Optimal. Leaf size=128 \[ -\frac {2 a \sqrt {a x^3+b x^n}}{c^4 (3-n) (c x)^{3/2}}-\frac {2 \left (a x^3+b x^n\right )^{3/2}}{3 c (3-n) (c x)^{9/2}}+\frac {2 a^{3/2} \sqrt {c x} \tanh ^{-1}\left (\frac {\sqrt {a} x^{3/2}}{\sqrt {a x^3+b x^n}}\right )}{c^6 (3-n) \sqrt {x}} \]
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Rubi [A]
time = 0.15, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2053, 2056,
2054, 212} \begin {gather*} \frac {2 a^{3/2} \sqrt {c x} \tanh ^{-1}\left (\frac {\sqrt {a} x^{3/2}}{\sqrt {a x^3+b x^n}}\right )}{c^6 (3-n) \sqrt {x}}-\frac {2 a \sqrt {a x^3+b x^n}}{c^4 (3-n) (c x)^{3/2}}-\frac {2 \left (a x^3+b x^n\right )^{3/2}}{3 c (3-n) (c x)^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2053
Rule 2054
Rule 2056
Rubi steps
\begin {align*} \int \frac {\left (a x^3+b x^n\right )^{3/2}}{(c x)^{11/2}} \, dx &=-\frac {2 \left (a x^3+b x^n\right )^{3/2}}{3 c (3-n) (c x)^{9/2}}+\frac {a \int \frac {\sqrt {a x^3+b x^n}}{(c x)^{5/2}} \, dx}{c^3}\\ &=-\frac {2 a \sqrt {a x^3+b x^n}}{c^4 (3-n) (c x)^{3/2}}-\frac {2 \left (a x^3+b x^n\right )^{3/2}}{3 c (3-n) (c x)^{9/2}}+\frac {a^2 \int \frac {\sqrt {c x}}{\sqrt {a x^3+b x^n}} \, dx}{c^6}\\ &=-\frac {2 a \sqrt {a x^3+b x^n}}{c^4 (3-n) (c x)^{3/2}}-\frac {2 \left (a x^3+b x^n\right )^{3/2}}{3 c (3-n) (c x)^{9/2}}+\frac {\left (a^2 \sqrt {c x}\right ) \int \frac {\sqrt {x}}{\sqrt {a x^3+b x^n}} \, dx}{c^6 \sqrt {x}}\\ &=-\frac {2 a \sqrt {a x^3+b x^n}}{c^4 (3-n) (c x)^{3/2}}-\frac {2 \left (a x^3+b x^n\right )^{3/2}}{3 c (3-n) (c x)^{9/2}}+\frac {\left (2 a^2 \sqrt {c x}\right ) \text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {x^{3/2}}{\sqrt {a x^3+b x^n}}\right )}{c^6 (3-n) \sqrt {x}}\\ &=-\frac {2 a \sqrt {a x^3+b x^n}}{c^4 (3-n) (c x)^{3/2}}-\frac {2 \left (a x^3+b x^n\right )^{3/2}}{3 c (3-n) (c x)^{9/2}}+\frac {2 a^{3/2} \sqrt {c x} \tanh ^{-1}\left (\frac {\sqrt {a} x^{3/2}}{\sqrt {a x^3+b x^n}}\right )}{c^6 (3-n) \sqrt {x}}\\ \end {align*}
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Mathematica [A]
time = 0.55, size = 126, normalized size = 0.98 \begin {gather*} \frac {2 \sqrt {c x} \left (4 a^2 x^6+b^2 x^{2 n}+5 a b x^{3+n}-3 a^{3/2} \sqrt {b} x^{\frac {9+n}{2}} \sqrt {1+\frac {a x^{3-n}}{b}} \sinh ^{-1}\left (\frac {\sqrt {a} x^{\frac {3}{2}-\frac {n}{2}}}{\sqrt {b}}\right )\right )}{3 c^6 (-3+n) x^5 \sqrt {a x^3+b x^n}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.10, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{3}+b \,x^{n}\right )^{\frac {3}{2}}}{\left (c x \right )^{\frac {11}{2}}}\, dx\]
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 [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 (b\,x^n+a\,x^3\right )}^{3/2}}{{\left (c\,x\right )}^{11/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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