Optimal. Leaf size=139 \[ -\frac {c \sqrt {b x+c x^2}}{8 x^{7/2}}-\frac {c^2 \sqrt {b x+c x^2}}{32 b x^{5/2}}+\frac {3 c^3 \sqrt {b x+c x^2}}{64 b^2 x^{3/2}}-\frac {\left (b x+c x^2\right )^{3/2}}{4 x^{11/2}}-\frac {3 c^4 \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{64 b^{5/2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {676, 686, 674,
213} \begin {gather*} -\frac {3 c^4 \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{64 b^{5/2}}+\frac {3 c^3 \sqrt {b x+c x^2}}{64 b^2 x^{3/2}}-\frac {c^2 \sqrt {b x+c x^2}}{32 b x^{5/2}}-\frac {c \sqrt {b x+c x^2}}{8 x^{7/2}}-\frac {\left (b x+c x^2\right )^{3/2}}{4 x^{11/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 213
Rule 674
Rule 676
Rule 686
Rubi steps
\begin {align*} \int \frac {\left (b x+c x^2\right )^{3/2}}{x^{13/2}} \, dx &=-\frac {\left (b x+c x^2\right )^{3/2}}{4 x^{11/2}}+\frac {1}{8} (3 c) \int \frac {\sqrt {b x+c x^2}}{x^{9/2}} \, dx\\ &=-\frac {c \sqrt {b x+c x^2}}{8 x^{7/2}}-\frac {\left (b x+c x^2\right )^{3/2}}{4 x^{11/2}}+\frac {1}{16} c^2 \int \frac {1}{x^{5/2} \sqrt {b x+c x^2}} \, dx\\ &=-\frac {c \sqrt {b x+c x^2}}{8 x^{7/2}}-\frac {c^2 \sqrt {b x+c x^2}}{32 b x^{5/2}}-\frac {\left (b x+c x^2\right )^{3/2}}{4 x^{11/2}}-\frac {\left (3 c^3\right ) \int \frac {1}{x^{3/2} \sqrt {b x+c x^2}} \, dx}{64 b}\\ &=-\frac {c \sqrt {b x+c x^2}}{8 x^{7/2}}-\frac {c^2 \sqrt {b x+c x^2}}{32 b x^{5/2}}+\frac {3 c^3 \sqrt {b x+c x^2}}{64 b^2 x^{3/2}}-\frac {\left (b x+c x^2\right )^{3/2}}{4 x^{11/2}}+\frac {\left (3 c^4\right ) \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx}{128 b^2}\\ &=-\frac {c \sqrt {b x+c x^2}}{8 x^{7/2}}-\frac {c^2 \sqrt {b x+c x^2}}{32 b x^{5/2}}+\frac {3 c^3 \sqrt {b x+c x^2}}{64 b^2 x^{3/2}}-\frac {\left (b x+c x^2\right )^{3/2}}{4 x^{11/2}}+\frac {\left (3 c^4\right ) \text {Subst}\left (\int \frac {1}{-b+x^2} \, dx,x,\frac {\sqrt {b x+c x^2}}{\sqrt {x}}\right )}{64 b^2}\\ &=-\frac {c \sqrt {b x+c x^2}}{8 x^{7/2}}-\frac {c^2 \sqrt {b x+c x^2}}{32 b x^{5/2}}+\frac {3 c^3 \sqrt {b x+c x^2}}{64 b^2 x^{3/2}}-\frac {\left (b x+c x^2\right )^{3/2}}{4 x^{11/2}}-\frac {3 c^4 \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{64 b^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 104, normalized size = 0.75 \begin {gather*} -\frac {\sqrt {x (b+c x)} \left (\sqrt {b} \sqrt {b+c x} \left (16 b^3+24 b^2 c x+2 b c^2 x^2-3 c^3 x^3\right )+3 c^4 x^4 \tanh ^{-1}\left (\frac {\sqrt {b+c x}}{\sqrt {b}}\right )\right )}{64 b^{5/2} x^{9/2} \sqrt {b+c x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.44, size = 108, normalized size = 0.78
method | result | size |
risch | \(-\frac {\left (c x +b \right ) \left (-3 c^{3} x^{3}+2 b \,c^{2} x^{2}+24 b^{2} c x +16 b^{3}\right )}{64 x^{\frac {7}{2}} b^{2} \sqrt {x \left (c x +b \right )}}-\frac {3 c^{4} \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right ) \sqrt {c x +b}\, \sqrt {x}}{64 b^{\frac {5}{2}} \sqrt {x \left (c x +b \right )}}\) | \(93\) |
default | \(-\frac {\sqrt {x \left (c x +b \right )}\, \left (3 \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right ) c^{4} x^{4}-3 c^{3} x^{3} \sqrt {b}\, \sqrt {c x +b}+2 b^{\frac {3}{2}} c^{2} x^{2} \sqrt {c x +b}+24 b^{\frac {5}{2}} c x \sqrt {c x +b}+16 b^{\frac {7}{2}} \sqrt {c x +b}\right )}{64 b^{\frac {5}{2}} x^{\frac {9}{2}} \sqrt {c x +b}}\) | \(108\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.85, size = 196, normalized size = 1.41 \begin {gather*} \left [\frac {3 \, \sqrt {b} c^{4} x^{5} \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 (3 \, b c^{3} x^{3} - 2 \, b^{2} c^{2} x^{2} - 24 \, b^{3} c x - 16 \, b^{4}\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{128 \, b^{3} x^{5}}, \frac {3 \, \sqrt {-b} c^{4} x^{5} \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {c x^{2} + b x}}\right ) + {\left (3 \, b c^{3} x^{3} - 2 \, b^{2} c^{2} x^{2} - 24 \, b^{3} c x - 16 \, b^{4}\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{64 \, b^{3} x^{5}}\right ] \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 {13}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.98, size = 99, normalized size = 0.71 \begin {gather*} \frac {\frac {3 \, c^{5} \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{2}} + \frac {3 \, {\left (c x + b\right )}^{\frac {7}{2}} c^{5} - 11 \, {\left (c x + b\right )}^{\frac {5}{2}} b c^{5} - 11 \, {\left (c x + b\right )}^{\frac {3}{2}} b^{2} c^{5} + 3 \, \sqrt {c x + b} b^{3} c^{5}}{b^{2} c^{4} x^{4}}}{64 \, c} \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^{13/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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