3.1771 \(\int \frac{(a+b x)^3}{(c+d x) (e+f x)^{7/2}} \, dx\)

Optimal. Leaf size=227 \[ -\frac{2 (b e-a f) \left (a^2 d^2 f^2+a b d f (d e-3 c f)+b^2 \left (3 c^2 f^2-3 c d e f+d^2 e^2\right )\right )}{f^3 \sqrt{e+f x} (d e-c f)^3}+\frac{2 (b e-a f)^2 (a d f-3 b c f+2 b d e)}{3 f^3 (e+f x)^{3/2} (d e-c f)^2}-\frac{2 (b e-a f)^3}{5 f^3 (e+f x)^{5/2} (d e-c f)}+\frac{2 (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{\sqrt{d} (d e-c f)^{7/2}} \]

[Out]

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

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Rubi [A]  time = 0.694216, antiderivative size = 227, 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 \[ -\frac{2 (b e-a f) \left (a^2 d^2 f^2+a b d f (d e-3 c f)+b^2 \left (3 c^2 f^2-3 c d e f+d^2 e^2\right )\right )}{f^3 \sqrt{e+f x} (d e-c f)^3}+\frac{2 (b e-a f)^2 (a d f-3 b c f+2 b d e)}{3 f^3 (e+f x)^{3/2} (d e-c f)^2}-\frac{2 (b e-a f)^3}{5 f^3 (e+f x)^{5/2} (d e-c f)}+\frac{2 (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{\sqrt{d} (d e-c f)^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^3/((c + d*x)*(e + f*x)^(7/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 159.692, size = 226, normalized size = 1. \[ - \frac{2 \left (a f - b e\right ) \left (a^{2} d^{2} f^{2} - 3 a b c d f^{2} + a b d^{2} e f + 3 b^{2} c^{2} f^{2} - 3 b^{2} c d e f + b^{2} d^{2} e^{2}\right )}{f^{3} \sqrt{e + f x} \left (c f - d e\right )^{3}} + \frac{2 \left (a f - b e\right )^{2} \left (a d f - 3 b c f + 2 b d e\right )}{3 f^{3} \left (e + f x\right )^{\frac{3}{2}} \left (c f - d e\right )^{2}} - \frac{2 \left (a f - b e\right )^{3}}{5 f^{3} \left (e + f x\right )^{\frac{5}{2}} \left (c f - d e\right )} - \frac{2 \left (a d - b c\right )^{3} \operatorname{atan}{\left (\frac{\sqrt{d} \sqrt{e + f x}}{\sqrt{c f - d e}} \right )}}{\sqrt{d} \left (c f - d e\right )^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**3/(d*x+c)/(f*x+e)**(7/2),x)

[Out]

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

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Mathematica [A]  time = 0.935301, size = 212, normalized size = 0.93 \[ \frac{2 (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{\sqrt{d} (d e-c f)^{7/2}}-\frac{2 (b e-a f) \left (15 (e+f x)^2 \left (a^2 d^2 f^2+a b d f (d e-3 c f)+b^2 \left (3 c^2 f^2-3 c d e f+d^2 e^2\right )\right )-5 (e+f x) (b e-a f) (d e-c f) (a d f-3 b c f+2 b d e)+3 (b e-a f)^2 (d e-c f)^2\right )}{15 f^3 (e+f x)^{5/2} (d e-c f)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^3/((c + d*x)*(e + f*x)^(7/2)),x]

[Out]

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

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Maple [B]  time = 0.028, size = 649, normalized size = 2.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3/(d*x+c)/(f*x+e)^(7/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.25766, 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/((d*x + c)*(f*x + e)^(7/2)),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^5*x^2 + 2*(b^3*
c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*e*f^4*x + (b^3*c^3 - 3*a*b^2*c^2*
d + 3*a^2*b*c*d^2 - a^3*d^3)*e^2*f^3)*sqrt(f*x + e)*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*(8*b^3*d^2*
e^5 - 3*a^3*c^2*f^5 - 2*(13*b^3*c*d - 3*a*b^2*d^2)*e^4*f + 3*(11*b^3*c^2 - 9*a*b
^2*c*d + 3*a^2*b*d^2)*e^3*f^2 - (24*a*b^2*c^2 - 42*a^2*b*c*d + 23*a^3*d^2)*e^2*f
^3 - (6*a^2*b*c^2 - 11*a^3*c*d)*e*f^4 + 15*(b^3*d^2*e^3*f^2 - 3*b^3*c*d*e^2*f^3
+ 3*b^3*c^2*e*f^4 - (3*a*b^2*c^2 - 3*a^2*b*c*d + a^3*d^2)*f^5)*x^2 + 5*(4*b^3*d^
2*e^4*f - (13*b^3*c*d - 3*a*b^2*d^2)*e^3*f^2 + 3*(5*b^3*c^2 - 3*a*b^2*c*d)*e^2*f
^3 - (12*a*b^2*c^2 - 21*a^2*b*c*d + 7*a^3*d^2)*e*f^4 - (3*a^2*b*c^2 - a^3*c*d)*f
^5)*x)*sqrt(d^2*e - c*d*f))/((d^3*e^5*f^3 - 3*c*d^2*e^4*f^4 + 3*c^2*d*e^3*f^5 -
c^3*e^2*f^6 + (d^3*e^3*f^5 - 3*c*d^2*e^2*f^6 + 3*c^2*d*e*f^7 - c^3*f^8)*x^2 + 2*
(d^3*e^4*f^4 - 3*c*d^2*e^3*f^5 + 3*c^2*d*e^2*f^6 - c^3*e*f^7)*x)*sqrt(d^2*e - c*
d*f)*sqrt(f*x + e)), 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^5*x^2 + 2*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*e*f^4*x + (b^
3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*e^2*f^3)*sqrt(f*x + e)*arctan(-
(d*e - c*f)/(sqrt(-d^2*e + c*d*f)*sqrt(f*x + e))) - (8*b^3*d^2*e^5 - 3*a^3*c^2*f
^5 - 2*(13*b^3*c*d - 3*a*b^2*d^2)*e^4*f + 3*(11*b^3*c^2 - 9*a*b^2*c*d + 3*a^2*b*
d^2)*e^3*f^2 - (24*a*b^2*c^2 - 42*a^2*b*c*d + 23*a^3*d^2)*e^2*f^3 - (6*a^2*b*c^2
- 11*a^3*c*d)*e*f^4 + 15*(b^3*d^2*e^3*f^2 - 3*b^3*c*d*e^2*f^3 + 3*b^3*c^2*e*f^4
- (3*a*b^2*c^2 - 3*a^2*b*c*d + a^3*d^2)*f^5)*x^2 + 5*(4*b^3*d^2*e^4*f - (13*b^3
*c*d - 3*a*b^2*d^2)*e^3*f^2 + 3*(5*b^3*c^2 - 3*a*b^2*c*d)*e^2*f^3 - (12*a*b^2*c^
2 - 21*a^2*b*c*d + 7*a^3*d^2)*e*f^4 - (3*a^2*b*c^2 - a^3*c*d)*f^5)*x)*sqrt(-d^2*
e + c*d*f))/((d^3*e^5*f^3 - 3*c*d^2*e^4*f^4 + 3*c^2*d*e^3*f^5 - c^3*e^2*f^6 + (d
^3*e^3*f^5 - 3*c*d^2*e^2*f^6 + 3*c^2*d*e*f^7 - c^3*f^8)*x^2 + 2*(d^3*e^4*f^4 - 3
*c*d^2*e^3*f^5 + 3*c^2*d*e^2*f^6 - c^3*e*f^7)*x)*sqrt(-d^2*e + c*d*f)*sqrt(f*x +
e))]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**3/(d*x+c)/(f*x+e)**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.232642, size = 826, normalized size = 3.64 \[ \frac{2 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{f x + e} d}{\sqrt{c d f - d^{2} e}}\right )}{{\left (c^{3} f^{3} - 3 \, c^{2} d f^{2} e + 3 \, c d^{2} f e^{2} - d^{3} e^{3}\right )} \sqrt{c d f - d^{2} e}} - \frac{2 \,{\left (45 \,{\left (f x + e\right )}^{2} a b^{2} c^{2} f^{3} - 45 \,{\left (f x + e\right )}^{2} a^{2} b c d f^{3} + 15 \,{\left (f x + e\right )}^{2} a^{3} d^{2} f^{3} + 15 \,{\left (f x + e\right )} a^{2} b c^{2} f^{4} - 5 \,{\left (f x + e\right )} a^{3} c d f^{4} + 3 \, a^{3} c^{2} f^{5} - 45 \,{\left (f x + e\right )}^{2} b^{3} c^{2} f^{2} e - 30 \,{\left (f x + e\right )} a b^{2} c^{2} f^{3} e - 15 \,{\left (f x + e\right )} a^{2} b c d f^{3} e + 5 \,{\left (f x + e\right )} a^{3} d^{2} f^{3} e - 9 \, a^{2} b c^{2} f^{4} e - 6 \, a^{3} c d f^{4} e + 45 \,{\left (f x + e\right )}^{2} b^{3} c d f e^{2} + 15 \,{\left (f x + e\right )} b^{3} c^{2} f^{2} e^{2} + 45 \,{\left (f x + e\right )} a b^{2} c d f^{2} e^{2} + 9 \, a b^{2} c^{2} f^{3} e^{2} + 18 \, a^{2} b c d f^{3} e^{2} + 3 \, a^{3} d^{2} f^{3} e^{2} - 15 \,{\left (f x + e\right )}^{2} b^{3} d^{2} e^{3} - 25 \,{\left (f x + e\right )} b^{3} c d f e^{3} - 15 \,{\left (f x + e\right )} a b^{2} d^{2} f e^{3} - 3 \, b^{3} c^{2} f^{2} e^{3} - 18 \, a b^{2} c d f^{2} e^{3} - 9 \, a^{2} b d^{2} f^{2} e^{3} + 10 \,{\left (f x + e\right )} b^{3} d^{2} e^{4} + 6 \, b^{3} c d f e^{4} + 9 \, a b^{2} d^{2} f e^{4} - 3 \, b^{3} d^{2} e^{5}\right )}}{15 \,{\left (c^{3} f^{6} - 3 \, c^{2} d f^{5} e + 3 \, c d^{2} f^{4} e^{2} - d^{3} f^{3} e^{3}\right )}{\left (f x + e\right )}^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3/((d*x + c)*(f*x + e)^(7/2)),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/sqr
t(c*d*f - d^2*e))/((c^3*f^3 - 3*c^2*d*f^2*e + 3*c*d^2*f*e^2 - d^3*e^3)*sqrt(c*d*
f - d^2*e)) - 2/15*(45*(f*x + e)^2*a*b^2*c^2*f^3 - 45*(f*x + e)^2*a^2*b*c*d*f^3
+ 15*(f*x + e)^2*a^3*d^2*f^3 + 15*(f*x + e)*a^2*b*c^2*f^4 - 5*(f*x + e)*a^3*c*d*
f^4 + 3*a^3*c^2*f^5 - 45*(f*x + e)^2*b^3*c^2*f^2*e - 30*(f*x + e)*a*b^2*c^2*f^3*
e - 15*(f*x + e)*a^2*b*c*d*f^3*e + 5*(f*x + e)*a^3*d^2*f^3*e - 9*a^2*b*c^2*f^4*e
- 6*a^3*c*d*f^4*e + 45*(f*x + e)^2*b^3*c*d*f*e^2 + 15*(f*x + e)*b^3*c^2*f^2*e^2
+ 45*(f*x + e)*a*b^2*c*d*f^2*e^2 + 9*a*b^2*c^2*f^3*e^2 + 18*a^2*b*c*d*f^3*e^2 +
3*a^3*d^2*f^3*e^2 - 15*(f*x + e)^2*b^3*d^2*e^3 - 25*(f*x + e)*b^3*c*d*f*e^3 - 1
5*(f*x + e)*a*b^2*d^2*f*e^3 - 3*b^3*c^2*f^2*e^3 - 18*a*b^2*c*d*f^2*e^3 - 9*a^2*b
*d^2*f^2*e^3 + 10*(f*x + e)*b^3*d^2*e^4 + 6*b^3*c*d*f*e^4 + 9*a*b^2*d^2*f*e^4 -
3*b^3*d^2*e^5)/((c^3*f^6 - 3*c^2*d*f^5*e + 3*c*d^2*f^4*e^2 - d^3*f^3*e^3)*(f*x +
e)^(5/2))