3.102 \(\int \frac{(A+B x) \left (b x+c x^2\right )^{5/2}}{x^6} \, dx\)

Optimal. Leaf size=155 \[ c^{3/2} (2 A c+5 b B) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )+\frac{c^2 \sqrt{b x+c x^2} (2 A c+5 b B)}{b}-\frac{2 c \left (b x+c x^2\right )^{3/2} (2 A c+5 b B)}{3 b x^2}-\frac{2 \left (b x+c x^2\right )^{5/2} (2 A c+5 b B)}{15 b x^4}-\frac{2 A \left (b x+c x^2\right )^{7/2}}{5 b x^6} \]

[Out]

(c^2*(5*b*B + 2*A*c)*Sqrt[b*x + c*x^2])/b - (2*c*(5*b*B + 2*A*c)*(b*x + c*x^2)^(
3/2))/(3*b*x^2) - (2*(5*b*B + 2*A*c)*(b*x + c*x^2)^(5/2))/(15*b*x^4) - (2*A*(b*x
+ c*x^2)^(7/2))/(5*b*x^6) + c^(3/2)*(5*b*B + 2*A*c)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*
x + c*x^2]]

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Rubi [A]  time = 0.352904, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ c^{3/2} (2 A c+5 b B) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )+\frac{c^2 \sqrt{b x+c x^2} (2 A c+5 b B)}{b}-\frac{2 c \left (b x+c x^2\right )^{3/2} (2 A c+5 b B)}{3 b x^2}-\frac{2 \left (b x+c x^2\right )^{5/2} (2 A c+5 b B)}{15 b x^4}-\frac{2 A \left (b x+c x^2\right )^{7/2}}{5 b x^6} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(b*x + c*x^2)^(5/2))/x^6,x]

[Out]

(c^2*(5*b*B + 2*A*c)*Sqrt[b*x + c*x^2])/b - (2*c*(5*b*B + 2*A*c)*(b*x + c*x^2)^(
3/2))/(3*b*x^2) - (2*(5*b*B + 2*A*c)*(b*x + c*x^2)^(5/2))/(15*b*x^4) - (2*A*(b*x
+ c*x^2)^(7/2))/(5*b*x^6) + c^(3/2)*(5*b*B + 2*A*c)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*
x + c*x^2]]

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Rubi in Sympy [A]  time = 20.8812, size = 150, normalized size = 0.97 \[ - \frac{2 A \left (b x + c x^{2}\right )^{\frac{7}{2}}}{5 b x^{6}} + 2 c^{\frac{3}{2}} \left (A c + \frac{5 B b}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )} + \frac{2 c^{2} \left (A c + \frac{5 B b}{2}\right ) \sqrt{b x + c x^{2}}}{b} - \frac{2 c \left (2 A c + 5 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{3 b x^{2}} - \frac{4 \left (A c + \frac{5 B b}{2}\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}}}{15 b x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**(5/2)/x**6,x)

[Out]

-2*A*(b*x + c*x**2)**(7/2)/(5*b*x**6) + 2*c**(3/2)*(A*c + 5*B*b/2)*atanh(sqrt(c)
*x/sqrt(b*x + c*x**2)) + 2*c**2*(A*c + 5*B*b/2)*sqrt(b*x + c*x**2)/b - 2*c*(2*A*
c + 5*B*b)*(b*x + c*x**2)**(3/2)/(3*b*x**2) - 4*(A*c + 5*B*b/2)*(b*x + c*x**2)**
(5/2)/(15*b*x**4)

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Mathematica [A]  time = 0.286627, size = 118, normalized size = 0.76 \[ \frac{\sqrt{x (b+c x)} \left (-2 A \left (3 b^2+11 b c x+23 c^2 x^2\right )+\frac{15 c^{3/2} x^{5/2} (2 A c+5 b B) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{b+c x}}+5 B x \left (-2 b^2-14 b c x+3 c^2 x^2\right )\right )}{15 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(b*x + c*x^2)^(5/2))/x^6,x]

[Out]

(Sqrt[x*(b + c*x)]*(5*B*x*(-2*b^2 - 14*b*c*x + 3*c^2*x^2) - 2*A*(3*b^2 + 11*b*c*
x + 23*c^2*x^2) + (15*c^(3/2)*(5*b*B + 2*A*c)*x^(5/2)*Log[c*Sqrt[x] + Sqrt[c]*Sq
rt[b + c*x]])/Sqrt[b + c*x]))/(15*x^3)

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Maple [B]  time = 0.016, size = 460, normalized size = 3. \[ -{\frac{2\,A}{5\,b{x}^{6}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}-{\frac{4\,Ac}{15\,{b}^{2}{x}^{5}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}-{\frac{16\,A{c}^{2}}{15\,{b}^{3}{x}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}+{\frac{32\,A{c}^{3}}{5\,{b}^{4}{x}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}-{\frac{256\,A{c}^{4}}{15\,{b}^{5}{x}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}+{\frac{256\,A{c}^{5}}{15\,{b}^{5}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{32\,A{c}^{5}x}{3\,{b}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{16\,A{c}^{4}}{3\,{b}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-4\,{\frac{A{c}^{4}\sqrt{c{x}^{2}+bx}x}{{b}^{2}}}-2\,{\frac{A{c}^{3}\sqrt{c{x}^{2}+bx}}{b}}+A{c}^{{\frac{5}{2}}}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ) -{\frac{2\,B}{3\,b{x}^{5}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}-{\frac{8\,Bc}{3\,{b}^{2}{x}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}+16\,{\frac{B{c}^{2} \left ( c{x}^{2}+bx \right ) ^{7/2}}{{b}^{3}{x}^{3}}}-{\frac{128\,B{c}^{3}}{3\,{b}^{4}{x}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}+{\frac{128\,B{c}^{4}}{3\,{b}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{80\,B{c}^{4}x}{3\,{b}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{40\,B{c}^{3}}{3\,{b}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-10\,{\frac{B{c}^{3}\sqrt{c{x}^{2}+bx}x}{b}}-5\,B{c}^{2}\sqrt{c{x}^{2}+bx}+{\frac{5\,Bb}{2}{c}^{{\frac{3}{2}}}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^(5/2)/x^6,x)

[Out]

-2/5*A*(c*x^2+b*x)^(7/2)/b/x^6-4/15*A*c/b^2/x^5*(c*x^2+b*x)^(7/2)-16/15*A*c^2/b^
3/x^4*(c*x^2+b*x)^(7/2)+32/5*A*c^3/b^4/x^3*(c*x^2+b*x)^(7/2)-256/15*A*c^4/b^5/x^
2*(c*x^2+b*x)^(7/2)+256/15*A*c^5/b^5*(c*x^2+b*x)^(5/2)+32/3*A*c^5/b^4*(c*x^2+b*x
)^(3/2)*x+16/3*A*c^4/b^3*(c*x^2+b*x)^(3/2)-4*A*c^4/b^2*(c*x^2+b*x)^(1/2)*x-2*A*c
^3/b*(c*x^2+b*x)^(1/2)+A*c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))-2/3*B
/b/x^5*(c*x^2+b*x)^(7/2)-8/3*B*c/b^2/x^4*(c*x^2+b*x)^(7/2)+16*B*c^2/b^3/x^3*(c*x
^2+b*x)^(7/2)-128/3*B*c^3/b^4/x^2*(c*x^2+b*x)^(7/2)+128/3*B*c^4/b^4*(c*x^2+b*x)^
(5/2)+80/3*B*c^4/b^3*(c*x^2+b*x)^(3/2)*x+40/3*B*c^3/b^2*(c*x^2+b*x)^(3/2)-10*B*c
^3/b*(c*x^2+b*x)^(1/2)*x-5*B*c^2*(c*x^2+b*x)^(1/2)+5/2*B*c^(3/2)*b*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^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.301104, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (5 \, B b c + 2 \, A c^{2}\right )} \sqrt{c} x^{3} \log \left (2 \, c x + b + 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) + 2 \,{\left (15 \, B c^{2} x^{3} - 6 \, A b^{2} - 2 \,{\left (35 \, B b c + 23 \, A c^{2}\right )} x^{2} - 2 \,{\left (5 \, B b^{2} + 11 \, A b c\right )} x\right )} \sqrt{c x^{2} + b x}}{30 \, x^{3}}, \frac{15 \,{\left (5 \, B b c + 2 \, A c^{2}\right )} \sqrt{-c} x^{3} \arctan \left (\frac{\sqrt{c x^{2} + b x}}{\sqrt{-c} x}\right ) +{\left (15 \, B c^{2} x^{3} - 6 \, A b^{2} - 2 \,{\left (35 \, B b c + 23 \, A c^{2}\right )} x^{2} - 2 \,{\left (5 \, B b^{2} + 11 \, A b c\right )} x\right )} \sqrt{c x^{2} + b x}}{15 \, x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(B*x + A)/x^6,x, algorithm="fricas")

[Out]

[1/30*(15*(5*B*b*c + 2*A*c^2)*sqrt(c)*x^3*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sq
rt(c)) + 2*(15*B*c^2*x^3 - 6*A*b^2 - 2*(35*B*b*c + 23*A*c^2)*x^2 - 2*(5*B*b^2 +
11*A*b*c)*x)*sqrt(c*x^2 + b*x))/x^3, 1/15*(15*(5*B*b*c + 2*A*c^2)*sqrt(-c)*x^3*a
rctan(sqrt(c*x^2 + b*x)/(sqrt(-c)*x)) + (15*B*c^2*x^3 - 6*A*b^2 - 2*(35*B*b*c +
23*A*c^2)*x^2 - 2*(5*B*b^2 + 11*A*b*c)*x)*sqrt(c*x^2 + b*x))/x^3]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (b + c x\right )\right )^{\frac{5}{2}} \left (A + B x\right )}{x^{6}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)**(5/2)/x**6,x)

[Out]

Integral((x*(b + c*x))**(5/2)*(A + B*x)/x**6, x)

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GIAC/XCAS [A]  time = 0.294839, size = 410, normalized size = 2.65 \[ \sqrt{c x^{2} + b x} B c^{2} - \frac{{\left (5 \, B b c^{2} + 2 \, A c^{3}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{2 \, \sqrt{c}} + \frac{2 \,{\left (45 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} B b^{2} c^{\frac{3}{2}} + 45 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} A b c^{\frac{5}{2}} + 15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} B b^{3} c + 45 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} A b^{2} c^{2} + 5 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} B b^{4} \sqrt{c} + 35 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} A b^{3} c^{\frac{3}{2}} + 15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} A b^{4} c + 3 \, A b^{5} \sqrt{c}\right )}}{15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(B*x + A)/x^6,x, algorithm="giac")

[Out]

sqrt(c*x^2 + b*x)*B*c^2 - 1/2*(5*B*b*c^2 + 2*A*c^3)*ln(abs(-2*(sqrt(c)*x - sqrt(
c*x^2 + b*x))*sqrt(c) - b))/sqrt(c) + 2/15*(45*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4
*B*b^2*c^(3/2) + 45*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b*c^(5/2) + 15*(sqrt(c)*
x - sqrt(c*x^2 + b*x))^3*B*b^3*c + 45*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^2*c^
2 + 5*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^4*sqrt(c) + 35*(sqrt(c)*x - sqrt(c*x
^2 + b*x))^2*A*b^3*c^(3/2) + 15*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^4*c + 3*A*b^
5*sqrt(c))/((sqrt(c)*x - sqrt(c*x^2 + b*x))^5*sqrt(c))