3.314 \(\int \frac{x^{7/2} (A+B x)}{a+b x} \, dx\)

Optimal. Leaf size=136 \[ \frac{2 a^{7/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{11/2}}-\frac{2 a^3 \sqrt{x} (A b-a B)}{b^5}+\frac{2 a^2 x^{3/2} (A b-a B)}{3 b^4}-\frac{2 a x^{5/2} (A b-a B)}{5 b^3}+\frac{2 x^{7/2} (A b-a B)}{7 b^2}+\frac{2 B x^{9/2}}{9 b} \]

[Out]

(-2*a^3*(A*b - a*B)*Sqrt[x])/b^5 + (2*a^2*(A*b - a*B)*x^(3/2))/(3*b^4) - (2*a*(A
*b - a*B)*x^(5/2))/(5*b^3) + (2*(A*b - a*B)*x^(7/2))/(7*b^2) + (2*B*x^(9/2))/(9*
b) + (2*a^(7/2)*(A*b - a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/b^(11/2)

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Rubi [A]  time = 0.211681, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{2 a^{7/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{11/2}}-\frac{2 a^3 \sqrt{x} (A b-a B)}{b^5}+\frac{2 a^2 x^{3/2} (A b-a B)}{3 b^4}-\frac{2 a x^{5/2} (A b-a B)}{5 b^3}+\frac{2 x^{7/2} (A b-a B)}{7 b^2}+\frac{2 B x^{9/2}}{9 b} \]

Antiderivative was successfully verified.

[In]  Int[(x^(7/2)*(A + B*x))/(a + b*x),x]

[Out]

(-2*a^3*(A*b - a*B)*Sqrt[x])/b^5 + (2*a^2*(A*b - a*B)*x^(3/2))/(3*b^4) - (2*a*(A
*b - a*B)*x^(5/2))/(5*b^3) + (2*(A*b - a*B)*x^(7/2))/(7*b^2) + (2*B*x^(9/2))/(9*
b) + (2*a^(7/2)*(A*b - a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/b^(11/2)

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Rubi in Sympy [A]  time = 23.7586, size = 128, normalized size = 0.94 \[ \frac{2 B x^{\frac{9}{2}}}{9 b} + \frac{2 a^{\frac{7}{2}} \left (A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{b^{\frac{11}{2}}} - \frac{2 a^{3} \sqrt{x} \left (A b - B a\right )}{b^{5}} + \frac{2 a^{2} x^{\frac{3}{2}} \left (A b - B a\right )}{3 b^{4}} - \frac{2 a x^{\frac{5}{2}} \left (A b - B a\right )}{5 b^{3}} + \frac{2 x^{\frac{7}{2}} \left (A b - B a\right )}{7 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(7/2)*(B*x+A)/(b*x+a),x)

[Out]

2*B*x**(9/2)/(9*b) + 2*a**(7/2)*(A*b - B*a)*atan(sqrt(b)*sqrt(x)/sqrt(a))/b**(11
/2) - 2*a**3*sqrt(x)*(A*b - B*a)/b**5 + 2*a**2*x**(3/2)*(A*b - B*a)/(3*b**4) - 2
*a*x**(5/2)*(A*b - B*a)/(5*b**3) + 2*x**(7/2)*(A*b - B*a)/(7*b**2)

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Mathematica [A]  time = 0.182093, size = 120, normalized size = 0.88 \[ \frac{2 \sqrt{x} \left (315 a^4 B-105 a^3 b (3 A+B x)+21 a^2 b^2 x (5 A+3 B x)-9 a b^3 x^2 (7 A+5 B x)+5 b^4 x^3 (9 A+7 B x)\right )}{315 b^5}-\frac{2 a^{7/2} (a B-A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{11/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(7/2)*(A + B*x))/(a + b*x),x]

[Out]

(2*Sqrt[x]*(315*a^4*B - 105*a^3*b*(3*A + B*x) + 21*a^2*b^2*x*(5*A + 3*B*x) - 9*a
*b^3*x^2*(7*A + 5*B*x) + 5*b^4*x^3*(9*A + 7*B*x)))/(315*b^5) - (2*a^(7/2)*(-(A*b
) + a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/b^(11/2)

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Maple [A]  time = 0.013, size = 150, normalized size = 1.1 \[{\frac{2\,B}{9\,b}{x}^{{\frac{9}{2}}}}+{\frac{2\,A}{7\,b}{x}^{{\frac{7}{2}}}}-{\frac{2\,Ba}{7\,{b}^{2}}{x}^{{\frac{7}{2}}}}-{\frac{2\,Aa}{5\,{b}^{2}}{x}^{{\frac{5}{2}}}}+{\frac{2\,B{a}^{2}}{5\,{b}^{3}}{x}^{{\frac{5}{2}}}}+{\frac{2\,A{a}^{2}}{3\,{b}^{3}}{x}^{{\frac{3}{2}}}}-{\frac{2\,B{a}^{3}}{3\,{b}^{4}}{x}^{{\frac{3}{2}}}}-2\,{\frac{{a}^{3}A\sqrt{x}}{{b}^{4}}}+2\,{\frac{B{a}^{4}\sqrt{x}}{{b}^{5}}}+2\,{\frac{{a}^{4}A}{{b}^{4}\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) }-2\,{\frac{B{a}^{5}}{{b}^{5}\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(7/2)*(B*x+A)/(b*x+a),x)

[Out]

2/9*B*x^(9/2)/b+2/7/b*A*x^(7/2)-2/7/b^2*B*x^(7/2)*a-2/5/b^2*A*x^(5/2)*a+2/5/b^3*
B*x^(5/2)*a^2+2/3/b^3*A*x^(3/2)*a^2-2/3/b^4*B*x^(3/2)*a^3-2/b^4*A*a^3*x^(1/2)+2/
b^5*B*a^4*x^(1/2)+2*a^4/b^4/(a*b)^(1/2)*arctan(x^(1/2)*b/(a*b)^(1/2))*A-2*a^5/b^
5/(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^(7/2)/(b*x + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221586, size = 1, normalized size = 0.01 \[ \left [-\frac{315 \,{\left (B a^{4} - A a^{3} b\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x + 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) - 2 \,{\left (35 \, B b^{4} x^{4} + 315 \, B a^{4} - 315 \, A a^{3} b - 45 \,{\left (B a b^{3} - A b^{4}\right )} x^{3} + 63 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} x^{2} - 105 \,{\left (B a^{3} b - A a^{2} b^{2}\right )} x\right )} \sqrt{x}}{315 \, b^{5}}, -\frac{2 \,{\left (315 \,{\left (B a^{4} - A a^{3} b\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{a}{b}}}\right ) -{\left (35 \, B b^{4} x^{4} + 315 \, B a^{4} - 315 \, A a^{3} b - 45 \,{\left (B a b^{3} - A b^{4}\right )} x^{3} + 63 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} x^{2} - 105 \,{\left (B a^{3} b - A a^{2} b^{2}\right )} x\right )} \sqrt{x}\right )}}{315 \, b^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^(7/2)/(b*x + a),x, algorithm="fricas")

[Out]

[-1/315*(315*(B*a^4 - A*a^3*b)*sqrt(-a/b)*log((b*x + 2*b*sqrt(x)*sqrt(-a/b) - a)
/(b*x + a)) - 2*(35*B*b^4*x^4 + 315*B*a^4 - 315*A*a^3*b - 45*(B*a*b^3 - A*b^4)*x
^3 + 63*(B*a^2*b^2 - A*a*b^3)*x^2 - 105*(B*a^3*b - A*a^2*b^2)*x)*sqrt(x))/b^5, -
2/315*(315*(B*a^4 - A*a^3*b)*sqrt(a/b)*arctan(sqrt(x)/sqrt(a/b)) - (35*B*b^4*x^4
+ 315*B*a^4 - 315*A*a^3*b - 45*(B*a*b^3 - A*b^4)*x^3 + 63*(B*a^2*b^2 - A*a*b^3)
*x^2 - 105*(B*a^3*b - A*a^2*b^2)*x)*sqrt(x))/b^5]

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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**(7/2)*(B*x+A)/(b*x+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.259712, size = 188, normalized size = 1.38 \[ -\frac{2 \,{\left (B a^{5} - A a^{4} b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{5}} + \frac{2 \,{\left (35 \, B b^{8} x^{\frac{9}{2}} - 45 \, B a b^{7} x^{\frac{7}{2}} + 45 \, A b^{8} x^{\frac{7}{2}} + 63 \, B a^{2} b^{6} x^{\frac{5}{2}} - 63 \, A a b^{7} x^{\frac{5}{2}} - 105 \, B a^{3} b^{5} x^{\frac{3}{2}} + 105 \, A a^{2} b^{6} x^{\frac{3}{2}} + 315 \, B a^{4} b^{4} \sqrt{x} - 315 \, A a^{3} b^{5} \sqrt{x}\right )}}{315 \, b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^(7/2)/(b*x + a),x, algorithm="giac")

[Out]

-2*(B*a^5 - A*a^4*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^5) + 2/315*(35*B*b
^8*x^(9/2) - 45*B*a*b^7*x^(7/2) + 45*A*b^8*x^(7/2) + 63*B*a^2*b^6*x^(5/2) - 63*A
*a*b^7*x^(5/2) - 105*B*a^3*b^5*x^(3/2) + 105*A*a^2*b^6*x^(3/2) + 315*B*a^4*b^4*s
qrt(x) - 315*A*a^3*b^5*sqrt(x))/b^9