3.334 \(\int \frac{x^{5/2} (A+B x)}{(a+b x)^3} \, dx\)

Optimal. Leaf size=147 \[ -\frac{5 \sqrt{a} (3 A b-7 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 b^{9/2}}+\frac{5 \sqrt{x} (3 A b-7 a B)}{4 b^4}-\frac{5 x^{3/2} (3 A b-7 a B)}{12 a b^3}+\frac{x^{5/2} (3 A b-7 a B)}{4 a b^2 (a+b x)}+\frac{x^{7/2} (A b-a B)}{2 a b (a+b x)^2} \]

[Out]

(5*(3*A*b - 7*a*B)*Sqrt[x])/(4*b^4) - (5*(3*A*b - 7*a*B)*x^(3/2))/(12*a*b^3) + (
(A*b - a*B)*x^(7/2))/(2*a*b*(a + b*x)^2) + ((3*A*b - 7*a*B)*x^(5/2))/(4*a*b^2*(a
+ b*x)) - (5*Sqrt[a]*(3*A*b - 7*a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*b^(9
/2))

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Rubi [A]  time = 0.170531, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ -\frac{5 \sqrt{a} (3 A b-7 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 b^{9/2}}+\frac{5 \sqrt{x} (3 A b-7 a B)}{4 b^4}-\frac{5 x^{3/2} (3 A b-7 a B)}{12 a b^3}+\frac{x^{5/2} (3 A b-7 a B)}{4 a b^2 (a+b x)}+\frac{x^{7/2} (A b-a B)}{2 a b (a+b x)^2} \]

Antiderivative was successfully verified.

[In]  Int[(x^(5/2)*(A + B*x))/(a + b*x)^3,x]

[Out]

(5*(3*A*b - 7*a*B)*Sqrt[x])/(4*b^4) - (5*(3*A*b - 7*a*B)*x^(3/2))/(12*a*b^3) + (
(A*b - a*B)*x^(7/2))/(2*a*b*(a + b*x)^2) + ((3*A*b - 7*a*B)*x^(5/2))/(4*a*b^2*(a
+ b*x)) - (5*Sqrt[a]*(3*A*b - 7*a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*b^(9
/2))

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Rubi in Sympy [A]  time = 21.7829, size = 136, normalized size = 0.93 \[ - \frac{5 \sqrt{a} \left (3 A b - 7 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 b^{\frac{9}{2}}} + \frac{5 \sqrt{x} \left (3 A b - 7 B a\right )}{4 b^{4}} + \frac{x^{\frac{7}{2}} \left (A b - B a\right )}{2 a b \left (a + b x\right )^{2}} + \frac{x^{\frac{5}{2}} \left (3 A b - 7 B a\right )}{4 a b^{2} \left (a + b x\right )} - \frac{5 x^{\frac{3}{2}} \left (3 A b - 7 B a\right )}{12 a b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)*(B*x+A)/(b*x+a)**3,x)

[Out]

-5*sqrt(a)*(3*A*b - 7*B*a)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(4*b**(9/2)) + 5*sqrt(x
)*(3*A*b - 7*B*a)/(4*b**4) + x**(7/2)*(A*b - B*a)/(2*a*b*(a + b*x)**2) + x**(5/2
)*(3*A*b - 7*B*a)/(4*a*b**2*(a + b*x)) - 5*x**(3/2)*(3*A*b - 7*B*a)/(12*a*b**3)

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Mathematica [A]  time = 0.19429, size = 110, normalized size = 0.75 \[ \frac{\sqrt{x} \left (-105 a^3 B+5 a^2 b (9 A-35 B x)+a b^2 x (75 A-56 B x)+8 b^3 x^2 (3 A+B x)\right )}{12 b^4 (a+b x)^2}+\frac{5 \sqrt{a} (7 a B-3 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 b^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(5/2)*(A + B*x))/(a + b*x)^3,x]

[Out]

(Sqrt[x]*(-105*a^3*B + a*b^2*x*(75*A - 56*B*x) + 5*a^2*b*(9*A - 35*B*x) + 8*b^3*
x^2*(3*A + B*x)))/(12*b^4*(a + b*x)^2) + (5*Sqrt[a]*(-3*A*b + 7*a*B)*ArcTan[(Sqr
t[b]*Sqrt[x])/Sqrt[a]])/(4*b^(9/2))

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Maple [A]  time = 0.022, size = 152, normalized size = 1. \[{\frac{2\,B}{3\,{b}^{3}}{x}^{{\frac{3}{2}}}}+2\,{\frac{A\sqrt{x}}{{b}^{3}}}-6\,{\frac{Ba\sqrt{x}}{{b}^{4}}}+{\frac{9\,Aa}{4\,{b}^{2} \left ( bx+a \right ) ^{2}}{x}^{{\frac{3}{2}}}}-{\frac{13\,B{a}^{2}}{4\,{b}^{3} \left ( bx+a \right ) ^{2}}{x}^{{\frac{3}{2}}}}+{\frac{7\,A{a}^{2}}{4\,{b}^{3} \left ( bx+a \right ) ^{2}}\sqrt{x}}-{\frac{11\,B{a}^{3}}{4\,{b}^{4} \left ( bx+a \right ) ^{2}}\sqrt{x}}-{\frac{15\,Aa}{4\,{b}^{3}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{35\,B{a}^{2}}{4\,{b}^{4}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)*(B*x+A)/(b*x+a)^3,x)

[Out]

2/3/b^3*B*x^(3/2)+2/b^3*A*x^(1/2)-6/b^4*B*a*x^(1/2)+9/4*a/b^2/(b*x+a)^2*x^(3/2)*
A-13/4*a^2/b^3/(b*x+a)^2*x^(3/2)*B+7/4*a^2/b^3/(b*x+a)^2*A*x^(1/2)-11/4*a^3/b^4/
(b*x+a)^2*B*x^(1/2)-15/4*a/b^3/(a*b)^(1/2)*arctan(x^(1/2)*b/(a*b)^(1/2))*A+35/4*
a^2/b^4/(a*b)^(1/2)*arctan(x^(1/2)*b/(a*b)^(1/2))*B

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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)*x^(5/2)/(b*x + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.225116, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (7 \, B a^{3} - 3 \, A a^{2} b +{\left (7 \, B a b^{2} - 3 \, A b^{3}\right )} x^{2} + 2 \,{\left (7 \, B a^{2} b - 3 \, A a b^{2}\right )} x\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x - 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) - 2 \,{\left (8 \, B b^{3} x^{3} - 105 \, B a^{3} + 45 \, A a^{2} b - 8 \,{\left (7 \, B a b^{2} - 3 \, A b^{3}\right )} x^{2} - 25 \,{\left (7 \, B a^{2} b - 3 \, A a b^{2}\right )} x\right )} \sqrt{x}}{24 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}, \frac{15 \,{\left (7 \, B a^{3} - 3 \, A a^{2} b +{\left (7 \, B a b^{2} - 3 \, A b^{3}\right )} x^{2} + 2 \,{\left (7 \, B a^{2} b - 3 \, A a b^{2}\right )} x\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{a}{b}}}\right ) +{\left (8 \, B b^{3} x^{3} - 105 \, B a^{3} + 45 \, A a^{2} b - 8 \,{\left (7 \, B a b^{2} - 3 \, A b^{3}\right )} x^{2} - 25 \,{\left (7 \, B a^{2} b - 3 \, A a b^{2}\right )} x\right )} \sqrt{x}}{12 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^(5/2)/(b*x + a)^3,x, algorithm="fricas")

[Out]

[-1/24*(15*(7*B*a^3 - 3*A*a^2*b + (7*B*a*b^2 - 3*A*b^3)*x^2 + 2*(7*B*a^2*b - 3*A
*a*b^2)*x)*sqrt(-a/b)*log((b*x - 2*b*sqrt(x)*sqrt(-a/b) - a)/(b*x + a)) - 2*(8*B
*b^3*x^3 - 105*B*a^3 + 45*A*a^2*b - 8*(7*B*a*b^2 - 3*A*b^3)*x^2 - 25*(7*B*a^2*b
- 3*A*a*b^2)*x)*sqrt(x))/(b^6*x^2 + 2*a*b^5*x + a^2*b^4), 1/12*(15*(7*B*a^3 - 3*
A*a^2*b + (7*B*a*b^2 - 3*A*b^3)*x^2 + 2*(7*B*a^2*b - 3*A*a*b^2)*x)*sqrt(a/b)*arc
tan(sqrt(x)/sqrt(a/b)) + (8*B*b^3*x^3 - 105*B*a^3 + 45*A*a^2*b - 8*(7*B*a*b^2 -
3*A*b^3)*x^2 - 25*(7*B*a^2*b - 3*A*a*b^2)*x)*sqrt(x))/(b^6*x^2 + 2*a*b^5*x + a^2
*b^4)]

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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(x**(5/2)*(B*x+A)/(b*x+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.215067, size = 161, normalized size = 1.1 \[ \frac{5 \,{\left (7 \, B a^{2} - 3 \, A a b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} b^{4}} - \frac{13 \, B a^{2} b x^{\frac{3}{2}} - 9 \, A a b^{2} x^{\frac{3}{2}} + 11 \, B a^{3} \sqrt{x} - 7 \, A a^{2} b \sqrt{x}}{4 \,{\left (b x + a\right )}^{2} b^{4}} + \frac{2 \,{\left (B b^{6} x^{\frac{3}{2}} - 9 \, B a b^{5} \sqrt{x} + 3 \, A b^{6} \sqrt{x}\right )}}{3 \, b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^(5/2)/(b*x + a)^3,x, algorithm="giac")

[Out]

5/4*(7*B*a^2 - 3*A*a*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^4) - 1/4*(13*B*
a^2*b*x^(3/2) - 9*A*a*b^2*x^(3/2) + 11*B*a^3*sqrt(x) - 7*A*a^2*b*sqrt(x))/((b*x
+ a)^2*b^4) + 2/3*(B*b^6*x^(3/2) - 9*B*a*b^5*sqrt(x) + 3*A*b^6*sqrt(x))/b^9