Optimal. Leaf size=102 \[ -\frac {5 b^3 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 c^{7/2}}+\frac {5 b^2 \sqrt {b x+c x^2}}{8 c^3}-\frac {5 b x \sqrt {b x+c x^2}}{12 c^2}+\frac {x^2 \sqrt {b x+c x^2}}{3 c} \]
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Rubi [A] time = 0.04, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {670, 640, 620, 206} \[ \frac {5 b^2 \sqrt {b x+c x^2}}{8 c^3}-\frac {5 b^3 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 c^{7/2}}-\frac {5 b x \sqrt {b x+c x^2}}{12 c^2}+\frac {x^2 \sqrt {b x+c x^2}}{3 c} \]
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 640
Rule 670
Rubi steps
\begin {align*} \int \frac {x^3}{\sqrt {b x+c x^2}} \, dx &=\frac {x^2 \sqrt {b x+c x^2}}{3 c}-\frac {(5 b) \int \frac {x^2}{\sqrt {b x+c x^2}} \, dx}{6 c}\\ &=-\frac {5 b x \sqrt {b x+c x^2}}{12 c^2}+\frac {x^2 \sqrt {b x+c x^2}}{3 c}+\frac {\left (5 b^2\right ) \int \frac {x}{\sqrt {b x+c x^2}} \, dx}{8 c^2}\\ &=\frac {5 b^2 \sqrt {b x+c x^2}}{8 c^3}-\frac {5 b x \sqrt {b x+c x^2}}{12 c^2}+\frac {x^2 \sqrt {b x+c x^2}}{3 c}-\frac {\left (5 b^3\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{16 c^3}\\ &=\frac {5 b^2 \sqrt {b x+c x^2}}{8 c^3}-\frac {5 b x \sqrt {b x+c x^2}}{12 c^2}+\frac {x^2 \sqrt {b x+c x^2}}{3 c}-\frac {\left (5 b^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{8 c^3}\\ &=\frac {5 b^2 \sqrt {b x+c x^2}}{8 c^3}-\frac {5 b x \sqrt {b x+c x^2}}{12 c^2}+\frac {x^2 \sqrt {b x+c x^2}}{3 c}-\frac {5 b^3 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 c^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 87, normalized size = 0.85 \[ \frac {\sqrt {x (b+c x)} \left (\sqrt {c} \left (15 b^2-10 b c x+8 c^2 x^2\right )-\frac {15 b^{5/2} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}\right )}{24 c^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.99, size = 147, normalized size = 1.44 \[ \left [\frac {15 \, b^{3} \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (8 \, c^{3} x^{2} - 10 \, b c^{2} x + 15 \, b^{2} c\right )} \sqrt {c x^{2} + b x}}{48 \, c^{4}}, \frac {15 \, b^{3} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (8 \, c^{3} x^{2} - 10 \, b c^{2} x + 15 \, b^{2} c\right )} \sqrt {c x^{2} + b x}}{24 \, c^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 77, normalized size = 0.75 \[ \frac {1}{24} \, \sqrt {c x^{2} + b x} {\left (2 \, x {\left (\frac {4 \, x}{c} - \frac {5 \, b}{c^{2}}\right )} + \frac {15 \, b^{2}}{c^{3}}\right )} + \frac {5 \, b^{3} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{16 \, c^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 90, normalized size = 0.88 \[ \frac {\sqrt {c \,x^{2}+b x}\, x^{2}}{3 c}-\frac {5 b^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{16 c^{\frac {7}{2}}}-\frac {5 \sqrt {c \,x^{2}+b x}\, b x}{12 c^{2}}+\frac {5 \sqrt {c \,x^{2}+b x}\, b^{2}}{8 c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 88, normalized size = 0.86 \[ \frac {\sqrt {c x^{2} + b x} x^{2}}{3 \, c} - \frac {5 \, \sqrt {c x^{2} + b x} b x}{12 \, c^{2}} - \frac {5 \, b^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{16 \, c^{\frac {7}{2}}} + \frac {5 \, \sqrt {c x^{2} + b x} b^{2}}{8 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^3}{\sqrt {c\,x^2+b\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\sqrt {x \left (b + c x\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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