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

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

[Out]

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

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Rubi [A]  time = 0.159226, 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 \[ -\frac{2 c^{5/2} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{b^{9/2}}-\frac{2 c^2 (b B-A c)}{b^4 \sqrt{x}}+\frac{2 c (b B-A c)}{3 b^3 x^{3/2}}-\frac{2 (b B-A c)}{5 b^2 x^{5/2}}-\frac{2 A}{7 b x^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 20.1947, size = 107, normalized size = 0.95 \[ - \frac{2 A}{7 b x^{\frac{7}{2}}} + \frac{2 \left (A c - B b\right )}{5 b^{2} x^{\frac{5}{2}}} - \frac{2 c \left (A c - B b\right )}{3 b^{3} x^{\frac{3}{2}}} + \frac{2 c^{2} \left (A c - B b\right )}{b^{4} \sqrt{x}} + \frac{2 c^{\frac{5}{2}} \left (A c - B b\right ) \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}} \right )}}{b^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.235231, size = 107, normalized size = 0.95 \[ \frac{2 c^{5/2} (A c-b B) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{b^{9/2}}+\frac{A \left (-30 b^3+42 b^2 c x-70 b c^2 x^2+210 c^3 x^3\right )-14 b B x \left (3 b^2-5 b c x+15 c^2 x^2\right )}{105 b^4 x^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-14*b*B*x*(3*b^2 - 5*b*c*x + 15*c^2*x^2) + A*(-30*b^3 + 42*b^2*c*x - 70*b*c^2*x
^2 + 210*c^3*x^3))/(105*b^4*x^(7/2)) + (2*c^(5/2)*(-(b*B) + A*c)*ArcTan[(Sqrt[c]
*Sqrt[x])/Sqrt[b]])/b^(9/2)

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Maple [A]  time = 0.018, size = 126, normalized size = 1.1 \[ -{\frac{2\,A}{7\,b}{x}^{-{\frac{7}{2}}}}+{\frac{2\,Ac}{5\,{b}^{2}}{x}^{-{\frac{5}{2}}}}-{\frac{2\,B}{5\,b}{x}^{-{\frac{5}{2}}}}-{\frac{2\,A{c}^{2}}{3\,{b}^{3}}{x}^{-{\frac{3}{2}}}}+{\frac{2\,Bc}{3\,{b}^{2}}{x}^{-{\frac{3}{2}}}}+2\,{\frac{A{c}^{3}}{{b}^{4}\sqrt{x}}}-2\,{\frac{B{c}^{2}}{{b}^{3}\sqrt{x}}}+2\,{\frac{A{c}^{4}}{{b}^{4}\sqrt{bc}}\arctan \left ({\frac{c\sqrt{x}}{\sqrt{bc}}} \right ) }-2\,{\frac{B{c}^{3}}{{b}^{3}\sqrt{bc}}\arctan \left ({\frac{c\sqrt{x}}{\sqrt{bc}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.300907, size = 1, normalized size = 0.01 \[ \left [-\frac{105 \,{\left (B b c^{2} - A c^{3}\right )} x^{\frac{7}{2}} \sqrt{-\frac{c}{b}} \log \left (\frac{c x + 2 \, b \sqrt{x} \sqrt{-\frac{c}{b}} - b}{c x + b}\right ) + 30 \, A b^{3} + 210 \,{\left (B b c^{2} - A c^{3}\right )} x^{3} - 70 \,{\left (B b^{2} c - A b c^{2}\right )} x^{2} + 42 \,{\left (B b^{3} - A b^{2} c\right )} x}{105 \, b^{4} x^{\frac{7}{2}}}, \frac{2 \,{\left (105 \,{\left (B b c^{2} - A c^{3}\right )} x^{\frac{7}{2}} \sqrt{\frac{c}{b}} \arctan \left (\frac{b \sqrt{\frac{c}{b}}}{c \sqrt{x}}\right ) - 15 \, A b^{3} - 105 \,{\left (B b c^{2} - A c^{3}\right )} x^{3} + 35 \,{\left (B b^{2} c - A b c^{2}\right )} x^{2} - 21 \,{\left (B b^{3} - A b^{2} c\right )} x\right )}}{105 \, b^{4} x^{\frac{7}{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/105*(105*(B*b*c^2 - A*c^3)*x^(7/2)*sqrt(-c/b)*log((c*x + 2*b*sqrt(x)*sqrt(-c
/b) - b)/(c*x + b)) + 30*A*b^3 + 210*(B*b*c^2 - A*c^3)*x^3 - 70*(B*b^2*c - A*b*c
^2)*x^2 + 42*(B*b^3 - A*b^2*c)*x)/(b^4*x^(7/2)), 2/105*(105*(B*b*c^2 - A*c^3)*x^
(7/2)*sqrt(c/b)*arctan(b*sqrt(c/b)/(c*sqrt(x))) - 15*A*b^3 - 105*(B*b*c^2 - A*c^
3)*x^3 + 35*(B*b^2*c - A*b*c^2)*x^2 - 21*(B*b^3 - A*b^2*c)*x)/(b^4*x^(7/2))]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.271709, size = 140, normalized size = 1.24 \[ -\frac{2 \,{\left (B b c^{3} - A c^{4}\right )} \arctan \left (\frac{c \sqrt{x}}{\sqrt{b c}}\right )}{\sqrt{b c} b^{4}} - \frac{2 \,{\left (105 \, B b c^{2} x^{3} - 105 \, A c^{3} x^{3} - 35 \, B b^{2} c x^{2} + 35 \, A b c^{2} x^{2} + 21 \, B b^{3} x - 21 \, A b^{2} c x + 15 \, A b^{3}\right )}}{105 \, b^{4} x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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