Optimal. Leaf size=133 \[ \frac {b x \left (c+d x^n\right )^{3-\frac {1}{n}}}{3 a (b c-a d) n \left (a+b x^n\right )^3}-\frac {c^2 (b c (1-3 n)+3 a d n) x \left (c+d x^n\right )^{-1/n} \, _2F_1\left (3,\frac {1}{n};1+\frac {1}{n};-\frac {(b c-a d) x^n}{a \left (c+d x^n\right )}\right )}{3 a^4 (b c-a d) n} \]
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Rubi [A]
time = 0.04, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {390, 387}
\begin {gather*} \frac {b x \left (c+d x^n\right )^{3-\frac {1}{n}}}{3 a n (b c-a d) \left (a+b x^n\right )^3}-\frac {c^2 x \left (c+d x^n\right )^{-1/n} (3 a d n+b c (1-3 n)) \, _2F_1\left (3,\frac {1}{n};1+\frac {1}{n};-\frac {(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{3 a^4 n (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 387
Rule 390
Rubi steps
\begin {align*} \int \frac {\left (c+d x^n\right )^{2-\frac {1}{n}}}{\left (a+b x^n\right )^4} \, dx &=\frac {b x \left (c+d x^n\right )^{3-\frac {1}{n}}}{3 a (b c-a d) n \left (a+b x^n\right )^3}-\frac {(b c-3 (b c-a d) n) \int \frac {\left (c+d x^n\right )^{2-\frac {1}{n}}}{\left (a+b x^n\right )^3} \, dx}{3 a (b c-a d) n}\\ &=\frac {b x \left (c+d x^n\right )^{3-\frac {1}{n}}}{3 a (b c-a d) n \left (a+b x^n\right )^3}-\frac {c^2 (b c (1-3 n)+3 a d n) x \left (c+d x^n\right )^{-1/n} \, _2F_1\left (3,\frac {1}{n};1+\frac {1}{n};-\frac {(b c-a d) x^n}{a \left (c+d x^n\right )}\right )}{3 a^4 (b c-a d) n}\\ \end {align*}
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Mathematica [F]
time = 180.01, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.10, size = 0, normalized size = 0.00 \[\int \frac {\left (c +d \,x^{n}\right )^{2-\frac {1}{n}}}{\left (a +b \,x^{n}\right )^{4}}\, dx\]
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]
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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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \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+d\,x^n\right )}^{2-\frac {1}{n}}}{{\left (a+b\,x^n\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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