Optimal. Leaf size=90 \[ -\frac {c (b c-a d)^2 \log \left (c+d x^n\right )}{d^4 n}+\frac {x^n (b c-a d)^2}{d^3 n}-\frac {b x^{2 n} (b c-2 a d)}{2 d^2 n}+\frac {b^2 x^{3 n}}{3 d n} \]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {446, 77} \begin {gather*} \frac {x^n (b c-a d)^2}{d^3 n}-\frac {b x^{2 n} (b c-2 a d)}{2 d^2 n}-\frac {c (b c-a d)^2 \log \left (c+d x^n\right )}{d^4 n}+\frac {b^2 x^{3 n}}{3 d n} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 77
Rule 446
Rubi steps
\begin {align*} \int \frac {x^{-1+2 n} \left (a+b x^n\right )^2}{c+d x^n} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x (a+b x)^2}{c+d x} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {(-b c+a d)^2}{d^3}-\frac {b (b c-2 a d) x}{d^2}+\frac {b^2 x^2}{d}-\frac {c (b c-a d)^2}{d^3 (c+d x)}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {(b c-a d)^2 x^n}{d^3 n}-\frac {b (b c-2 a d) x^{2 n}}{2 d^2 n}+\frac {b^2 x^{3 n}}{3 d n}-\frac {c (b c-a d)^2 \log \left (c+d x^n\right )}{d^4 n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.14, size = 82, normalized size = 0.91 \begin {gather*} \frac {-\frac {c (b c-a d)^2 \log \left (c+d x^n\right )}{d^4}+\frac {x^n (b c-a d)^2}{d^3}-\frac {b x^{2 n} (b c-2 a d)}{2 d^2}+\frac {b^2 x^{3 n}}{3 d}}{n} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.08, size = 97, normalized size = 1.08 \begin {gather*} \frac {x^n \left (6 a^2 d^2-12 a b c d+6 a b d^2 x^n+6 b^2 c^2-3 b^2 c d x^n+2 b^2 d^2 x^{2 n}\right )}{6 d^3 n}-\frac {c (b c-a d)^2 \log \left (c+d x^n\right )}{d^4 n} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 108, normalized size = 1.20 \begin {gather*} \frac {2 \, b^{2} d^{3} x^{3 \, n} - 3 \, {\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )} x^{2 \, n} + 6 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{n} - 6 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \log \left (d x^{n} + c\right )}{6 \, d^{4} n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (b x^{n} + a\right )}^{2} x^{2 \, n - 1}}{d x^{n} + c}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.08, size = 173, normalized size = 1.92 \begin {gather*} -\frac {a^{2} c \ln \left (d \,{\mathrm e}^{n \ln \relax (x )}+c \right )}{d^{2} n}+\frac {a^{2} {\mathrm e}^{n \ln \relax (x )}}{d n}+\frac {2 a b \,c^{2} \ln \left (d \,{\mathrm e}^{n \ln \relax (x )}+c \right )}{d^{3} n}-\frac {2 a b c \,{\mathrm e}^{n \ln \relax (x )}}{d^{2} n}+\frac {a b \,{\mathrm e}^{2 n \ln \relax (x )}}{d n}-\frac {b^{2} c^{3} \ln \left (d \,{\mathrm e}^{n \ln \relax (x )}+c \right )}{d^{4} n}+\frac {b^{2} c^{2} {\mathrm e}^{n \ln \relax (x )}}{d^{3} n}-\frac {b^{2} c \,{\mathrm e}^{2 n \ln \relax (x )}}{2 d^{2} n}+\frac {b^{2} {\mathrm e}^{3 n \ln \relax (x )}}{3 d n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.70, size = 150, normalized size = 1.67 \begin {gather*} a^{2} {\left (\frac {x^{n}}{d n} - \frac {c \log \left (\frac {d x^{n} + c}{d}\right )}{d^{2} n}\right )} - \frac {1}{6} \, b^{2} {\left (\frac {6 \, c^{3} \log \left (\frac {d x^{n} + c}{d}\right )}{d^{4} n} - \frac {2 \, d^{2} x^{3 \, n} - 3 \, c d x^{2 \, n} + 6 \, c^{2} x^{n}}{d^{3} n}\right )} + a b {\left (\frac {2 \, c^{2} \log \left (\frac {d x^{n} + c}{d}\right )}{d^{3} n} + \frac {d x^{2 \, n} - 2 \, c x^{n}}{d^{2} n}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^{2\,n-1}\,{\left (a+b\,x^n\right )}^2}{c+d\,x^n} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 63.51, size = 202, normalized size = 2.24 \begin {gather*} \begin {cases} \frac {\left (a + b\right )^{2} \log {\relax (x )}}{c} & \text {for}\: d = 0 \wedge n = 0 \\\frac {\frac {a^{2} x^{2 n}}{2 n} + \frac {2 a b x^{3 n}}{3 n} + \frac {b^{2} x^{4 n}}{4 n}}{c} & \text {for}\: d = 0 \\\frac {\left (a + b\right )^{2} \log {\relax (x )}}{c + d} & \text {for}\: n = 0 \\- \frac {a^{2} c \log {\left (\frac {c}{d} + x^{n} \right )}}{d^{2} n} + \frac {a^{2} x^{n}}{d n} + \frac {2 a b c^{2} \log {\left (\frac {c}{d} + x^{n} \right )}}{d^{3} n} - \frac {2 a b c x^{n}}{d^{2} n} + \frac {a b x^{2 n}}{d n} - \frac {b^{2} c^{3} \log {\left (\frac {c}{d} + x^{n} \right )}}{d^{4} n} + \frac {b^{2} c^{2} x^{n}}{d^{3} n} - \frac {b^{2} c x^{2 n}}{2 d^{2} n} + \frac {b^{2} x^{3 n}}{3 d n} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________