Optimal. Leaf size=179 \[ -\frac{3 \left (A \left (4 a c+b^2\right )+4 a b B\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{8 \sqrt{a}}+\frac{3 \left (4 a B c+4 A b c+b^2 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 \sqrt{c}}-\frac{(A-B x) \left (a+b x+c x^2\right )^{3/2}}{2 x^2}-\frac{3 \sqrt{a+b x+c x^2} (2 a B-x (2 A c+b B)+A b)}{4 x} \]
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Rubi [A] time = 0.44945, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ -\frac{3 \left (A \left (4 a c+b^2\right )+4 a b B\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{8 \sqrt{a}}+\frac{3 \left (4 a B c+4 A b c+b^2 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 \sqrt{c}}-\frac{(A-B x) \left (a+b x+c x^2\right )^{3/2}}{2 x^2}-\frac{3 \sqrt{a+b x+c x^2} (2 a B-x (2 A c+b B)+A b)}{4 x} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^3,x]
[Out]
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Rubi in Sympy [A] time = 54.2547, size = 178, normalized size = 0.99 \[ - \frac{3 \sqrt{a + b x + c x^{2}} \left (2 A b + 4 B a - x \left (4 A c + 2 B b\right )\right )}{8 x} - \frac{\left (2 A - 2 B x\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{4 x^{2}} + \frac{3 \left (4 A b c + 4 B a c + B b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{8 \sqrt{c}} - \frac{3 \left (4 A a c + A b^{2} + 4 B a b\right ) \operatorname{atanh}{\left (\frac{2 a + b x}{2 \sqrt{a} \sqrt{a + b x + c x^{2}}} \right )}}{8 \sqrt{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+b*x+a)**(3/2)/x**3,x)
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Mathematica [A] time = 0.770672, size = 183, normalized size = 1.02 \[ \frac{1}{8} \left (\frac{3 \log (x) \left (A \left (4 a c+b^2\right )+4 a b B\right )}{\sqrt{a}}-\frac{3 \left (A \left (4 a c+b^2\right )+4 a b B\right ) \log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )}{\sqrt{a}}+\frac{3 \left (4 a B c+4 A b c+b^2 B\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{\sqrt{c}}+\frac{2 \sqrt{a+x (b+c x)} (x (A (4 c x-5 b)+B x (5 b+2 c x))-2 a (A+2 B x))}{x^2}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^3,x]
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Maple [B] time = 0.016, size = 463, normalized size = 2.6 \[ -{\frac{A}{2\,a{x}^{2}} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{5}{2}}}}-{\frac{Ab}{4\,{a}^{2}x} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{5}{2}}}}+{\frac{{b}^{2}A}{4\,{a}^{2}} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{b}^{2}A}{4\,a}\sqrt{c{x}^{2}+bx+a}}-{\frac{3\,{b}^{2}A}{8}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}+{\frac{Abcx}{4\,{a}^{2}} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,Abcx}{4\,a}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,Ab}{2}\sqrt{c}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ) }+{\frac{Ac}{2\,a} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,Ac}{2}\sqrt{c{x}^{2}+bx+a}}-{\frac{3\,Ac}{2}\sqrt{a}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ) }-{\frac{B}{ax} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{5}{2}}}}+{\frac{Bb}{a} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{b}^{2}B}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{9\,Bb}{4}\sqrt{c{x}^{2}+bx+a}}-{\frac{3\,Bb}{2}\sqrt{a}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ) }+{\frac{Bcx}{a} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,Bcx}{2}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,Ba}{2}\sqrt{c}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+b*x+a)^(3/2)/x^3,x)
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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 + a)^(3/2)*(B*x + A)/x^3,x, algorithm="maxima")
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Fricas [A] time = 1.34766, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^(3/2)*(B*x + A)/x^3,x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+b*x+a)**(3/2)/x**3,x)
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GIAC/XCAS [A] time = 0.92563, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^(3/2)*(B*x + A)/x^3,x, algorithm="giac")
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