{\frac{- c f + d e}{d}}} & \text{for}\: e + f x > \frac{- c f + d e}{d} \wedge \frac{c f - d e}{d} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{\sqrt{e + f x}}{\sqrt{\frac{- c f + d e}{d}}} \right )}}{d \sqrt{\frac{- c f + d e}{d}}} & \text{for}\: \frac{c f - d e}{d} < 0 \wedge e + f x < \frac{- c f + d e}{d} \end{cases}\right )}{d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b**2*(e + f*x)**(7/2)/(7*d*f**2) + (e + f*x)**(5/2)*(4*a*b*d*f - 2*b**2*c*f -
2*b**2*d*e)/(5*d**2*f**2) + (e + f*x)**(3/2)*(2*a**2*d**2 - 4*a*b*c*d + 2*b**2*c
**2)/(3*d**3) + sqrt(e + f*x)*(-2*a**2*c*d**2*f + 2*a**2*d**3*e + 4*a*b*c**2*d*f
- 4*a*b*c*d**2*e - 2*b**2*c**3*f + 2*b**2*c**2*d*e)/d**4 + 2*(a*d - b*c)**2*(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(sqr
t(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 = 0.223798, size = 572, normalized size = 3.33 \[ \frac{2 \,{\left (b^{2} c^{4} f^{2} - 2 \, a b c^{3} d f^{2} + a^{2} c^{2} d^{2} f^{2} - 2 \, b^{2} c^{3} d f e + 4 \, a b c^{2} d^{2} f e - 2 \, a^{2} c d^{3} f e + b^{2} c^{2} d^{2} e^{2} - 2 \, a b c d^{3} e^{2} + a^{2} d^{4} e^{2}\right )} \arctan \left (\frac{\sqrt{f x + e} d}{\sqrt{c d f - d^{2} e}}\right )}{\sqrt{c d f - d^{2} e} d^{4}} + \frac{2 \,{\left (15 \,{\left (f x + e\right )}^{\frac{7}{2}} b^{2} d^{6} f^{12} - 21 \,{\left (f x + e\right )}^{\frac{5}{2}} b^{2} c d^{5} f^{13} + 42 \,{\left (f x + e\right )}^{\frac{5}{2}} a b d^{6} f^{13} + 35 \,{\left (f x + e\right )}^{\frac{3}{2}} b^{2} c^{2} d^{4} f^{14} - 70 \,{\left (f x + e\right )}^{\frac{3}{2}} a b c d^{5} f^{14} + 35 \,{\left (f x + e\right )}^{\frac{3}{2}} a^{2} d^{6} f^{14} - 105 \, \sqrt{f x + e} b^{2} c^{3} d^{3} f^{15} + 210 \, \sqrt{f x + e} a b c^{2} d^{4} f^{15} - 105 \, \sqrt{f x + e} a^{2} c d^{5} f^{15} - 21 \,{\left (f x + e\right )}^{\frac{5}{2}} b^{2} d^{6} f^{12} e + 105 \, \sqrt{f x + e} b^{2} c^{2} d^{4} f^{14} e - 210 \, \sqrt{f x + e} a b c d^{5} f^{14} e + 105 \, \sqrt{f x + e} a^{2} d^{6} f^{14} e\right )}}{105 \, d^{7} f^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(b^2*c^4*f^2 - 2*a*b*c^3*d*f^2 + a^2*c^2*d^2*f^2 - 2*b^2*c^3*d*f*e + 4*a*b*c^2
*d^2*f*e - 2*a^2*c*d^3*f*e + b^2*c^2*d^2*e^2 - 2*a*b*c*d^3*e^2 + a^2*d^4*e^2)*ar
ctan(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^2*d^6*f^12 - 21*(f*x + e)^(5/2)*b^2*c*d^5*f^13 + 42*(f*x + e)^
(5/2)*a*b*d^6*f^13 + 35*(f*x + e)^(3/2)*b^2*c^2*d^4*f^14 - 70*(f*x + e)^(3/2)*a*
b*c*d^5*f^14 + 35*(f*x + e)^(3/2)*a^2*d^6*f^14 - 105*sqrt(f*x + e)*b^2*c^3*d^3*f
^15 + 210*sqrt(f*x + e)*a*b*c^2*d^4*f^15 - 105*sqrt(f*x + e)*a^2*c*d^5*f^15 - 21
*(f*x + e)^(5/2)*b^2*d^6*f^12*e + 105*sqrt(f*x + e)*b^2*c^2*d^4*f^14*e - 210*sqr
t(f*x + e)*a*b*c*d^5*f^14*e + 105*sqrt(f*x + e)*a^2*d^6*f^14*e)/(d^7*f^14)