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

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

[Out]

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

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Rubi [A]  time = 0.17119, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{2 b^{5/2} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{c^{9/2}}-\frac{2 b^2 \sqrt{x} (b B-A c)}{c^4}+\frac{2 b x^{3/2} (b B-A c)}{3 c^3}-\frac{2 x^{5/2} (b B-A c)}{5 c^2}+\frac{2 B x^{7/2}}{7 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 19.8925, size = 105, normalized size = 0.93 \[ \frac{2 B x^{\frac{7}{2}}}{7 c} - \frac{2 b^{\frac{5}{2}} \left (A c - B b\right ) \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}} \right )}}{c^{\frac{9}{2}}} + \frac{2 b^{2} \sqrt{x} \left (A c - B b\right )}{c^{4}} - \frac{2 b x^{\frac{3}{2}} \left (A c - B b\right )}{3 c^{3}} + \frac{2 x^{\frac{5}{2}} \left (A c - B b\right )}{5 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.139656, size = 101, normalized size = 0.89 \[ \frac{2 b^{5/2} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{c^{9/2}}+\frac{2 \sqrt{x} \left (35 b^2 c (3 A+B x)-7 b c^2 x (5 A+3 B x)+3 c^3 x^2 (7 A+5 B x)-105 b^3 B\right )}{105 c^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.015, size = 126, normalized size = 1.1 \[{\frac{2\,B}{7\,c}{x}^{{\frac{7}{2}}}}+{\frac{2\,A}{5\,c}{x}^{{\frac{5}{2}}}}-{\frac{2\,Bb}{5\,{c}^{2}}{x}^{{\frac{5}{2}}}}-{\frac{2\,Ab}{3\,{c}^{2}}{x}^{{\frac{3}{2}}}}+{\frac{2\,{b}^{2}B}{3\,{c}^{3}}{x}^{{\frac{3}{2}}}}+2\,{\frac{A{b}^{2}\sqrt{x}}{{c}^{3}}}-2\,{\frac{B\sqrt{x}{b}^{3}}{{c}^{4}}}-2\,{\frac{A{b}^{3}}{{c}^{3}\sqrt{bc}}\arctan \left ({\frac{c\sqrt{x}}{\sqrt{bc}}} \right ) }+2\,{\frac{{b}^{4}B}{{c}^{4}\sqrt{bc}}\arctan \left ({\frac{c\sqrt{x}}{\sqrt{bc}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.301763, size = 1, normalized size = 0.01 \[ \left [-\frac{105 \,{\left (B b^{3} - A b^{2} c\right )} \sqrt{-\frac{b}{c}} \log \left (\frac{c x - 2 \, c \sqrt{x} \sqrt{-\frac{b}{c}} - b}{c x + b}\right ) - 2 \,{\left (15 \, B c^{3} x^{3} - 105 \, B b^{3} + 105 \, A b^{2} c - 21 \,{\left (B b c^{2} - A c^{3}\right )} x^{2} + 35 \,{\left (B b^{2} c - A b c^{2}\right )} x\right )} \sqrt{x}}{105 \, c^{4}}, \frac{2 \,{\left (105 \,{\left (B b^{3} - A b^{2} c\right )} \sqrt{\frac{b}{c}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{b}{c}}}\right ) +{\left (15 \, B c^{3} x^{3} - 105 \, B b^{3} + 105 \, A b^{2} c - 21 \,{\left (B b c^{2} - A c^{3}\right )} x^{2} + 35 \,{\left (B b^{2} c - A b c^{2}\right )} x\right )} \sqrt{x}\right )}}{105 \, c^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 38.9883, size = 162, normalized size = 1.43 \[ - \frac{2 A b^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}} \right )}}{c^{\frac{7}{2}}} + \frac{2 A b^{2} \sqrt{x}}{c^{3}} - \frac{2 A b x^{\frac{3}{2}}}{3 c^{2}} + \frac{2 A x^{\frac{5}{2}}}{5 c} + \frac{2 B b^{\frac{7}{2}} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}} \right )}}{c^{\frac{9}{2}}} - \frac{2 B b^{3} \sqrt{x}}{c^{4}} + \frac{2 B b^{2} x^{\frac{3}{2}}}{3 c^{3}} - \frac{2 B b x^{\frac{5}{2}}}{5 c^{2}} + \frac{2 B x^{\frac{7}{2}}}{7 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(7/2)*(B*x+A)/(c*x**2+b*x),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.274961, size = 155, normalized size = 1.37 \[ \frac{2 \,{\left (B b^{4} - A b^{3} c\right )} \arctan \left (\frac{c \sqrt{x}}{\sqrt{b c}}\right )}{\sqrt{b c} c^{4}} + \frac{2 \,{\left (15 \, B c^{6} x^{\frac{7}{2}} - 21 \, B b c^{5} x^{\frac{5}{2}} + 21 \, A c^{6} x^{\frac{5}{2}} + 35 \, B b^{2} c^{4} x^{\frac{3}{2}} - 35 \, A b c^{5} x^{\frac{3}{2}} - 105 \, B b^{3} c^{3} \sqrt{x} + 105 \, A b^{2} c^{4} \sqrt{x}\right )}}{105 \, c^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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