Optimal. Leaf size=288 \[ -\frac {e (1-n) x^{n+1} \left (3 c d^2-a e^2\right ) \, _2F_1\left (1,\frac {n+1}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{2 a^2 c n (n+1)}-\frac {d (1-2 n) x \left (c d^2-3 a e^2\right ) \, _2F_1\left (1,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{2 a^2 c n}+\frac {x \left (e x^n \left (3 c d^2-a e^2\right )+d \left (c d^2-3 a e^2\right )\right )}{2 a c n \left (a+c x^{2 n}\right )}+\frac {3 d e^2 x \, _2F_1\left (1,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a c}+\frac {e^3 x^{n+1} \, _2F_1\left (1,\frac {n+1}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a c (n+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.25, antiderivative size = 288, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {1437, 1431, 1418, 245, 364} \[ -\frac {e (1-n) x^{n+1} \left (3 c d^2-a e^2\right ) \, _2F_1\left (1,\frac {n+1}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{2 a^2 c n (n+1)}-\frac {d (1-2 n) x \left (c d^2-3 a e^2\right ) \, _2F_1\left (1,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{2 a^2 c n}+\frac {x \left (e x^n \left (3 c d^2-a e^2\right )+d \left (c d^2-3 a e^2\right )\right )}{2 a c n \left (a+c x^{2 n}\right )}+\frac {3 d e^2 x \, _2F_1\left (1,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a c}+\frac {e^3 x^{n+1} \, _2F_1\left (1,\frac {n+1}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a c (n+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 245
Rule 364
Rule 1418
Rule 1431
Rule 1437
Rubi steps
\begin {align*} \int \frac {\left (d+e x^n\right )^3}{\left (a+c x^{2 n}\right )^2} \, dx &=\int \left (\frac {c d^3-3 a d e^2+\left (3 c d^2 e-a e^3\right ) x^n}{c \left (a+c x^{2 n}\right )^2}+\frac {e^2 \left (3 d+e x^n\right )}{c \left (a+c x^{2 n}\right )}\right ) \, dx\\ &=\frac {\int \frac {c d^3-3 a d e^2+\left (3 c d^2 e-a e^3\right ) x^n}{\left (a+c x^{2 n}\right )^2} \, dx}{c}+\frac {e^2 \int \frac {3 d+e x^n}{a+c x^{2 n}} \, dx}{c}\\ &=\frac {x \left (d \left (c d^2-3 a e^2\right )+e \left (3 c d^2-a e^2\right ) x^n\right )}{2 a c n \left (a+c x^{2 n}\right )}+\frac {\left (3 d e^2\right ) \int \frac {1}{a+c x^{2 n}} \, dx}{c}+\frac {e^3 \int \frac {x^n}{a+c x^{2 n}} \, dx}{c}-\frac {\int \frac {\left (c d^3-3 a d e^2\right ) (1-2 n)+\left (3 c d^2 e-a e^3\right ) (1-n) x^n}{a+c x^{2 n}} \, dx}{2 a c n}\\ &=\frac {x \left (d \left (c d^2-3 a e^2\right )+e \left (3 c d^2-a e^2\right ) x^n\right )}{2 a c n \left (a+c x^{2 n}\right )}+\frac {3 d e^2 x \, _2F_1\left (1,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a c}+\frac {e^3 x^{1+n} \, _2F_1\left (1,\frac {1+n}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a c (1+n)}-\frac {\left (d \left (c d^2-3 a e^2\right ) (1-2 n)\right ) \int \frac {1}{a+c x^{2 n}} \, dx}{2 a c n}-\frac {\left (e \left (3 c d^2-a e^2\right ) (1-n)\right ) \int \frac {x^n}{a+c x^{2 n}} \, dx}{2 a c n}\\ &=\frac {x \left (d \left (c d^2-3 a e^2\right )+e \left (3 c d^2-a e^2\right ) x^n\right )}{2 a c n \left (a+c x^{2 n}\right )}+\frac {3 d e^2 x \, _2F_1\left (1,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a c}-\frac {d \left (c d^2-3 a e^2\right ) (1-2 n) x \, _2F_1\left (1,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{2 a^2 c n}+\frac {e^3 x^{1+n} \, _2F_1\left (1,\frac {1+n}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a c (1+n)}-\frac {e \left (3 c d^2-a e^2\right ) (1-n) x^{1+n} \, _2F_1\left (1,\frac {1+n}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{2 a^2 c n (1+n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.27, size = 188, normalized size = 0.65 \[ \frac {x \left (d \left (c d^2-3 a e^2\right ) \, _2F_1\left (2,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )+\frac {e x^n \left (3 c d^2-a e^2\right ) \, _2F_1\left (2,\frac {n+1}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{n+1}+3 a d e^2 \, _2F_1\left (1,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )+\frac {a e^3 x^n \, _2F_1\left (1,\frac {n+1}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{n+1}\right )}{a^2 c} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.11, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {e^{3} x^{3 \, n} + 3 \, d e^{2} x^{2 \, n} + 3 \, d^{2} e x^{n} + d^{3}}{c^{2} x^{4 \, n} + 2 \, a c x^{2 \, n} + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x^{n} + d\right )}^{3}}{{\left (c x^{2 \, n} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \,x^{n}+d \right )^{3}}{\left (c \,x^{2 n}+a \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (3 \, c d^{2} e - a e^{3}\right )} x x^{n} + {\left (c d^{3} - 3 \, a d e^{2}\right )} x}{2 \, {\left (a c^{2} n x^{2 \, n} + a^{2} c n\right )}} + \int \frac {c d^{3} {\left (2 \, n - 1\right )} + 3 \, a d e^{2} + {\left (a e^{3} {\left (n + 1\right )} + 3 \, c d^{2} e {\left (n - 1\right )}\right )} x^{n}}{2 \, {\left (a c^{2} n x^{2 \, n} + a^{2} c n\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (d+e\,x^n\right )}^3}{{\left (a+c\,x^{2\,n}\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________