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

Optimal. Leaf size=113 \[ -\frac{2 a^{5/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{9/2}}+\frac{2 a^2 \sqrt{x} (A b-a B)}{b^4}-\frac{2 a x^{3/2} (A b-a B)}{3 b^3}+\frac{2 x^{5/2} (A b-a B)}{5 b^2}+\frac{2 B x^{7/2}}{7 b} \]

[Out]

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

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Rubi [A]  time = 0.142805, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{2 a^{5/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{9/2}}+\frac{2 a^2 \sqrt{x} (A b-a B)}{b^4}-\frac{2 a x^{3/2} (A b-a B)}{3 b^3}+\frac{2 x^{5/2} (A b-a B)}{5 b^2}+\frac{2 B x^{7/2}}{7 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 18.8372, size = 105, normalized size = 0.93 \[ \frac{2 B x^{\frac{7}{2}}}{7 b} - \frac{2 a^{\frac{5}{2}} \left (A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{b^{\frac{9}{2}}} + \frac{2 a^{2} \sqrt{x} \left (A b - B a\right )}{b^{4}} - \frac{2 a x^{\frac{3}{2}} \left (A b - B a\right )}{3 b^{3}} + \frac{2 x^{\frac{5}{2}} \left (A b - B a\right )}{5 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.119589, size = 101, normalized size = 0.89 \[ \frac{2 a^{5/2} (a B-A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{9/2}}+\frac{2 \sqrt{x} \left (-105 a^3 B+35 a^2 b (3 A+B x)-7 a b^2 x (5 A+3 B x)+3 b^3 x^2 (7 A+5 B x)\right )}{105 b^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 126, normalized size = 1.1 \[{\frac{2\,B}{7\,b}{x}^{{\frac{7}{2}}}}+{\frac{2\,A}{5\,b}{x}^{{\frac{5}{2}}}}-{\frac{2\,Ba}{5\,{b}^{2}}{x}^{{\frac{5}{2}}}}-{\frac{2\,Aa}{3\,{b}^{2}}{x}^{{\frac{3}{2}}}}+{\frac{2\,B{a}^{2}}{3\,{b}^{3}}{x}^{{\frac{3}{2}}}}+2\,{\frac{{a}^{2}A\sqrt{x}}{{b}^{3}}}-2\,{\frac{B{a}^{3}\sqrt{x}}{{b}^{4}}}-2\,{\frac{{a}^{3}A}{{b}^{3}\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) }+2\,{\frac{B{a}^{4}}{{b}^{4}\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)*(B*x+A)/(b*x+a),x)

[Out]

2/7*B*x^(7/2)/b+2/5/b*A*x^(5/2)-2/5/b^2*B*x^(5/2)*a-2/3/b^2*A*x^(3/2)*a+2/3/b^3*
B*x^(3/2)*a^2+2/b^3*a^2*A*x^(1/2)-2/b^4*a^3*B*x^(1/2)-2*a^3/b^3/(a*b)^(1/2)*arct
an(x^(1/2)*b/(a*b)^(1/2))*A+2*a^4/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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220915, size = 1, normalized size = 0.01 \[ \left [-\frac{105 \,{\left (B a^{3} - A a^{2} b\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x - 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) - 2 \,{\left (15 \, B b^{3} x^{3} - 105 \, B a^{3} + 105 \, A a^{2} b - 21 \,{\left (B a b^{2} - A b^{3}\right )} x^{2} + 35 \,{\left (B a^{2} b - A a b^{2}\right )} x\right )} \sqrt{x}}{105 \, b^{4}}, \frac{2 \,{\left (105 \,{\left (B a^{3} - A a^{2} b\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{a}{b}}}\right ) +{\left (15 \, B b^{3} x^{3} - 105 \, B a^{3} + 105 \, A a^{2} b - 21 \,{\left (B a b^{2} - A b^{3}\right )} x^{2} + 35 \,{\left (B a^{2} b - A a b^{2}\right )} x\right )} \sqrt{x}\right )}}{105 \, b^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/105*(105*(B*a^3 - A*a^2*b)*sqrt(-a/b)*log((b*x - 2*b*sqrt(x)*sqrt(-a/b) - a)
/(b*x + a)) - 2*(15*B*b^3*x^3 - 105*B*a^3 + 105*A*a^2*b - 21*(B*a*b^2 - A*b^3)*x
^2 + 35*(B*a^2*b - A*a*b^2)*x)*sqrt(x))/b^4, 2/105*(105*(B*a^3 - A*a^2*b)*sqrt(a
/b)*arctan(sqrt(x)/sqrt(a/b)) + (15*B*b^3*x^3 - 105*B*a^3 + 105*A*a^2*b - 21*(B*
a*b^2 - A*b^3)*x^2 + 35*(B*a^2*b - A*a*b^2)*x)*sqrt(x))/b^4]

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Sympy [A]  time = 52.3262, size = 162, normalized size = 1.43 \[ - \frac{2 A a^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{b^{\frac{7}{2}}} + \frac{2 A a^{2} \sqrt{x}}{b^{3}} - \frac{2 A a x^{\frac{3}{2}}}{3 b^{2}} + \frac{2 A x^{\frac{5}{2}}}{5 b} + \frac{2 B a^{\frac{7}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{b^{\frac{9}{2}}} - \frac{2 B a^{3} \sqrt{x}}{b^{4}} + \frac{2 B a^{2} x^{\frac{3}{2}}}{3 b^{3}} - \frac{2 B a x^{\frac{5}{2}}}{5 b^{2}} + \frac{2 B x^{\frac{7}{2}}}{7 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.256701, size = 155, normalized size = 1.37 \[ \frac{2 \,{\left (B a^{4} - A a^{3} b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{4}} + \frac{2 \,{\left (15 \, B b^{6} x^{\frac{7}{2}} - 21 \, B a b^{5} x^{\frac{5}{2}} + 21 \, A b^{6} x^{\frac{5}{2}} + 35 \, B a^{2} b^{4} x^{\frac{3}{2}} - 35 \, A a b^{5} x^{\frac{3}{2}} - 105 \, B a^{3} b^{3} \sqrt{x} + 105 \, A a^{2} b^{4} \sqrt{x}\right )}}{105 \, b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(B*a^4 - A*a^3*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^4) + 2/105*(15*B*b^
6*x^(7/2) - 21*B*a*b^5*x^(5/2) + 21*A*b^6*x^(5/2) + 35*B*a^2*b^4*x^(3/2) - 35*A*
a*b^5*x^(3/2) - 105*B*a^3*b^3*sqrt(x) + 105*A*a^2*b^4*sqrt(x))/b^7