Optimal. Leaf size=111 \[ \frac {2 x^{1-2 n} (c x)^m \sqrt {1+\frac {a x^{j-n}}{b}} \, _2F_1\left (\frac {5}{2},\frac {1+m-\frac {5 n}{2}}{j-n};1+\frac {1+m-\frac {5 n}{2}}{j-n};-\frac {a x^{j-n}}{b}\right )}{b^2 (2+2 m-5 n) \sqrt {a x^j+b x^n}} \]
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Rubi [A]
time = 0.07, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2057, 372, 371}
\begin {gather*} \frac {2 x^{1-2 n} (c x)^m \sqrt {\frac {a x^{j-n}}{b}+1} \, _2F_1\left (\frac {5}{2},\frac {m-\frac {5 n}{2}+1}{j-n};\frac {m-\frac {5 n}{2}+1}{j-n}+1;-\frac {a x^{j-n}}{b}\right )}{b^2 (2 m-5 n+2) \sqrt {a x^j+b x^n}} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 372
Rule 2057
Rubi steps
\begin {align*} \int \frac {(c x)^m}{\left (a x^j+b x^n\right )^{5/2}} \, dx &=\frac {\left (x^{-m+\frac {n}{2}} (c x)^m \sqrt {b+a x^{j-n}}\right ) \int \frac {x^{m-\frac {5 n}{2}}}{\left (b+a x^{j-n}\right )^{5/2}} \, dx}{\sqrt {a x^j+b x^n}}\\ &=\frac {\left (x^{-m+\frac {n}{2}} (c x)^m \sqrt {1+\frac {a x^{j-n}}{b}}\right ) \int \frac {x^{m-\frac {5 n}{2}}}{\left (1+\frac {a x^{j-n}}{b}\right )^{5/2}} \, dx}{b^2 \sqrt {a x^j+b x^n}}\\ &=\frac {2 x^{1-2 n} (c x)^m \sqrt {1+\frac {a x^{j-n}}{b}} \, _2F_1\left (\frac {5}{2},\frac {1+m-\frac {5 n}{2}}{j-n};1+\frac {1+m-\frac {5 n}{2}}{j-n};-\frac {a x^{j-n}}{b}\right )}{b^2 (2+2 m-5 n) \sqrt {a x^j+b x^n}}\\ \end {align*}
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Mathematica [A]
time = 0.36, size = 166, normalized size = 1.50 \begin {gather*} \frac {2 x^{1-2 j} (c x)^m \left (-2+2 j-2 m+3 n-\frac {a (j-n) x^j}{a x^j+b x^n}-(-2+2 j-2 m+3 n) \sqrt {1+\frac {a x^{j-n}}{b}} \, _2F_1\left (\frac {1}{2},\frac {2-4 j+2 m-n}{2 j-2 n};\frac {2-2 j+2 m-3 n}{2 j-2 n};-\frac {a x^{j-n}}{b}\right )\right )}{3 a^2 (j-n)^2 \sqrt {a x^j+b x^n}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.11, size = 0, normalized size = 0.00 \[\int \frac {\left (c x \right )^{m}}{\left (a \,x^{j}+b \,x^{n}\right )^{\frac {5}{2}}}\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c x\right )^{m}}{\left (a x^{j} + b x^{n}\right )^{\frac {5}{2}}}\, dx \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 (c\,x\right )}^m}{{\left (a\,x^j+b\,x^n\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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