Optimal. Leaf size=83 \[ -\frac {3 c^2 \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{4 \sqrt {b}}-\frac {3 c \sqrt {b x+c x^2}}{4 x^{3/2}}-\frac {\left (b x+c x^2\right )^{3/2}}{2 x^{7/2}} \]
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Rubi [A] time = 0.03, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {662, 660, 207} \begin {gather*} -\frac {3 c^2 \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{4 \sqrt {b}}-\frac {3 c \sqrt {b x+c x^2}}{4 x^{3/2}}-\frac {\left (b x+c x^2\right )^{3/2}}{2 x^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 207
Rule 660
Rule 662
Rubi steps
\begin {align*} \int \frac {\left (b x+c x^2\right )^{3/2}}{x^{9/2}} \, dx &=-\frac {\left (b x+c x^2\right )^{3/2}}{2 x^{7/2}}+\frac {1}{4} (3 c) \int \frac {\sqrt {b x+c x^2}}{x^{5/2}} \, dx\\ &=-\frac {3 c \sqrt {b x+c x^2}}{4 x^{3/2}}-\frac {\left (b x+c x^2\right )^{3/2}}{2 x^{7/2}}+\frac {1}{8} \left (3 c^2\right ) \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx\\ &=-\frac {3 c \sqrt {b x+c x^2}}{4 x^{3/2}}-\frac {\left (b x+c x^2\right )^{3/2}}{2 x^{7/2}}+\frac {1}{4} \left (3 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{-b+x^2} \, dx,x,\frac {\sqrt {b x+c x^2}}{\sqrt {x}}\right )\\ &=-\frac {3 c \sqrt {b x+c x^2}}{4 x^{3/2}}-\frac {\left (b x+c x^2\right )^{3/2}}{2 x^{7/2}}-\frac {3 c^2 \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{4 \sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 72, normalized size = 0.87 \begin {gather*} -\frac {2 b^2+3 c^2 x^2 \sqrt {\frac {c x}{b}+1} \tanh ^{-1}\left (\sqrt {\frac {c x}{b}+1}\right )+7 b c x+5 c^2 x^2}{4 x^{3/2} \sqrt {x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.56, size = 68, normalized size = 0.82 \begin {gather*} \frac {(-2 b-5 c x) \sqrt {b x+c x^2}}{4 x^{5/2}}-\frac {3 c^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {b x+c x^2}}\right )}{4 \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 153, normalized size = 1.84 \begin {gather*} \left [\frac {3 \, \sqrt {b} c^{2} x^{3} \log \left (-\frac {c x^{2} + 2 \, b x - 2 \, \sqrt {c x^{2} + b x} \sqrt {b} \sqrt {x}}{x^{2}}\right ) - 2 \, {\left (5 \, b c x + 2 \, b^{2}\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{8 \, b x^{3}}, \frac {3 \, \sqrt {-b} c^{2} x^{3} \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {c x^{2} + b x}}\right ) - {\left (5 \, b c x + 2 \, b^{2}\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{4 \, b x^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 64, normalized size = 0.77 \begin {gather*} \frac {\frac {3 \, c^{3} \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{\sqrt {-b}} - \frac {5 \, {\left (c x + b\right )}^{\frac {3}{2}} c^{3} - 3 \, \sqrt {c x + b} b c^{3}}{c^{2} x^{2}}}{4 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 72, normalized size = 0.87 \begin {gather*} -\frac {\sqrt {\left (c x +b \right ) x}\, \left (3 c^{2} x^{2} \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )+5 \sqrt {c x +b}\, \sqrt {b}\, c x +2 \sqrt {c x +b}\, b^{\frac {3}{2}}\right )}{4 \sqrt {c x +b}\, \sqrt {b}\, x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}}}{x^{\frac {9}{2}}}\,{d x} \end {gather*}
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 \begin {gather*} \int \frac {{\left (c\,x^2+b\,x\right )}^{3/2}}{x^{9/2}} \,d x \end {gather*}
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 \begin {gather*} \int \frac {\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}{x^{\frac {9}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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