3.1738 \(\int \frac{A+B x}{(a+b x)^2 (d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=140 \[ -\frac{A b-a B}{b (a+b x) \sqrt{d+e x} (b d-a e)}+\frac{a B e-3 A b e+2 b B d}{b \sqrt{d+e x} (b d-a e)^2}-\frac{(a B e-3 A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{b} (b d-a e)^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.30329, antiderivative size = 140, 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 \[ -\frac{A b-a B}{b (a+b x) \sqrt{d+e x} (b d-a e)}+\frac{a B e-3 A b e+2 b B d}{b \sqrt{d+e x} (b d-a e)^2}-\frac{(a B e-3 A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{b} (b d-a e)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 30.8975, size = 124, normalized size = 0.89 \[ - \frac{3 A b e - B a e - 2 B b d}{b \sqrt{d + e x} \left (a e - b d\right )^{2}} + \frac{A b - B a}{b \left (a + b x\right ) \sqrt{d + e x} \left (a e - b d\right )} - \frac{\left (3 A b e - B a e - 2 B b d\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{\sqrt{b} \left (a e - b d\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.322992, size = 123, normalized size = 0.88 \[ \frac{B (3 a d+a e x+2 b d x)-A (2 a e+b (d+3 e x))}{(a+b x) \sqrt{d+e x} (b d-a e)^2}-\frac{(a B e-3 A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{b} (b d-a e)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.027, size = 253, normalized size = 1.8 \[ -2\,{\frac{Ae}{ \left ( ae-bd \right ) ^{2}\sqrt{ex+d}}}+2\,{\frac{Bd}{ \left ( ae-bd \right ) ^{2}\sqrt{ex+d}}}-{\frac{Abe}{ \left ( ae-bd \right ) ^{2} \left ( bxe+ae \right ) }\sqrt{ex+d}}+{\frac{Bae}{ \left ( ae-bd \right ) ^{2} \left ( bxe+ae \right ) }\sqrt{ex+d}}-3\,{\frac{Abe}{ \left ( ae-bd \right ) ^{2}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }+{\frac{Bae}{ \left ( ae-bd \right ) ^{2}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}}+2\,{\frac{Bbd}{ \left ( ae-bd \right ) ^{2}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)^2*(e*x + d)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224561, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (2 \, B a b d +{\left (B a^{2} - 3 \, A a b\right )} e +{\left (2 \, B b^{2} d +{\left (B a b - 3 \, A b^{2}\right )} e\right )} x\right )} \sqrt{e x + d} \log \left (\frac{\sqrt{b^{2} d - a b e}{\left (b e x + 2 \, b d - a e\right )} + 2 \,{\left (b^{2} d - a b e\right )} \sqrt{e x + d}}{b x + a}\right ) + 2 \, \sqrt{b^{2} d - a b e}{\left (2 \, A a e -{\left (3 \, B a - A b\right )} d -{\left (2 \, B b d +{\left (B a - 3 \, A b\right )} e\right )} x\right )}}{2 \,{\left (a b^{2} d^{2} - 2 \, a^{2} b d e + a^{3} e^{2} +{\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )} x\right )} \sqrt{b^{2} d - a b e} \sqrt{e x + d}}, -\frac{{\left (2 \, B a b d +{\left (B a^{2} - 3 \, A a b\right )} e +{\left (2 \, B b^{2} d +{\left (B a b - 3 \, A b^{2}\right )} e\right )} x\right )} \sqrt{e x + d} \arctan \left (-\frac{b d - a e}{\sqrt{-b^{2} d + a b e} \sqrt{e x + d}}\right ) + \sqrt{-b^{2} d + a b e}{\left (2 \, A a e -{\left (3 \, B a - A b\right )} d -{\left (2 \, B b d +{\left (B a - 3 \, A b\right )} e\right )} x\right )}}{{\left (a b^{2} d^{2} - 2 \, a^{2} b d e + a^{3} e^{2} +{\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )} x\right )} \sqrt{-b^{2} d + a b e} \sqrt{e x + d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)/(b*x+a)**2/(e*x+d)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.234022, size = 275, normalized size = 1.96 \[ \frac{{\left (2 \, B b d + B a e - 3 \, A b e\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \sqrt{-b^{2} d + a b e}} + \frac{2 \,{\left (x e + d\right )} B b d - 2 \, B b d^{2} +{\left (x e + d\right )} B a e - 3 \,{\left (x e + d\right )} A b e + 2 \, B a d e + 2 \, A b d e - 2 \, A a e^{2}}{{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )}{\left ({\left (x e + d\right )}^{\frac{3}{2}} b - \sqrt{x e + d} b d + \sqrt{x e + d} a e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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