Optimal. Leaf size=210 \[ -\frac{d x \left (a^2 d^2 \left (2 n^2-3 n+1\right )-2 a b c d \left (3 n^2-4 n+1\right )+b^2 c^2 \left (6 n^2-5 n+1\right )\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{2 c^3 n^2 (b c-a d)^3}+\frac{b^3 x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a (b c-a d)^3}-\frac{d x (a d (1-2 n)-b (c-4 c n))}{2 c^2 n^2 (b c-a d)^2 \left (c+d x^n\right )}-\frac{d x}{2 c n (b c-a d) \left (c+d x^n\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.328671, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {414, 527, 522, 245} \[ -\frac{d x \left (a^2 d^2 \left (2 n^2-3 n+1\right )-2 a b c d \left (3 n^2-4 n+1\right )+b^2 c^2 \left (6 n^2-5 n+1\right )\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{2 c^3 n^2 (b c-a d)^3}+\frac{b^3 x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a (b c-a d)^3}-\frac{d x (a d (1-2 n)-b (c-4 c n))}{2 c^2 n^2 (b c-a d)^2 \left (c+d x^n\right )}-\frac{d x}{2 c n (b c-a d) \left (c+d x^n\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 414
Rule 527
Rule 522
Rule 245
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b x^n\right ) \left (c+d x^n\right )^3} \, dx &=-\frac{d x}{2 c (b c-a d) n \left (c+d x^n\right )^2}+\frac{\int \frac{2 b c n+a (d-2 d n)+b d (1-2 n) x^n}{\left (a+b x^n\right ) \left (c+d x^n\right )^2} \, dx}{2 c (b c-a d) n}\\ &=-\frac{d x}{2 c (b c-a d) n \left (c+d x^n\right )^2}+\frac{d (b c (1-4 n)-a d (1-2 n)) x}{2 c^2 (b c-a d)^2 n^2 \left (c+d x^n\right )}+\frac{\int \frac{2 b^2 c^2 n^2+a^2 d^2 \left (1-3 n+2 n^2\right )-a b c d \left (1-5 n+4 n^2\right )-b d (b c (1-4 n)-a d (1-2 n)) (1-n) x^n}{\left (a+b x^n\right ) \left (c+d x^n\right )} \, dx}{2 c^2 (b c-a d)^2 n^2}\\ &=-\frac{d x}{2 c (b c-a d) n \left (c+d x^n\right )^2}+\frac{d (b c (1-4 n)-a d (1-2 n)) x}{2 c^2 (b c-a d)^2 n^2 \left (c+d x^n\right )}+\frac{b^3 \int \frac{1}{a+b x^n} \, dx}{(b c-a d)^3}-\frac{\left (d \left (a^2 d^2 \left (1-3 n+2 n^2\right )-2 a b c d \left (1-4 n+3 n^2\right )+b^2 c^2 \left (1-5 n+6 n^2\right )\right )\right ) \int \frac{1}{c+d x^n} \, dx}{2 c^2 (b c-a d)^3 n^2}\\ &=-\frac{d x}{2 c (b c-a d) n \left (c+d x^n\right )^2}+\frac{d (b c (1-4 n)-a d (1-2 n)) x}{2 c^2 (b c-a d)^2 n^2 \left (c+d x^n\right )}+\frac{b^3 x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a (b c-a d)^3}-\frac{d \left (a^2 d^2 \left (1-3 n+2 n^2\right )-2 a b c d \left (1-4 n+3 n^2\right )+b^2 c^2 \left (1-5 n+6 n^2\right )\right ) x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{2 c^3 (b c-a d)^3 n^2}\\ \end{align*}
Mathematica [A] time = 0.197811, size = 210, normalized size = 1. \[ \frac{x \left (-a d \left (c+d x^n\right )^2 \left (a^2 d^2 \left (2 n^2-3 n+1\right )-2 a b c d \left (3 n^2-4 n+1\right )+b^2 c^2 \left (6 n^2-5 n+1\right )\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )+2 b^3 c^3 n^2 \left (c+d x^n\right )^2 \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )-a c^2 d n (b c-a d)^2+a c d (b c-a d) \left (c+d x^n\right ) (a d (2 n-1)+b (c-4 c n))\right )}{2 a c^3 n^2 (b c-a d)^3 \left (c+d x^n\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.757, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( a+b{x}^{n} \right ) \left ( c+d{x}^{n} \right ) ^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -b^{3} \int -\frac{1}{a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3} +{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{n}}\,{d x} +{\left ({\left (6 \, n^{2} - 5 \, n + 1\right )} b^{2} c^{2} d - 2 \,{\left (3 \, n^{2} - 4 \, n + 1\right )} a b c d^{2} +{\left (2 \, n^{2} - 3 \, n + 1\right )} a^{2} d^{3}\right )} \int -\frac{1}{2 \,{\left (b^{3} c^{6} n^{2} - 3 \, a b^{2} c^{5} d n^{2} + 3 \, a^{2} b c^{4} d^{2} n^{2} - a^{3} c^{3} d^{3} n^{2} +{\left (b^{3} c^{5} d n^{2} - 3 \, a b^{2} c^{4} d^{2} n^{2} + 3 \, a^{2} b c^{3} d^{3} n^{2} - a^{3} c^{2} d^{4} n^{2}\right )} x^{n}\right )}}\,{d x} - \frac{{\left (b c d^{2}{\left (4 \, n - 1\right )} - a d^{3}{\left (2 \, n - 1\right )}\right )} x x^{n} +{\left (b c^{2} d{\left (5 \, n - 1\right )} - a c d^{2}{\left (3 \, n - 1\right )}\right )} x}{2 \,{\left (b^{2} c^{6} n^{2} - 2 \, a b c^{5} d n^{2} + a^{2} c^{4} d^{2} n^{2} +{\left (b^{2} c^{4} d^{2} n^{2} - 2 \, a b c^{3} d^{3} n^{2} + a^{2} c^{2} d^{4} n^{2}\right )} x^{2 \, n} + 2 \,{\left (b^{2} c^{5} d n^{2} - 2 \, a b c^{4} d^{2} n^{2} + a^{2} c^{3} d^{3} n^{2}\right )} x^{n}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{b d^{3} x^{4 \, n} + a c^{3} +{\left (3 \, b c d^{2} + a d^{3}\right )} x^{3 \, n} + 3 \,{\left (b c^{2} d + a c d^{2}\right )} x^{2 \, n} +{\left (b c^{3} + 3 \, a c^{2} d\right )} x^{n}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b x^{n}\right ) \left (c + d x^{n}\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{n} + a\right )}{\left (d x^{n} + c\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]