3.103 \(\int \frac{(A+B x) \left (b x+c x^2\right )^{5/2}}{x^7} \, dx\)

Optimal. Leaf size=119 \[ -\frac{2 A \left (b x+c x^2\right )^{7/2}}{7 b x^7}+2 B c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )-\frac{2 B c^2 \sqrt{b x+c x^2}}{x}-\frac{2 B \left (b x+c x^2\right )^{5/2}}{5 x^5}-\frac{2 B c \left (b x+c x^2\right )^{3/2}}{3 x^3} \]

[Out]

(-2*B*c^2*Sqrt[b*x + c*x^2])/x - (2*B*c*(b*x + c*x^2)^(3/2))/(3*x^3) - (2*B*(b*x
+ c*x^2)^(5/2))/(5*x^5) - (2*A*(b*x + c*x^2)^(7/2))/(7*b*x^7) + 2*B*c^(5/2)*Arc
Tanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]]

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Rubi [A]  time = 0.298727, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{2 A \left (b x+c x^2\right )^{7/2}}{7 b x^7}+2 B c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )-\frac{2 B c^2 \sqrt{b x+c x^2}}{x}-\frac{2 B \left (b x+c x^2\right )^{5/2}}{5 x^5}-\frac{2 B c \left (b x+c x^2\right )^{3/2}}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(b*x + c*x^2)^(5/2))/x^7,x]

[Out]

(-2*B*c^2*Sqrt[b*x + c*x^2])/x - (2*B*c*(b*x + c*x^2)^(3/2))/(3*x^3) - (2*B*(b*x
+ c*x^2)^(5/2))/(5*x^5) - (2*A*(b*x + c*x^2)^(7/2))/(7*b*x^7) + 2*B*c^(5/2)*Arc
Tanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]]

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Rubi in Sympy [A]  time = 17.8514, size = 112, normalized size = 0.94 \[ - \frac{2 A \left (b x + c x^{2}\right )^{\frac{7}{2}}}{7 b x^{7}} + 2 B c^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )} - \frac{2 B c^{2} \sqrt{b x + c x^{2}}}{x} - \frac{2 B c \left (b x + c x^{2}\right )^{\frac{3}{2}}}{3 x^{3}} - \frac{2 B \left (b x + c x^{2}\right )^{\frac{5}{2}}}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**(5/2)/x**7,x)

[Out]

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

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Mathematica [A]  time = 0.348266, size = 101, normalized size = 0.85 \[ \frac{2 \sqrt{x (b+c x)} \left (-\frac{15 A (b+c x)^3}{b}-7 B x \left (3 b^2+11 b c x+23 c^2 x^2\right )+\frac{105 B c^{5/2} x^{7/2} \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{b+c x}}\right )}{105 x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(b*x + c*x^2)^(5/2))/x^7,x]

[Out]

(2*Sqrt[x*(b + c*x)]*((-15*A*(b + c*x)^3)/b - 7*B*x*(3*b^2 + 11*b*c*x + 23*c^2*x
^2) + (105*B*c^(5/2)*x^(7/2)*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]])/Sqrt[b + c*
x]))/(105*x^4)

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Maple [B]  time = 0.015, size = 263, normalized size = 2.2 \[ -{\frac{2\,A}{7\,b{x}^{7}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}-{\frac{2\,B}{5\,b{x}^{6}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}-{\frac{4\,Bc}{15\,{b}^{2}{x}^{5}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}-{\frac{16\,B{c}^{2}}{15\,{b}^{3}{x}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}+{\frac{32\,B{c}^{3}}{5\,{b}^{4}{x}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}-{\frac{256\,B{c}^{4}}{15\,{b}^{5}{x}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}+{\frac{256\,B{c}^{5}}{15\,{b}^{5}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{32\,B{c}^{5}x}{3\,{b}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{16\,B{c}^{4}}{3\,{b}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-4\,{\frac{B{c}^{4}\sqrt{c{x}^{2}+bx}x}{{b}^{2}}}-2\,{\frac{B{c}^{3}\sqrt{c{x}^{2}+bx}}{b}}+B{c}^{{\frac{5}{2}}}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^(5/2)/x^7,x)

[Out]

-2/7*A*(c*x^2+b*x)^(7/2)/b/x^7-2/5*B/b/x^6*(c*x^2+b*x)^(7/2)-4/15*B*c/b^2/x^5*(c
*x^2+b*x)^(7/2)-16/15*B*c^2/b^3/x^4*(c*x^2+b*x)^(7/2)+32/5*B*c^3/b^4/x^3*(c*x^2+
b*x)^(7/2)-256/15*B*c^4/b^5/x^2*(c*x^2+b*x)^(7/2)+256/15*B*c^5/b^5*(c*x^2+b*x)^(
5/2)+32/3*B*c^5/b^4*(c*x^2+b*x)^(3/2)*x+16/3*B*c^4/b^3*(c*x^2+b*x)^(3/2)-4*B*c^4
/b^2*(c*x^2+b*x)^(1/2)*x-2*B*c^3/b*(c*x^2+b*x)^(1/2)+B*c^(5/2)*ln((1/2*b+c*x)/c^
(1/2)+(c*x^2+b*x)^(1/2))

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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((c*x^2 + b*x)^(5/2)*(B*x + A)/x^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.293969, size = 1, normalized size = 0.01 \[ \left [\frac{105 \, B b c^{\frac{5}{2}} x^{4} \log \left (2 \, c x + b + 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) - 2 \,{\left (15 \, A b^{3} +{\left (161 \, B b c^{2} + 15 \, A c^{3}\right )} x^{3} +{\left (77 \, B b^{2} c + 45 \, A b c^{2}\right )} x^{2} + 3 \,{\left (7 \, B b^{3} + 15 \, A b^{2} c\right )} x\right )} \sqrt{c x^{2} + b x}}{105 \, b x^{4}}, \frac{2 \,{\left (105 \, B b \sqrt{-c} c^{2} x^{4} \arctan \left (\frac{\sqrt{c x^{2} + b x}}{\sqrt{-c} x}\right ) -{\left (15 \, A b^{3} +{\left (161 \, B b c^{2} + 15 \, A c^{3}\right )} x^{3} +{\left (77 \, B b^{2} c + 45 \, A b c^{2}\right )} x^{2} + 3 \,{\left (7 \, B b^{3} + 15 \, A b^{2} c\right )} x\right )} \sqrt{c x^{2} + b x}\right )}}{105 \, b x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(B*x + A)/x^7,x, algorithm="fricas")

[Out]

[1/105*(105*B*b*c^(5/2)*x^4*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) - 2*(15
*A*b^3 + (161*B*b*c^2 + 15*A*c^3)*x^3 + (77*B*b^2*c + 45*A*b*c^2)*x^2 + 3*(7*B*b
^3 + 15*A*b^2*c)*x)*sqrt(c*x^2 + b*x))/(b*x^4), 2/105*(105*B*b*sqrt(-c)*c^2*x^4*
arctan(sqrt(c*x^2 + b*x)/(sqrt(-c)*x)) - (15*A*b^3 + (161*B*b*c^2 + 15*A*c^3)*x^
3 + (77*B*b^2*c + 45*A*b*c^2)*x^2 + 3*(7*B*b^3 + 15*A*b^2*c)*x)*sqrt(c*x^2 + b*x
))/(b*x^4)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (b + c x\right )\right )^{\frac{5}{2}} \left (A + B x\right )}{x^{7}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)**(5/2)/x**7,x)

[Out]

Integral((x*(b + c*x))**(5/2)*(A + B*x)/x**7, x)

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GIAC/XCAS [A]  time = 0.296678, size = 527, normalized size = 4.43 \[ -B c^{\frac{5}{2}}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right ) + \frac{2 \,{\left (315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{6} B b c^{\frac{5}{2}} + 105 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{6} A c^{\frac{7}{2}} + 315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} B b^{2} c^{2} + 315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} A b c^{3} + 245 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} B b^{3} c^{\frac{3}{2}} + 525 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} A b^{2} c^{\frac{5}{2}} + 105 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} B b^{4} c + 525 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} A b^{3} c^{2} + 21 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} B b^{5} \sqrt{c} + 315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} A b^{4} c^{\frac{3}{2}} + 105 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} A b^{5} c + 15 \, A b^{6} \sqrt{c}\right )}}{105 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{7} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(B*x + A)/x^7,x, algorithm="giac")

[Out]

-B*c^(5/2)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b)) + 2/105*(315*
(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b*c^(5/2) + 105*(sqrt(c)*x - sqrt(c*x^2 + b*
x))^6*A*c^(7/2) + 315*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^2*c^2 + 315*(sqrt(c)
*x - sqrt(c*x^2 + b*x))^5*A*b*c^3 + 245*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^3*
c^(3/2) + 525*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^2*c^(5/2) + 105*(sqrt(c)*x -
sqrt(c*x^2 + b*x))^3*B*b^4*c + 525*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^3*c^2
+ 21*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^5*sqrt(c) + 315*(sqrt(c)*x - sqrt(c*x
^2 + b*x))^2*A*b^4*c^(3/2) + 105*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^5*c + 15*A*
b^6*sqrt(c))/((sqrt(c)*x - sqrt(c*x^2 + b*x))^7*sqrt(c))