3.1756 \(\int \frac{a+b x}{(c+d x) (e+f x)^{9/2}} \, dx\)

Optimal. Leaf size=185 \[ \frac{2 d^{5/2} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{(d e-c f)^{9/2}}-\frac{2 d^2 (b c-a d)}{\sqrt{e+f x} (d e-c f)^4}-\frac{2 d (b c-a d)}{3 (e+f x)^{3/2} (d e-c f)^3}-\frac{2 (b c-a d)}{5 (e+f x)^{5/2} (d e-c f)^2}-\frac{2 (b e-a f)}{7 f (e+f x)^{7/2} (d e-c f)} \]

[Out]

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

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Rubi [A]  time = 0.400514, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{2 d^{5/2} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{(d e-c f)^{9/2}}-\frac{2 d^2 (b c-a d)}{\sqrt{e+f x} (d e-c f)^4}-\frac{2 d (b c-a d)}{3 (e+f x)^{3/2} (d e-c f)^3}-\frac{2 (b c-a d)}{5 (e+f x)^{5/2} (d e-c f)^2}-\frac{2 (b e-a f)}{7 f (e+f x)^{7/2} (d e-c f)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)/((c + d*x)*(e + f*x)^(9/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 40.5435, size = 163, normalized size = 0.88 \[ \frac{2 d^{\frac{5}{2}} \left (a d - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{d} \sqrt{e + f x}}{\sqrt{c f - d e}} \right )}}{\left (c f - d e\right )^{\frac{9}{2}}} + \frac{2 d^{2} \left (a d - b c\right )}{\sqrt{e + f x} \left (c f - d e\right )^{4}} - \frac{2 d \left (a d - b c\right )}{3 \left (e + f x\right )^{\frac{3}{2}} \left (c f - d e\right )^{3}} + \frac{2 \left (a d - b c\right )}{5 \left (e + f x\right )^{\frac{5}{2}} \left (c f - d e\right )^{2}} - \frac{2 \left (a f - b e\right )}{7 f \left (e + f x\right )^{\frac{7}{2}} \left (c f - d e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)/(d*x+c)/(f*x+e)**(9/2),x)

[Out]

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

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Mathematica [A]  time = 0.772054, size = 185, normalized size = 1. \[ -\frac{2 d^{5/2} (a d-b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{(d e-c f)^{9/2}}+\frac{2 d^2 (a d-b c)}{\sqrt{e+f x} (d e-c f)^4}+\frac{2 d (a d-b c)}{3 (e+f x)^{3/2} (d e-c f)^3}-\frac{2 (b c-a d)}{5 (e+f x)^{5/2} (d e-c f)^2}-\frac{2 (a f-b e)}{7 f (e+f x)^{7/2} (c f-d e)} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)/((c + d*x)*(e + f*x)^(9/2)),x]

[Out]

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

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Maple [A]  time = 0.024, size = 281, normalized size = 1.5 \[ -{\frac{2\,a}{7\,cf-7\,de} \left ( fx+e \right ) ^{-{\frac{7}{2}}}}+{\frac{2\,be}{7\, \left ( cf-de \right ) f} \left ( fx+e \right ) ^{-{\frac{7}{2}}}}-{\frac{2\,{d}^{2}a}{3\, \left ( cf-de \right ) ^{3}} \left ( fx+e \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,bdc}{3\, \left ( cf-de \right ) ^{3}} \left ( fx+e \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,ad}{5\, \left ( cf-de \right ) ^{2}} \left ( fx+e \right ) ^{-{\frac{5}{2}}}}-{\frac{2\,bc}{5\, \left ( cf-de \right ) ^{2}} \left ( fx+e \right ) ^{-{\frac{5}{2}}}}+2\,{\frac{{d}^{3}a}{ \left ( cf-de \right ) ^{4}\sqrt{fx+e}}}-2\,{\frac{{d}^{2}bc}{ \left ( cf-de \right ) ^{4}\sqrt{fx+e}}}+2\,{\frac{{d}^{4}a}{ \left ( cf-de \right ) ^{4}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }-2\,{\frac{b{d}^{3}c}{ \left ( cf-de \right ) ^{4}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)/(d*x+c)/(f*x+e)^(9/2),x)

[Out]

-2/7/(c*f-d*e)/(f*x+e)^(7/2)*a+2/7/f/(c*f-d*e)/(f*x+e)^(7/2)*b*e-2/3/(c*f-d*e)^3
*d^2/(f*x+e)^(3/2)*a+2/3/(c*f-d*e)^3*d/(f*x+e)^(3/2)*b*c+2/5/(c*f-d*e)^2/(f*x+e)
^(5/2)*a*d-2/5/(c*f-d*e)^2/(f*x+e)^(5/2)*b*c+2/(c*f-d*e)^4*d^3/(f*x+e)^(1/2)*a-2
/(c*f-d*e)^4*d^2/(f*x+e)^(1/2)*b*c+2*d^4/(c*f-d*e)^4/((c*f-d*e)*d)^(1/2)*arctan(
(f*x+e)^(1/2)*d/((c*f-d*e)*d)^(1/2))*a-2*d^3/(c*f-d*e)^4/((c*f-d*e)*d)^(1/2)*arc
tan((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)^(9/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.236608, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)/((d*x + c)*(f*x + e)^(9/2)),x, algorithm="fricas")

[Out]

[-1/105*(30*b*d^3*e^4 + 30*a*c^3*f^4 + 210*(b*c*d^2 - a*d^3)*f^4*x^3 + 8*(29*b*c
*d^2 - 44*a*d^3)*e^3*f - 4*(16*b*c^2*d - 61*a*c*d^2)*e^2*f^2 + 12*(b*c^3 - 11*a*
c^2*d)*e*f^3 + 70*(10*(b*c*d^2 - a*d^3)*e*f^3 - (b*c^2*d - a*c*d^2)*f^4)*x^2 + 1
05*((b*c*d^2 - a*d^3)*f^4*x^3 + 3*(b*c*d^2 - a*d^3)*e*f^3*x^2 + 3*(b*c*d^2 - a*d
^3)*e^2*f^2*x + (b*c*d^2 - a*d^3)*e^3*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)
) + 14*(58*(b*c*d^2 - a*d^3)*e^2*f^2 - 16*(b*c^2*d - a*c*d^2)*e*f^3 + 3*(b*c^3 -
a*c^2*d)*f^4)*x)/((d^4*e^7*f - 4*c*d^3*e^6*f^2 + 6*c^2*d^2*e^5*f^3 - 4*c^3*d*e^
4*f^4 + c^4*e^3*f^5 + (d^4*e^4*f^4 - 4*c*d^3*e^3*f^5 + 6*c^2*d^2*e^2*f^6 - 4*c^3
*d*e*f^7 + c^4*f^8)*x^3 + 3*(d^4*e^5*f^3 - 4*c*d^3*e^4*f^4 + 6*c^2*d^2*e^3*f^5 -
4*c^3*d*e^2*f^6 + c^4*e*f^7)*x^2 + 3*(d^4*e^6*f^2 - 4*c*d^3*e^5*f^3 + 6*c^2*d^2
*e^4*f^4 - 4*c^3*d*e^3*f^5 + c^4*e^2*f^6)*x)*sqrt(f*x + e)), -2/105*(15*b*d^3*e^
4 + 15*a*c^3*f^4 + 105*(b*c*d^2 - a*d^3)*f^4*x^3 + 4*(29*b*c*d^2 - 44*a*d^3)*e^3
*f - 2*(16*b*c^2*d - 61*a*c*d^2)*e^2*f^2 + 6*(b*c^3 - 11*a*c^2*d)*e*f^3 + 35*(10
*(b*c*d^2 - a*d^3)*e*f^3 - (b*c^2*d - a*c*d^2)*f^4)*x^2 - 105*((b*c*d^2 - a*d^3)
*f^4*x^3 + 3*(b*c*d^2 - a*d^3)*e*f^3*x^2 + 3*(b*c*d^2 - a*d^3)*e^2*f^2*x + (b*c*
d^2 - a*d^3)*e^3*f)*sqrt(f*x + e)*sqrt(-d/(d*e - c*f))*arctan(-(d*e - c*f)*sqrt(
-d/(d*e - c*f))/(sqrt(f*x + e)*d)) + 7*(58*(b*c*d^2 - a*d^3)*e^2*f^2 - 16*(b*c^2
*d - a*c*d^2)*e*f^3 + 3*(b*c^3 - a*c^2*d)*f^4)*x)/((d^4*e^7*f - 4*c*d^3*e^6*f^2
+ 6*c^2*d^2*e^5*f^3 - 4*c^3*d*e^4*f^4 + c^4*e^3*f^5 + (d^4*e^4*f^4 - 4*c*d^3*e^3
*f^5 + 6*c^2*d^2*e^2*f^6 - 4*c^3*d*e*f^7 + c^4*f^8)*x^3 + 3*(d^4*e^5*f^3 - 4*c*d
^3*e^4*f^4 + 6*c^2*d^2*e^3*f^5 - 4*c^3*d*e^2*f^6 + c^4*e*f^7)*x^2 + 3*(d^4*e^6*f
^2 - 4*c*d^3*e^5*f^3 + 6*c^2*d^2*e^4*f^4 - 4*c^3*d*e^3*f^5 + c^4*e^2*f^6)*x)*sqr
t(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)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.226167, size = 608, normalized size = 3.29 \[ -\frac{2 \,{\left (b c d^{3} - a d^{4}\right )} \arctan \left (\frac{\sqrt{f x + e} d}{\sqrt{c d f - d^{2} e}}\right )}{{\left (c^{4} f^{4} - 4 \, c^{3} d f^{3} e + 6 \, c^{2} d^{2} f^{2} e^{2} - 4 \, c d^{3} f e^{3} + d^{4} e^{4}\right )} \sqrt{c d f - d^{2} e}} - \frac{2 \,{\left (105 \,{\left (f x + e\right )}^{3} b c d^{2} f - 105 \,{\left (f x + e\right )}^{3} a d^{3} f - 35 \,{\left (f x + e\right )}^{2} b c^{2} d f^{2} + 35 \,{\left (f x + e\right )}^{2} a c d^{2} f^{2} + 21 \,{\left (f x + e\right )} b c^{3} f^{3} - 21 \,{\left (f x + e\right )} a c^{2} d f^{3} + 15 \, a c^{3} f^{4} + 35 \,{\left (f x + e\right )}^{2} b c d^{2} f e - 35 \,{\left (f x + e\right )}^{2} a d^{3} f e - 42 \,{\left (f x + e\right )} b c^{2} d f^{2} e + 42 \,{\left (f x + e\right )} a c d^{2} f^{2} e - 15 \, b c^{3} f^{3} e - 45 \, a c^{2} d f^{3} e + 21 \,{\left (f x + e\right )} b c d^{2} f e^{2} - 21 \,{\left (f x + e\right )} a d^{3} f e^{2} + 45 \, b c^{2} d f^{2} e^{2} + 45 \, a c d^{2} f^{2} e^{2} - 45 \, b c d^{2} f e^{3} - 15 \, a d^{3} f e^{3} + 15 \, b d^{3} e^{4}\right )}}{105 \,{\left (c^{4} f^{5} - 4 \, c^{3} d f^{4} e + 6 \, c^{2} d^{2} f^{3} e^{2} - 4 \, c d^{3} f^{2} e^{3} + d^{4} f e^{4}\right )}{\left (f x + e\right )}^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)/((d*x + c)*(f*x + e)^(9/2)),x, algorithm="giac")

[Out]

-2*(b*c*d^3 - a*d^4)*arctan(sqrt(f*x + e)*d/sqrt(c*d*f - d^2*e))/((c^4*f^4 - 4*c
^3*d*f^3*e + 6*c^2*d^2*f^2*e^2 - 4*c*d^3*f*e^3 + d^4*e^4)*sqrt(c*d*f - d^2*e)) -
2/105*(105*(f*x + e)^3*b*c*d^2*f - 105*(f*x + e)^3*a*d^3*f - 35*(f*x + e)^2*b*c
^2*d*f^2 + 35*(f*x + e)^2*a*c*d^2*f^2 + 21*(f*x + e)*b*c^3*f^3 - 21*(f*x + e)*a*
c^2*d*f^3 + 15*a*c^3*f^4 + 35*(f*x + e)^2*b*c*d^2*f*e - 35*(f*x + e)^2*a*d^3*f*e
- 42*(f*x + e)*b*c^2*d*f^2*e + 42*(f*x + e)*a*c*d^2*f^2*e - 15*b*c^3*f^3*e - 45
*a*c^2*d*f^3*e + 21*(f*x + e)*b*c*d^2*f*e^2 - 21*(f*x + e)*a*d^3*f*e^2 + 45*b*c^
2*d*f^2*e^2 + 45*a*c*d^2*f^2*e^2 - 45*b*c*d^2*f*e^3 - 15*a*d^3*f*e^3 + 15*b*d^3*
e^4)/((c^4*f^5 - 4*c^3*d*f^4*e + 6*c^2*d^2*f^3*e^2 - 4*c*d^3*f^2*e^3 + d^4*f*e^4
)*(f*x + e)^(7/2))