Optimal. Leaf size=200 \[ -\frac {x (b c-a d)^2 (b c (1-n)-a d (2 n+1)) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a^2 b^3 n}-\frac {d x \left (a^2 d^2 \left (2 n^2+3 n+1\right )-a b c d \left (3 n^2+4 n+2\right )+b^2 c^2 (n+1)\right )}{a b^3 n (n+1)}-\frac {d x \left (c+d x^n\right ) (b c (n+1)-a d (2 n+1))}{a b^2 n (n+1)}+\frac {x (b c-a d) \left (c+d x^n\right )^2}{a b n \left (a+b x^n\right )} \]
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Rubi [A] time = 0.26, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {413, 528, 388, 245} \[ -\frac {d x \left (a^2 d^2 \left (2 n^2+3 n+1\right )-a b c d \left (3 n^2+4 n+2\right )+b^2 c^2 (n+1)\right )}{a b^3 n (n+1)}-\frac {x (b c-a d)^2 (b c (1-n)-a d (2 n+1)) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a^2 b^3 n}-\frac {d x \left (c+d x^n\right ) (b c (n+1)-a d (2 n+1))}{a b^2 n (n+1)}+\frac {x (b c-a d) \left (c+d x^n\right )^2}{a b n \left (a+b x^n\right )} \]
Antiderivative was successfully verified.
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Rule 245
Rule 388
Rule 413
Rule 528
Rubi steps
\begin {align*} \int \frac {\left (c+d x^n\right )^3}{\left (a+b x^n\right )^2} \, dx &=\frac {(b c-a d) x \left (c+d x^n\right )^2}{a b n \left (a+b x^n\right )}+\frac {\int \frac {\left (c+d x^n\right ) \left (c (a d-b c (1-n))-d (b c (1+n)-a d (1+2 n)) x^n\right )}{a+b x^n} \, dx}{a b n}\\ &=-\frac {d (b c (1+n)-a d (1+2 n)) x \left (c+d x^n\right )}{a b^2 n (1+n)}+\frac {(b c-a d) x \left (c+d x^n\right )^2}{a b n \left (a+b x^n\right )}+\frac {\int \frac {c \left (2 a b c d (1+n)-a^2 d^2 (1+2 n)-b^2 c^2 \left (1-n^2\right )\right )-d \left (b^2 c^2 (1+n)+a^2 d^2 \left (1+3 n+2 n^2\right )-a b c d \left (2+4 n+3 n^2\right )\right ) x^n}{a+b x^n} \, dx}{a b^2 n (1+n)}\\ &=-\frac {d \left (b^2 c^2 (1+n)+a^2 d^2 \left (1+3 n+2 n^2\right )-a b c d \left (2+4 n+3 n^2\right )\right ) x}{a b^3 n (1+n)}-\frac {d (b c (1+n)-a d (1+2 n)) x \left (c+d x^n\right )}{a b^2 n (1+n)}+\frac {(b c-a d) x \left (c+d x^n\right )^2}{a b n \left (a+b x^n\right )}-\frac {\left ((b c-a d)^2 (b c (1-n)-a d (1+2 n))\right ) \int \frac {1}{a+b x^n} \, dx}{a b^3 n}\\ &=-\frac {d \left (b^2 c^2 (1+n)+a^2 d^2 \left (1+3 n+2 n^2\right )-a b c d \left (2+4 n+3 n^2\right )\right ) x}{a b^3 n (1+n)}-\frac {d (b c (1+n)-a d (1+2 n)) x \left (c+d x^n\right )}{a b^2 n (1+n)}+\frac {(b c-a d) x \left (c+d x^n\right )^2}{a b n \left (a+b x^n\right )}-\frac {(b c-a d)^2 (b c (1-n)-a d (1+2 n)) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a^2 b^3 n}\\ \end {align*}
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Mathematica [A] time = 5.38, size = 167, normalized size = 0.84 \[ \frac {x \left (\frac {a \left (-a^3 d^3 \left (2 n^2+3 n+1\right )+a^2 b d^2 \left (3 c (n+1)^2-d n (2 n+1) x^n\right )+a b^2 d \left (-3 c^2 (n+1)+3 c d n (n+1) x^n+d^2 n x^{2 n}\right )+b^3 c^3 (n+1)\right )}{(n+1) \left (a+b x^n\right )}+(b c-a d)^2 (a d (2 n+1)+b c (n-1)) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )\right )}{a^2 b^3 n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {d^{3} x^{3 \, n} + 3 \, c d^{2} x^{2 \, n} + 3 \, c^{2} d x^{n} + c^{3}}{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}}, x\right ) \]
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 \[ \int \frac {{\left (d x^{n} + c\right )}^{3}}{{\left (b x^{n} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.68, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \,x^{n}+c \right )^{3}}{\left (b \,x^{n}+a \right )^{2}}\, dx \]
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 \[ {\left (a^{3} d^{3} {\left (2 \, n + 1\right )} - 3 \, a^{2} b c d^{2} {\left (n + 1\right )} + b^{3} c^{3} {\left (n - 1\right )} + 3 \, a b^{2} c^{2} d\right )} \int \frac {1}{a b^{4} n x^{n} + a^{2} b^{3} n}\,{d x} + \frac {a b^{2} d^{3} n x x^{2 \, n} + {\left (3 \, {\left (n^{2} + n\right )} a b^{2} c d^{2} - {\left (2 \, n^{2} + n\right )} a^{2} b d^{3}\right )} x x^{n} + {\left (3 \, {\left (n^{2} + 2 \, n + 1\right )} a^{2} b c d^{2} - {\left (2 \, n^{2} + 3 \, n + 1\right )} a^{3} d^{3} + b^{3} c^{3} {\left (n + 1\right )} - 3 \, a b^{2} c^{2} d {\left (n + 1\right )}\right )} x}{{\left (n^{2} + n\right )} a b^{4} x^{n} + {\left (n^{2} + n\right )} a^{2} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c+d\,x^n\right )}^3}{{\left (a+b\,x^n\right )}^2} \,d x \]
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 \[ \int \frac {\left (c + d x^{n}\right )^{3}}{\left (a + b x^{n}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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