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 = 0.03, antiderivative size = 78, normalized size of antiderivative = 1.00, 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 245
Rule 385
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*}
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Mathematica [A] time = 0.07, 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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fricas [F] time = 0.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{n} + a}{d^{4} x^{4 \, n} + 4 \, c d^{3} x^{3 \, n} + 6 \, c^{2} d^{2} x^{2 \, n} + 4 \, c^{3} d x^{n} + c^{4}}, x\right ) \]
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 \[ \int \frac {b x^{n} + a}{{\left (d x^{n} + c\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.58, size = 0, normalized size = 0.00 \[ \int \frac {b \,x^{n}+a}{\left (d \,x^{n}+c \right )^{4}}\, dx \]
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 \[ {\left ({\left (2 \, n^{2} - 3 \, n + 1\right )} b c + {\left (6 \, n^{3} - 11 \, n^{2} + 6 \, n - 1\right )} a d\right )} \int \frac {1}{6 \, {\left (c^{3} d^{2} n^{3} x^{n} + c^{4} d n^{3}\right )}}\,{d x} + \frac {{\left ({\left (6 \, n^{2} - 5 \, n + 1\right )} a d^{3} + b c d^{2} {\left (2 \, n - 1\right )}\right )} x x^{2 \, n} + {\left ({\left (15 \, n^{2} - 11 \, n + 2\right )} a c d^{2} + b c^{2} d {\left (5 \, n - 2\right )}\right )} x x^{n} - {\left ({\left (2 \, n^{2} - 3 \, n + 1\right )} b c^{3} - {\left (11 \, n^{2} - 6 \, n + 1\right )} a c^{2} d\right )} x}{6 \, {\left (c^{3} d^{4} n^{3} x^{3 \, n} + 3 \, c^{4} d^{3} n^{3} x^{2 \, n} + 3 \, c^{5} d^{2} n^{3} x^{n} + c^{6} d n^{3}\right )}} \]
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 \[ \int \frac {a+b\,x^n}{{\left (c+d\,x^n\right )}^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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