Optimal. Leaf size=115 \[ \frac {x (b c-a d) (a d (1-n)-b c (n+1)) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {d x^n}{c}\right )}{c^2 d^2 n}-\frac {b x (a d-b c (n+1))}{c d^2 n}-\frac {x (b c-a d) \left (a+b x^n\right )}{c d n \left (c+d x^n\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {413, 388, 245} \[ \frac {x (b c-a d) (a d (1-n)-b c (n+1)) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {d x^n}{c}\right )}{c^2 d^2 n}-\frac {b x (a d-b c (n+1))}{c d^2 n}-\frac {x (b c-a d) \left (a+b x^n\right )}{c d n \left (c+d x^n\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 245
Rule 388
Rule 413
Rubi steps
\begin {align*} \int \frac {\left (a+b x^n\right )^2}{\left (c+d x^n\right )^2} \, dx &=-\frac {(b c-a d) x \left (a+b x^n\right )}{c d n \left (c+d x^n\right )}+\frac {\int \frac {a (b c-a d (1-n))-b (a d-b c (1+n)) x^n}{c+d x^n} \, dx}{c d n}\\ &=-\frac {b (a d-b c (1+n)) x}{c d^2 n}-\frac {(b c-a d) x \left (a+b x^n\right )}{c d n \left (c+d x^n\right )}+\frac {((b c-a d) (a d (1-n)-b c (1+n))) \int \frac {1}{c+d x^n} \, dx}{c d^2 n}\\ &=-\frac {b (a d-b c (1+n)) x}{c d^2 n}-\frac {(b c-a d) x \left (a+b x^n\right )}{c d n \left (c+d x^n\right )}+\frac {(b c-a d) (a d (1-n)-b c (1+n)) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {d x^n}{c}\right )}{c^2 d^2 n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.16, size = 95, normalized size = 0.83 \[ \frac {x \left (\frac {c \left (a^2 d^2-2 a b c d+b^2 c \left (c n+c+d n x^n\right )\right )}{c+d x^n}-(b c-a d) (a d (n-1)+b c (n+1)) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {d x^n}{c}\right )\right )}{c^2 d^2 n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.97, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}}{d^{2} x^{2 \, n} + 2 \, c d x^{n} + c^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{n} + a\right )}^{2}}{{\left (d x^{n} + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.58, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{n}+a \right )^{2}}{\left (d \,x^{n}+c \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -{\left (b^{2} c^{2} {\left (n + 1\right )} - a^{2} d^{2} {\left (n - 1\right )} - 2 \, a b c d\right )} \int \frac {1}{c d^{3} n x^{n} + c^{2} d^{2} n}\,{d x} + \frac {b^{2} c d n x x^{n} + {\left (b^{2} c^{2} {\left (n + 1\right )} - 2 \, a b c d + a^{2} d^{2}\right )} x}{c d^{3} n x^{n} + c^{2} d^{2} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,x^n\right )}^2}{{\left (c+d\,x^n\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b x^{n}\right )^{2}}{\left (c + d x^{n}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________