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

Optimal. Leaf size=157 \[ \frac{5 c \left (b x+c x^2\right )^{3/2} (4 A c+3 b B)}{6 b x}+\frac{5}{4} c \sqrt{b x+c x^2} (4 A c+3 b B)+\frac{5}{4} b \sqrt{c} (4 A c+3 b B) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )-\frac{2 \left (b x+c x^2\right )^{5/2} (4 A c+3 b B)}{3 b x^3}-\frac{2 A \left (b x+c x^2\right )^{7/2}}{3 b x^5} \]

[Out]

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

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Rubi [A]  time = 0.348908, antiderivative size = 157, 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 \[ \frac{5 c \left (b x+c x^2\right )^{3/2} (4 A c+3 b B)}{6 b x}+\frac{5}{4} c \sqrt{b x+c x^2} (4 A c+3 b B)+\frac{5}{4} b \sqrt{c} (4 A c+3 b B) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )-\frac{2 \left (b x+c x^2\right )^{5/2} (4 A c+3 b B)}{3 b x^3}-\frac{2 A \left (b x+c x^2\right )^{7/2}}{3 b x^5} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 20.6478, size = 150, normalized size = 0.96 \[ - \frac{2 A \left (b x + c x^{2}\right )^{\frac{7}{2}}}{3 b x^{5}} + \frac{5 b \sqrt{c} \left (4 A c + 3 B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{4} + \frac{5 c \left (4 A c + 3 B b\right ) \sqrt{b x + c x^{2}}}{4} + \frac{5 c \left (4 A c + 3 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{6 b x} - \frac{2 \left (4 A c + 3 B b\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}}}{3 b x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.201212, size = 130, normalized size = 0.83 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{b+c x} \left (3 B x \left (-8 b^2+9 b c x+2 c^2 x^2\right )-4 A \left (2 b^2+14 b c x-3 c^2 x^2\right )\right )+15 b \sqrt{c} x^{3/2} (4 A c+3 b B) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )\right )}{12 x^2 \sqrt{b+c x}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[b + c*x]*(-4*A*(2*b^2 + 14*b*c*x - 3*c^2*x^2) + 3*B*x*(
-8*b^2 + 9*b*c*x + 2*c^2*x^2)) + 15*b*Sqrt[c]*(3*b*B + 4*A*c)*x^(3/2)*Log[c*Sqrt
[x] + Sqrt[c]*Sqrt[b + c*x]]))/(12*x^2*Sqrt[b + c*x])

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Maple [B]  time = 0.015, size = 411, normalized size = 2.6 \[ -{\frac{2\,A}{3\,b{x}^{5}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}-{\frac{8\,Ac}{3\,{b}^{2}{x}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}+16\,{\frac{A{c}^{2} \left ( c{x}^{2}+bx \right ) ^{7/2}}{{b}^{3}{x}^{3}}}-{\frac{128\,A{c}^{3}}{3\,{b}^{4}{x}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}+{\frac{128\,A{c}^{4}}{3\,{b}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{80\,A{c}^{4}x}{3\,{b}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{40\,A{c}^{3}}{3\,{b}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-10\,{\frac{A{c}^{3}\sqrt{c{x}^{2}+bx}x}{b}}-5\,A{c}^{2}\sqrt{c{x}^{2}+bx}+{\frac{5\,Ab}{2}{c}^{{\frac{3}{2}}}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ) }-2\,{\frac{B \left ( c{x}^{2}+bx \right ) ^{7/2}}{b{x}^{4}}}+12\,{\frac{Bc \left ( c{x}^{2}+bx \right ) ^{7/2}}{{b}^{2}{x}^{3}}}-32\,{\frac{B{c}^{2} \left ( c{x}^{2}+bx \right ) ^{7/2}}{{b}^{3}{x}^{2}}}+32\,{\frac{B{c}^{3} \left ( c{x}^{2}+bx \right ) ^{5/2}}{{b}^{3}}}+20\,{\frac{B{c}^{3} \left ( c{x}^{2}+bx \right ) ^{3/2}x}{{b}^{2}}}+10\,{\frac{B{c}^{2} \left ( c{x}^{2}+bx \right ) ^{3/2}}{b}}-{\frac{15\,B{c}^{2}x}{2}\sqrt{c{x}^{2}+bx}}-{\frac{15\,Bbc}{4}\sqrt{c{x}^{2}+bx}}+{\frac{15\,{b}^{2}B}{8}\sqrt{c}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*A*(c*x^2+b*x)^(7/2)/b/x^5-8/3*A*c/b^2/x^4*(c*x^2+b*x)^(7/2)+16*A*c^2/b^3/x^
3*(c*x^2+b*x)^(7/2)-128/3*A*c^3/b^4/x^2*(c*x^2+b*x)^(7/2)+128/3*A*c^4/b^4*(c*x^2
+b*x)^(5/2)+80/3*A*c^4/b^3*(c*x^2+b*x)^(3/2)*x+40/3*A*c^3/b^2*(c*x^2+b*x)^(3/2)-
10*A*c^3/b*(c*x^2+b*x)^(1/2)*x-5*A*c^2*(c*x^2+b*x)^(1/2)+5/2*A*c^(3/2)*b*ln((1/2
*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))-2*B/b/x^4*(c*x^2+b*x)^(7/2)+12*B*c/b^2/x^3*(c
*x^2+b*x)^(7/2)-32*B*c^2/b^3/x^2*(c*x^2+b*x)^(7/2)+32*B*c^3/b^3*(c*x^2+b*x)^(5/2
)+20*B*c^3/b^2*(c*x^2+b*x)^(3/2)*x+10*B*c^2/b*(c*x^2+b*x)^(3/2)-15/2*B*c^2*(c*x^
2+b*x)^(1/2)*x-15/4*B*c*b*(c*x^2+b*x)^(1/2)+15/8*B*c^(1/2)*b^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^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.299514, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (3 \, B b^{2} + 4 \, A b c\right )} \sqrt{c} x^{2} \log \left (2 \, c x + b + 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) + 2 \,{\left (6 \, B c^{2} x^{3} - 8 \, A b^{2} + 3 \,{\left (9 \, B b c + 4 \, A c^{2}\right )} x^{2} - 8 \,{\left (3 \, B b^{2} + 7 \, A b c\right )} x\right )} \sqrt{c x^{2} + b x}}{24 \, x^{2}}, \frac{15 \,{\left (3 \, B b^{2} + 4 \, A b c\right )} \sqrt{-c} x^{2} \arctan \left (\frac{\sqrt{c x^{2} + b x}}{\sqrt{-c} x}\right ) +{\left (6 \, B c^{2} x^{3} - 8 \, A b^{2} + 3 \,{\left (9 \, B b c + 4 \, A c^{2}\right )} x^{2} - 8 \,{\left (3 \, B b^{2} + 7 \, A b c\right )} x\right )} \sqrt{c x^{2} + b x}}{12 \, x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/24*(15*(3*B*b^2 + 4*A*b*c)*sqrt(c)*x^2*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sq
rt(c)) + 2*(6*B*c^2*x^3 - 8*A*b^2 + 3*(9*B*b*c + 4*A*c^2)*x^2 - 8*(3*B*b^2 + 7*A
*b*c)*x)*sqrt(c*x^2 + b*x))/x^2, 1/12*(15*(3*B*b^2 + 4*A*b*c)*sqrt(-c)*x^2*arcta
n(sqrt(c*x^2 + b*x)/(sqrt(-c)*x)) + (6*B*c^2*x^3 - 8*A*b^2 + 3*(9*B*b*c + 4*A*c^
2)*x^2 - 8*(3*B*b^2 + 7*A*b*c)*x)*sqrt(c*x^2 + b*x))/x^2]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.289487, size = 285, normalized size = 1.82 \[ \frac{1}{4} \,{\left (2 \, B c^{2} x + \frac{9 \, B b c^{2} + 4 \, A c^{3}}{c}\right )} \sqrt{c x^{2} + b x} - \frac{5 \,{\left (3 \, B b^{2} c + 4 \, A b c^{2}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{8 \, \sqrt{c}} + \frac{2 \,{\left (3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} B b^{3} \sqrt{c} + 9 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} A b^{2} c^{\frac{3}{2}} + 3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} A b^{3} c + A b^{4} \sqrt{c}\right )}}{3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(2*B*c^2*x + (9*B*b*c^2 + 4*A*c^3)/c)*sqrt(c*x^2 + b*x) - 5/8*(3*B*b^2*c + 4
*A*b*c^2)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/sqrt(c) + 2/3*
(3*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^3*sqrt(c) + 9*(sqrt(c)*x - sqrt(c*x^2 +
b*x))^2*A*b^2*c^(3/2) + 3*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^3*c + A*b^4*sqrt(
c))/((sqrt(c)*x - sqrt(c*x^2 + b*x))^3*sqrt(c))