3.84 \(\int \frac{(A+B x) \left (b x+c x^2\right )^{3/2}}{x^2} \, dx\)

Optimal. Leaf size=126 \[ -\frac{b^2 (b B-6 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{8 c^{3/2}}+\frac{\left (b x+c x^2\right )^{3/2} (b B-6 A c)}{3 b}+\frac{(b+2 c x) \sqrt{b x+c x^2} (b B-6 A c)}{8 c}+\frac{2 A \left (b x+c x^2\right )^{5/2}}{b x^2} \]

[Out]

((b*B - 6*A*c)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(8*c) + ((b*B - 6*A*c)*(b*x + c*x^
2)^(3/2))/(3*b) + (2*A*(b*x + c*x^2)^(5/2))/(b*x^2) - (b^2*(b*B - 6*A*c)*ArcTanh
[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(8*c^(3/2))

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Rubi [A]  time = 0.29099, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{b^2 (b B-6 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{8 c^{3/2}}+\frac{\left (b x+c x^2\right )^{3/2} (b B-6 A c)}{3 b}+\frac{(b+2 c x) \sqrt{b x+c x^2} (b B-6 A c)}{8 c}+\frac{2 A \left (b x+c x^2\right )^{5/2}}{b x^2} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(b*x + c*x^2)^(3/2))/x^2,x]

[Out]

((b*B - 6*A*c)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(8*c) + ((b*B - 6*A*c)*(b*x + c*x^
2)^(3/2))/(3*b) + (2*A*(b*x + c*x^2)^(5/2))/(b*x^2) - (b^2*(b*B - 6*A*c)*ArcTanh
[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(8*c^(3/2))

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Rubi in Sympy [A]  time = 15.3651, size = 112, normalized size = 0.89 \[ \frac{2 A \left (b x + c x^{2}\right )^{\frac{5}{2}}}{b x^{2}} + \frac{b^{2} \left (6 A c - B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{8 c^{\frac{3}{2}}} - \frac{\left (b + 2 c x\right ) \left (6 A c - B b\right ) \sqrt{b x + c x^{2}}}{8 c} - \frac{\left (6 A c - B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**(3/2)/x**2,x)

[Out]

2*A*(b*x + c*x**2)**(5/2)/(b*x**2) + b**2*(6*A*c - B*b)*atanh(sqrt(c)*x/sqrt(b*x
+ c*x**2))/(8*c**(3/2)) - (b + 2*c*x)*(6*A*c - B*b)*sqrt(b*x + c*x**2)/(8*c) -
(6*A*c - B*b)*(b*x + c*x**2)**(3/2)/(3*b)

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Mathematica [A]  time = 0.186308, size = 111, normalized size = 0.88 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (2 b c (15 A+7 B x)+4 c^2 x (3 A+2 B x)+3 b^2 B\right )-\frac{3 b^2 (b B-6 A c) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{x} \sqrt{b+c x}}\right )}{24 c^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(b*x + c*x^2)^(3/2))/x^2,x]

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[c]*(3*b^2*B + 4*c^2*x*(3*A + 2*B*x) + 2*b*c*(15*A + 7*B
*x)) - (3*b^2*(b*B - 6*A*c)*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]])/(Sqrt[x]*Sqr
t[b + c*x])))/(24*c^(3/2))

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Maple [A]  time = 0.013, size = 187, normalized size = 1.5 \[ 2\,{\frac{A \left ( c{x}^{2}+bx \right ) ^{5/2}}{b{x}^{2}}}-2\,{\frac{Ac \left ( c{x}^{2}+bx \right ) ^{3/2}}{b}}-{\frac{3\,Acx}{2}\sqrt{c{x}^{2}+bx}}-{\frac{3\,Ab}{4}\sqrt{c{x}^{2}+bx}}+{\frac{3\,{b}^{2}A}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){\frac{1}{\sqrt{c}}}}+{\frac{B}{3} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{xBb}{4}\sqrt{c{x}^{2}+bx}}+{\frac{{b}^{2}B}{8\,c}\sqrt{c{x}^{2}+bx}}-{\frac{B{b}^{3}}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^(3/2)/x^2,x)

[Out]

2*A*(c*x^2+b*x)^(5/2)/b/x^2-2*A*c/b*(c*x^2+b*x)^(3/2)-3/2*A*c*(c*x^2+b*x)^(1/2)*
x-3/4*A*b*(c*x^2+b*x)^(1/2)+3/8*A/c^(1/2)*b^2*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)
^(1/2))+1/3*B*(c*x^2+b*x)^(3/2)+1/4*B*b*(c*x^2+b*x)^(1/2)*x+1/8*B/c*(c*x^2+b*x)^
(1/2)*b^2-1/16*B*b^3/c^(3/2)*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)^(3/2)*(B*x + A)/x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.28203, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (8 \, B c^{2} x^{2} + 3 \, B b^{2} + 30 \, A b c + 2 \,{\left (7 \, B b c + 6 \, A c^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{c} - 3 \,{\left (B b^{3} - 6 \, A b^{2} c\right )} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right )}{48 \, c^{\frac{3}{2}}}, \frac{{\left (8 \, B c^{2} x^{2} + 3 \, B b^{2} + 30 \, A b c + 2 \,{\left (7 \, B b c + 6 \, A c^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{-c} - 3 \,{\left (B b^{3} - 6 \, A b^{2} c\right )} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right )}{24 \, \sqrt{-c} c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)*(B*x + A)/x^2,x, algorithm="fricas")

[Out]

[1/48*(2*(8*B*c^2*x^2 + 3*B*b^2 + 30*A*b*c + 2*(7*B*b*c + 6*A*c^2)*x)*sqrt(c*x^2
+ b*x)*sqrt(c) - 3*(B*b^3 - 6*A*b^2*c)*log((2*c*x + b)*sqrt(c) + 2*sqrt(c*x^2 +
b*x)*c))/c^(3/2), 1/24*((8*B*c^2*x^2 + 3*B*b^2 + 30*A*b*c + 2*(7*B*b*c + 6*A*c^
2)*x)*sqrt(c*x^2 + b*x)*sqrt(-c) - 3*(B*b^3 - 6*A*b^2*c)*arctan(sqrt(c*x^2 + b*x
)*sqrt(-c)/(c*x)))/(sqrt(-c)*c)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (b + c x\right )\right )^{\frac{3}{2}} \left (A + B x\right )}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)**(3/2)/x**2,x)

[Out]

Integral((x*(b + c*x))**(3/2)*(A + B*x)/x**2, x)

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GIAC/XCAS [A]  time = 0.282514, size = 147, normalized size = 1.17 \[ \frac{1}{24} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \, B c x + \frac{7 \, B b c^{2} + 6 \, A c^{3}}{c^{2}}\right )} x + \frac{3 \,{\left (B b^{2} c + 10 \, A b c^{2}\right )}}{c^{2}}\right )} + \frac{{\left (B b^{3} - 6 \, A b^{2} c\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{16 \, c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)*(B*x + A)/x^2,x, algorithm="giac")

[Out]

1/24*sqrt(c*x^2 + b*x)*(2*(4*B*c*x + (7*B*b*c^2 + 6*A*c^3)/c^2)*x + 3*(B*b^2*c +
10*A*b*c^2)/c^2) + 1/16*(B*b^3 - 6*A*b^2*c)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 +
b*x))*sqrt(c) - b))/c^(3/2)