3.1755 \(\int \frac{a+b x}{(c+d x) (e+f x)^{7/2}} \, dx\)

Optimal. Leaf size=151 \[ \frac{2 d^{3/2} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{(d e-c f)^{7/2}}-\frac{2 d (b c-a d)}{\sqrt{e+f x} (d e-c f)^3}-\frac{2 (b c-a d)}{3 (e+f x)^{3/2} (d e-c f)^2}-\frac{2 (b e-a f)}{5 f (e+f x)^{5/2} (d e-c f)} \]

[Out]

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

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Rubi [A]  time = 0.328917, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{2 d^{3/2} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{(d e-c f)^{7/2}}-\frac{2 d (b c-a d)}{\sqrt{e+f x} (d e-c f)^3}-\frac{2 (b c-a d)}{3 (e+f x)^{3/2} (d e-c f)^2}-\frac{2 (b e-a f)}{5 f (e+f x)^{5/2} (d e-c f)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 30.3367, size = 133, normalized size = 0.88 \[ - \frac{2 d^{\frac{3}{2}} \left (a d - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{d} \sqrt{e + f x}}{\sqrt{c f - d e}} \right )}}{\left (c f - d e\right )^{\frac{7}{2}}} - \frac{2 d \left (a d - b c\right )}{\sqrt{e + f x} \left (c f - d e\right )^{3}} + \frac{2 \left (a d - b c\right )}{3 \left (e + f x\right )^{\frac{3}{2}} \left (c f - d e\right )^{2}} - \frac{2 \left (a f - b e\right )}{5 f \left (e + f x\right )^{\frac{5}{2}} \left (c f - d e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.435367, size = 151, normalized size = 1. \[ -\frac{2 d^{3/2} (a d-b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{(d e-c f)^{7/2}}+\frac{2 d (a d-b c)}{\sqrt{e+f x} (d e-c f)^3}-\frac{2 (b c-a d)}{3 (e+f x)^{3/2} (d e-c f)^2}-\frac{2 (a f-b e)}{5 f (e+f x)^{5/2} (c f-d e)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.022, size = 234, normalized size = 1.6 \[ -{\frac{2\,a}{5\,cf-5\,de} \left ( fx+e \right ) ^{-{\frac{5}{2}}}}+{\frac{2\,be}{5\, \left ( cf-de \right ) f} \left ( fx+e \right ) ^{-{\frac{5}{2}}}}-2\,{\frac{{d}^{2}a}{ \left ( cf-de \right ) ^{3}\sqrt{fx+e}}}+2\,{\frac{bdc}{ \left ( cf-de \right ) ^{3}\sqrt{fx+e}}}+{\frac{2\,ad}{3\, \left ( cf-de \right ) ^{2}} \left ( fx+e \right ) ^{-{\frac{3}{2}}}}-{\frac{2\,bc}{3\, \left ( cf-de \right ) ^{2}} \left ( fx+e \right ) ^{-{\frac{3}{2}}}}-2\,{\frac{{d}^{3}a}{ \left ( cf-de \right ) ^{3}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }+2\,{\frac{b{d}^{2}c}{ \left ( cf-de \right ) ^{3}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/5/(c*f-d*e)/(f*x+e)^(5/2)*a+2/5/f/(c*f-d*e)/(f*x+e)^(5/2)*b*e-2/(c*f-d*e)^3*d
^2/(f*x+e)^(1/2)*a+2/(c*f-d*e)^3*d/(f*x+e)^(1/2)*b*c+2/3/(c*f-d*e)^2/(f*x+e)^(3/
2)*a*d-2/3/(c*f-d*e)^2/(f*x+e)^(3/2)*b*c-2*d^3/(c*f-d*e)^3/((c*f-d*e)*d)^(1/2)*a
rctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)^(1/2))*a+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*c

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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)/((d*x + c)*(f*x + e)^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.233718, size = 1, normalized size = 0.01 \[ \left [-\frac{6 \, b d^{2} e^{3} - 6 \, a c^{2} f^{3} + 30 \,{\left (b c d - a d^{2}\right )} f^{3} x^{2} + 2 \,{\left (14 \, b c d - 23 \, a d^{2}\right )} e^{2} f - 2 \,{\left (2 \, b c^{2} - 11 \, a c d\right )} e f^{2} - 15 \,{\left ({\left (b c d - a d^{2}\right )} f^{3} x^{2} + 2 \,{\left (b c d - a d^{2}\right )} e f^{2} x +{\left (b c d - a d^{2}\right )} e^{2} f\right )} \sqrt{f x + e} \sqrt{\frac{d}{d e - c f}} \log \left (\frac{d f x + 2 \, d e - c f + 2 \,{\left (d e - c f\right )} \sqrt{f x + e} \sqrt{\frac{d}{d e - c f}}}{d x + c}\right ) + 10 \,{\left (7 \,{\left (b c d - a d^{2}\right )} e f^{2} -{\left (b c^{2} - a c d\right )} f^{3}\right )} x}{15 \,{\left (d^{3} e^{5} f - 3 \, c d^{2} e^{4} f^{2} + 3 \, c^{2} d e^{3} f^{3} - c^{3} e^{2} f^{4} +{\left (d^{3} e^{3} f^{3} - 3 \, c d^{2} e^{2} f^{4} + 3 \, c^{2} d e f^{5} - c^{3} f^{6}\right )} x^{2} + 2 \,{\left (d^{3} e^{4} f^{2} - 3 \, c d^{2} e^{3} f^{3} + 3 \, c^{2} d e^{2} f^{4} - c^{3} e f^{5}\right )} x\right )} \sqrt{f x + e}}, -\frac{2 \,{\left (3 \, b d^{2} e^{3} - 3 \, a c^{2} f^{3} + 15 \,{\left (b c d - a d^{2}\right )} f^{3} x^{2} +{\left (14 \, b c d - 23 \, a d^{2}\right )} e^{2} f -{\left (2 \, b c^{2} - 11 \, a c d\right )} e f^{2} - 15 \,{\left ({\left (b c d - a d^{2}\right )} f^{3} x^{2} + 2 \,{\left (b c d - a d^{2}\right )} e f^{2} x +{\left (b c d - a d^{2}\right )} e^{2} f\right )} \sqrt{f x + e} \sqrt{-\frac{d}{d e - c f}} \arctan \left (-\frac{{\left (d e - c f\right )} \sqrt{-\frac{d}{d e - c f}}}{\sqrt{f x + e} d}\right ) + 5 \,{\left (7 \,{\left (b c d - a d^{2}\right )} e f^{2} -{\left (b c^{2} - a c d\right )} f^{3}\right )} x\right )}}{15 \,{\left (d^{3} e^{5} f - 3 \, c d^{2} e^{4} f^{2} + 3 \, c^{2} d e^{3} f^{3} - c^{3} e^{2} f^{4} +{\left (d^{3} e^{3} f^{3} - 3 \, c d^{2} e^{2} f^{4} + 3 \, c^{2} d e f^{5} - c^{3} f^{6}\right )} x^{2} + 2 \,{\left (d^{3} e^{4} f^{2} - 3 \, c d^{2} e^{3} f^{3} + 3 \, c^{2} d e^{2} f^{4} - c^{3} e f^{5}\right )} x\right )} \sqrt{f x + e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)/((d*x + c)*(f*x + e)^(7/2)),x, algorithm="fricas")

[Out]

[-1/15*(6*b*d^2*e^3 - 6*a*c^2*f^3 + 30*(b*c*d - a*d^2)*f^3*x^2 + 2*(14*b*c*d - 2
3*a*d^2)*e^2*f - 2*(2*b*c^2 - 11*a*c*d)*e*f^2 - 15*((b*c*d - a*d^2)*f^3*x^2 + 2*
(b*c*d - a*d^2)*e*f^2*x + (b*c*d - a*d^2)*e^2*f)*sqrt(f*x + e)*sqrt(d/(d*e - c*f
))*log((d*f*x + 2*d*e - c*f + 2*(d*e - c*f)*sqrt(f*x + e)*sqrt(d/(d*e - c*f)))/(
d*x + c)) + 10*(7*(b*c*d - a*d^2)*e*f^2 - (b*c^2 - a*c*d)*f^3)*x)/((d^3*e^5*f -
3*c*d^2*e^4*f^2 + 3*c^2*d*e^3*f^3 - c^3*e^2*f^4 + (d^3*e^3*f^3 - 3*c*d^2*e^2*f^4
+ 3*c^2*d*e*f^5 - c^3*f^6)*x^2 + 2*(d^3*e^4*f^2 - 3*c*d^2*e^3*f^3 + 3*c^2*d*e^2
*f^4 - c^3*e*f^5)*x)*sqrt(f*x + e)), -2/15*(3*b*d^2*e^3 - 3*a*c^2*f^3 + 15*(b*c*
d - a*d^2)*f^3*x^2 + (14*b*c*d - 23*a*d^2)*e^2*f - (2*b*c^2 - 11*a*c*d)*e*f^2 -
15*((b*c*d - a*d^2)*f^3*x^2 + 2*(b*c*d - a*d^2)*e*f^2*x + (b*c*d - a*d^2)*e^2*f)
*sqrt(f*x + e)*sqrt(-d/(d*e - c*f))*arctan(-(d*e - c*f)*sqrt(-d/(d*e - c*f))/(sq
rt(f*x + e)*d)) + 5*(7*(b*c*d - a*d^2)*e*f^2 - (b*c^2 - a*c*d)*f^3)*x)/((d^3*e^5
*f - 3*c*d^2*e^4*f^2 + 3*c^2*d*e^3*f^3 - c^3*e^2*f^4 + (d^3*e^3*f^3 - 3*c*d^2*e^
2*f^4 + 3*c^2*d*e*f^5 - c^3*f^6)*x^2 + 2*(d^3*e^4*f^2 - 3*c*d^2*e^3*f^3 + 3*c^2*
d*e^2*f^4 - c^3*e*f^5)*x)*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)/(d*x+c)/(f*x+e)**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.22161, size = 385, normalized size = 2.55 \[ \frac{2 \,{\left (b c d^{2} - a d^{3}\right )} \arctan \left (\frac{\sqrt{f x + e} d}{\sqrt{c d f - d^{2} e}}\right )}{{\left (c^{3} f^{3} - 3 \, c^{2} d f^{2} e + 3 \, c d^{2} f e^{2} - d^{3} e^{3}\right )} \sqrt{c d f - d^{2} e}} + \frac{2 \,{\left (15 \,{\left (f x + e\right )}^{2} b c d f - 15 \,{\left (f x + e\right )}^{2} a d^{2} f - 5 \,{\left (f x + e\right )} b c^{2} f^{2} + 5 \,{\left (f x + e\right )} a c d f^{2} - 3 \, a c^{2} f^{3} + 5 \,{\left (f x + e\right )} b c d f e - 5 \,{\left (f x + e\right )} a d^{2} f e + 3 \, b c^{2} f^{2} e + 6 \, a c d f^{2} e - 6 \, b c d f e^{2} - 3 \, a d^{2} f e^{2} + 3 \, b d^{2} e^{3}\right )}}{15 \,{\left (c^{3} f^{4} - 3 \, c^{2} d f^{3} e + 3 \, c d^{2} f^{2} e^{2} - d^{3} f e^{3}\right )}{\left (f x + e\right )}^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)/((d*x + c)*(f*x + e)^(7/2)),x, algorithm="giac")

[Out]

2*(b*c*d^2 - a*d^3)*arctan(sqrt(f*x + e)*d/sqrt(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*(15*(f*x + e)^2
*b*c*d*f - 15*(f*x + e)^2*a*d^2*f - 5*(f*x + e)*b*c^2*f^2 + 5*(f*x + e)*a*c*d*f^
2 - 3*a*c^2*f^3 + 5*(f*x + e)*b*c*d*f*e - 5*(f*x + e)*a*d^2*f*e + 3*b*c^2*f^2*e
+ 6*a*c*d*f^2*e - 6*b*c*d*f*e^2 - 3*a*d^2*f*e^2 + 3*b*d^2*e^3)/((c^3*f^4 - 3*c^2
*d*f^3*e + 3*c*d^2*f^2*e^2 - d^3*f*e^3)*(f*x + e)^(5/2))