Optimal. Leaf size=78 \[ \frac{x (b c-a d (1-3 n)) \, _2F_1\left (3,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{3 c^4 d n}-\frac{x (b c-a d)}{3 c d n \left (c+d x^n\right )^3} \]
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Rubi [A] time = 0.0307568, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {385, 245} \[ \frac{x (b c-a d (1-3 n)) \, _2F_1\left (3,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{3 c^4 d n}-\frac{x (b c-a d)}{3 c d n \left (c+d x^n\right )^3} \]
Antiderivative was successfully verified.
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Rule 385
Rule 245
Rubi steps
\begin{align*} \int \frac{a+b x^n}{\left (c+d x^n\right )^4} \, dx &=-\frac{(b c-a d) x}{3 c d n \left (c+d x^n\right )^3}+\frac{(b c-a d (1-3 n)) \int \frac{1}{\left (c+d x^n\right )^3} \, dx}{3 c d n}\\ &=-\frac{(b c-a d) x}{3 c d n \left (c+d x^n\right )^3}+\frac{(b c-a d (1-3 n)) x \, _2F_1\left (3,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{3 c^4 d n}\\ \end{align*}
Mathematica [A] time = 0.0486125, size = 58, normalized size = 0.74 \[ \frac{x \left (\frac{b}{\left (c+d x^n\right )^3}-\frac{(a d (3 n-1)+b c) \, _2F_1\left (4,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{c^4}\right )}{d-3 d n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.373, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b{x}^{n}}{ \left ( c+d{x}^{n} \right ) ^{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\left ({\left (2 \, n^{2} - 3 \, n + 1\right )} b c +{\left (6 \, n^{3} - 11 \, n^{2} + 6 \, n - 1\right )} a d\right )} \int \frac{1}{6 \,{\left (c^{3} d^{2} n^{3} x^{n} + c^{4} d n^{3}\right )}}\,{d x} + \frac{{\left ({\left (6 \, n^{2} - 5 \, n + 1\right )} a d^{3} + b c d^{2}{\left (2 \, n - 1\right )}\right )} x x^{2 \, n} +{\left ({\left (15 \, n^{2} - 11 \, n + 2\right )} a c d^{2} + b c^{2} d{\left (5 \, n - 2\right )}\right )} x x^{n} -{\left ({\left (2 \, n^{2} - 3 \, n + 1\right )} b c^{3} -{\left (11 \, n^{2} - 6 \, n + 1\right )} a c^{2} d\right )} x}{6 \,{\left (c^{3} d^{4} n^{3} x^{3 \, n} + 3 \, c^{4} d^{3} n^{3} x^{2 \, n} + 3 \, c^{5} d^{2} n^{3} x^{n} + c^{6} d n^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b x^{n} + a}{d^{4} x^{4 \, n} + 4 \, c d^{3} x^{3 \, n} + 6 \, c^{2} d^{2} x^{2 \, n} + 4 \, c^{3} d x^{n} + c^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b x^{n} + a}{{\left (d x^{n} + c\right )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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