3.1731 \(\int \frac{A+B x}{(a+b x) (d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=119 \[ \frac{2 (A b-a B)}{\sqrt{d+e x} (b d-a e)^2}-\frac{2 (B d-A e)}{3 e (d+e x)^{3/2} (b d-a e)}-\frac{2 \sqrt{b} (A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{5/2}} \]

[Out]

(-2*(B*d - A*e))/(3*e*(b*d - a*e)*(d + e*x)^(3/2)) + (2*(A*b - a*B))/((b*d - a*e
)^2*Sqrt[d + e*x]) - (2*Sqrt[b]*(A*b - a*B)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt
[b*d - a*e]])/(b*d - a*e)^(5/2)

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Rubi [A]  time = 0.220248, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{2 (A b-a B)}{\sqrt{d+e x} (b d-a e)^2}-\frac{2 (B d-A e)}{3 e (d+e x)^{3/2} (b d-a e)}-\frac{2 \sqrt{b} (A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/((a + b*x)*(d + e*x)^(5/2)),x]

[Out]

(-2*(B*d - A*e))/(3*e*(b*d - a*e)*(d + e*x)^(3/2)) + (2*(A*b - a*B))/((b*d - a*e
)^2*Sqrt[d + e*x]) - (2*Sqrt[b]*(A*b - a*B)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt
[b*d - a*e]])/(b*d - a*e)^(5/2)

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Rubi in Sympy [A]  time = 22.6272, size = 104, normalized size = 0.87 \[ \frac{2 \sqrt{b} \left (A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{\left (a e - b d\right )^{\frac{5}{2}}} + \frac{2 \left (A b - B a\right )}{\sqrt{d + e x} \left (a e - b d\right )^{2}} - \frac{2 \left (A e - B d\right )}{3 e \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/(b*x+a)/(e*x+d)**(5/2),x)

[Out]

2*sqrt(b)*(A*b - B*a)*atan(sqrt(b)*sqrt(d + e*x)/sqrt(a*e - b*d))/(a*e - b*d)**(
5/2) + 2*(A*b - B*a)/(sqrt(d + e*x)*(a*e - b*d)**2) - 2*(A*e - B*d)/(3*e*(d + e*
x)**(3/2)*(a*e - b*d))

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Mathematica [A]  time = 0.34649, size = 118, normalized size = 0.99 \[ -\frac{2 \left (a e (A e+2 B d+3 B e x)-A b e (4 d+3 e x)+b B d^2\right )}{3 e (d+e x)^{3/2} (b d-a e)^2}-\frac{2 \sqrt{b} (A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/((a + b*x)*(d + e*x)^(5/2)),x]

[Out]

(-2*(b*B*d^2 - A*b*e*(4*d + 3*e*x) + a*e*(2*B*d + A*e + 3*B*e*x)))/(3*e*(b*d - a
*e)^2*(d + e*x)^(3/2)) - (2*Sqrt[b]*(A*b - a*B)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/
Sqrt[b*d - a*e]])/(b*d - a*e)^(5/2)

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Maple [A]  time = 0.019, size = 187, normalized size = 1.6 \[ -{\frac{2\,A}{3\,ae-3\,bd} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,Bd}{3\,e \left ( ae-bd \right ) } \left ( ex+d \right ) ^{-{\frac{3}{2}}}}+2\,{\frac{Ab}{ \left ( ae-bd \right ) ^{2}\sqrt{ex+d}}}-2\,{\frac{Ba}{ \left ( ae-bd \right ) ^{2}\sqrt{ex+d}}}+2\,{\frac{{b}^{2}A}{ \left ( ae-bd \right ) ^{2}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }-2\,{\frac{Bba}{ \left ( ae-bd \right ) ^{2}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(b*x+a)/(e*x+d)^(5/2),x)

[Out]

-2/3/(a*e-b*d)/(e*x+d)^(3/2)*A+2/3/e/(a*e-b*d)/(e*x+d)^(3/2)*B*d+2/(a*e-b*d)^2/(
e*x+d)^(1/2)*A*b-2/(a*e-b*d)^2/(e*x+d)^(1/2)*B*a+2*b^2/(a*e-b*d)^2/((a*e-b*d)*b)
^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*A-2*b/(a*e-b*d)^2/((a*e-b*d)*
b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*B*a

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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)/((b*x + a)*(e*x + d)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.226166, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \, B b d^{2} + 2 \, A a e^{2} + 6 \,{\left (B a - A b\right )} e^{2} x + 4 \,{\left (B a - 2 \, A b\right )} d e + 3 \,{\left ({\left (B a - A b\right )} e^{2} x +{\left (B a - A b\right )} d e\right )} \sqrt{e x + d} \sqrt{\frac{b}{b d - a e}} \log \left (\frac{b e x + 2 \, b d - a e - 2 \,{\left (b d - a e\right )} \sqrt{e x + d} \sqrt{\frac{b}{b d - a e}}}{b x + a}\right )}{3 \,{\left (b^{2} d^{3} e - 2 \, a b d^{2} e^{2} + a^{2} d e^{3} +{\left (b^{2} d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} x\right )} \sqrt{e x + d}}, -\frac{2 \,{\left (B b d^{2} + A a e^{2} + 3 \,{\left (B a - A b\right )} e^{2} x + 2 \,{\left (B a - 2 \, A b\right )} d e - 3 \,{\left ({\left (B a - A b\right )} e^{2} x +{\left (B a - A b\right )} d e\right )} \sqrt{e x + d} \sqrt{-\frac{b}{b d - a e}} \arctan \left (-\frac{{\left (b d - a e\right )} \sqrt{-\frac{b}{b d - a e}}}{\sqrt{e x + d} b}\right )\right )}}{3 \,{\left (b^{2} d^{3} e - 2 \, a b d^{2} e^{2} + a^{2} d e^{3} +{\left (b^{2} d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} x\right )} \sqrt{e x + d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)*(e*x + d)^(5/2)),x, algorithm="fricas")

[Out]

[-1/3*(2*B*b*d^2 + 2*A*a*e^2 + 6*(B*a - A*b)*e^2*x + 4*(B*a - 2*A*b)*d*e + 3*((B
*a - A*b)*e^2*x + (B*a - A*b)*d*e)*sqrt(e*x + d)*sqrt(b/(b*d - a*e))*log((b*e*x
+ 2*b*d - a*e - 2*(b*d - a*e)*sqrt(e*x + d)*sqrt(b/(b*d - a*e)))/(b*x + a)))/((b
^2*d^3*e - 2*a*b*d^2*e^2 + a^2*d*e^3 + (b^2*d^2*e^2 - 2*a*b*d*e^3 + a^2*e^4)*x)*
sqrt(e*x + d)), -2/3*(B*b*d^2 + A*a*e^2 + 3*(B*a - A*b)*e^2*x + 2*(B*a - 2*A*b)*
d*e - 3*((B*a - A*b)*e^2*x + (B*a - A*b)*d*e)*sqrt(e*x + d)*sqrt(-b/(b*d - a*e))
*arctan(-(b*d - a*e)*sqrt(-b/(b*d - a*e))/(sqrt(e*x + d)*b)))/((b^2*d^3*e - 2*a*
b*d^2*e^2 + a^2*d*e^3 + (b^2*d^2*e^2 - 2*a*b*d*e^3 + a^2*e^4)*x)*sqrt(e*x + d))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x}{\left (a + b x\right ) \left (d + e x\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/(b*x+a)/(e*x+d)**(5/2),x)

[Out]

Integral((A + B*x)/((a + b*x)*(d + e*x)**(5/2)), x)

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GIAC/XCAS [A]  time = 0.216129, size = 217, normalized size = 1.82 \[ -\frac{2 \,{\left (B a b - A b^{2}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \sqrt{-b^{2} d + a b e}} - \frac{2 \,{\left (B b d^{2} + 3 \,{\left (x e + d\right )} B a e - 3 \,{\left (x e + d\right )} A b e - B a d e - A b d e + A a e^{2}\right )}}{3 \,{\left (b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}\right )}{\left (x e + d\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)*(e*x + d)^(5/2)),x, algorithm="giac")

[Out]

-2*(B*a*b - A*b^2)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/((b^2*d^2 - 2*a*
b*d*e + a^2*e^2)*sqrt(-b^2*d + a*b*e)) - 2/3*(B*b*d^2 + 3*(x*e + d)*B*a*e - 3*(x
*e + d)*A*b*e - B*a*d*e - A*b*d*e + A*a*e^2)/((b^2*d^2*e - 2*a*b*d*e^2 + a^2*e^3
)*(x*e + d)^(3/2))