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

Optimal. Leaf size=90 \[ \frac{2 a^{3/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{7/2}}-\frac{2 a \sqrt{x} (A b-a B)}{b^3}+\frac{2 x^{3/2} (A b-a B)}{3 b^2}+\frac{2 B x^{5/2}}{5 b} \]

[Out]

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

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Rubi [A]  time = 0.115959, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{2 a^{3/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{7/2}}-\frac{2 a \sqrt{x} (A b-a B)}{b^3}+\frac{2 x^{3/2} (A b-a B)}{3 b^2}+\frac{2 B x^{5/2}}{5 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.5253, size = 83, normalized size = 0.92 \[ \frac{2 B x^{\frac{5}{2}}}{5 b} + \frac{2 a^{\frac{3}{2}} \left (A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{b^{\frac{7}{2}}} - \frac{2 a \sqrt{x} \left (A b - B a\right )}{b^{3}} + \frac{2 x^{\frac{3}{2}} \left (A b - B a\right )}{3 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.115593, size = 81, normalized size = 0.9 \[ \frac{2 \sqrt{x} \left (15 a^2 B-5 a b (3 A+B x)+b^2 x (5 A+3 B x)\right )}{15 b^3}-\frac{2 a^{3/2} (a B-A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[x]*(15*a^2*B - 5*a*b*(3*A + B*x) + b^2*x*(5*A + 3*B*x)))/(15*b^3) - (2*a
^(3/2)*(-(A*b) + a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/b^(7/2)

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Maple [A]  time = 0.012, size = 102, normalized size = 1.1 \[{\frac{2\,B}{5\,b}{x}^{{\frac{5}{2}}}}+{\frac{2\,A}{3\,b}{x}^{{\frac{3}{2}}}}-{\frac{2\,Ba}{3\,{b}^{2}}{x}^{{\frac{3}{2}}}}-2\,{\frac{aA\sqrt{x}}{{b}^{2}}}+2\,{\frac{B{a}^{2}\sqrt{x}}{{b}^{3}}}+2\,{\frac{{a}^{2}A}{{b}^{2}\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) }-2\,{\frac{B{a}^{3}}{{b}^{3}\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)*(B*x+A)/(b*x+a),x)

[Out]

2/5*B*x^(5/2)/b+2/3/b*A*x^(3/2)-2/3/b^2*B*x^(3/2)*a-2/b^2*a*A*x^(1/2)+2/b^3*a^2*
B*x^(1/2)+2*a^2/b^2/(a*b)^(1/2)*arctan(x^(1/2)*b/(a*b)^(1/2))*A-2*a^3/b^3/(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^(3/2)/(b*x + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220483, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (B a^{2} - A a b\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x + 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) - 2 \,{\left (3 \, B b^{2} x^{2} + 15 \, B a^{2} - 15 \, A a b - 5 \,{\left (B a b - A b^{2}\right )} x\right )} \sqrt{x}}{15 \, b^{3}}, -\frac{2 \,{\left (15 \,{\left (B a^{2} - A a b\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{a}{b}}}\right ) -{\left (3 \, B b^{2} x^{2} + 15 \, B a^{2} - 15 \, A a b - 5 \,{\left (B a b - A b^{2}\right )} x\right )} \sqrt{x}\right )}}{15 \, b^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/15*(15*(B*a^2 - A*a*b)*sqrt(-a/b)*log((b*x + 2*b*sqrt(x)*sqrt(-a/b) - a)/(b*
x + a)) - 2*(3*B*b^2*x^2 + 15*B*a^2 - 15*A*a*b - 5*(B*a*b - A*b^2)*x)*sqrt(x))/b
^3, -2/15*(15*(B*a^2 - A*a*b)*sqrt(a/b)*arctan(sqrt(x)/sqrt(a/b)) - (3*B*b^2*x^2
+ 15*B*a^2 - 15*A*a*b - 5*(B*a*b - A*b^2)*x)*sqrt(x))/b^3]

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Sympy [A]  time = 28.2249, size = 128, normalized size = 1.42 \[ \frac{2 A a^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{b^{\frac{5}{2}}} - \frac{2 A a \sqrt{x}}{b^{2}} + \frac{2 A x^{\frac{3}{2}}}{3 b} - \frac{2 B a^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{b^{\frac{7}{2}}} + \frac{2 B a^{2} \sqrt{x}}{b^{3}} - \frac{2 B a x^{\frac{3}{2}}}{3 b^{2}} + \frac{2 B x^{\frac{5}{2}}}{5 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*a**(3/2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/b**(5/2) - 2*A*a*sqrt(x)/b**2 + 2*A*x
**(3/2)/(3*b) - 2*B*a**(5/2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/b**(7/2) + 2*B*a**2*s
qrt(x)/b**3 - 2*B*a*x**(3/2)/(3*b**2) + 2*B*x**(5/2)/(5*b)

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GIAC/XCAS [A]  time = 0.263589, size = 123, normalized size = 1.37 \[ -\frac{2 \,{\left (B a^{3} - A a^{2} b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{3}} + \frac{2 \,{\left (3 \, B b^{4} x^{\frac{5}{2}} - 5 \, B a b^{3} x^{\frac{3}{2}} + 5 \, A b^{4} x^{\frac{3}{2}} + 15 \, B a^{2} b^{2} \sqrt{x} - 15 \, A a b^{3} \sqrt{x}\right )}}{15 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(B*a^3 - A*a^2*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^3) + 2/15*(3*B*b^4
*x^(5/2) - 5*B*a*b^3*x^(3/2) + 5*A*b^4*x^(3/2) + 15*B*a^2*b^2*sqrt(x) - 15*A*a*b
^3*sqrt(x))/b^5