Optimal. Leaf size=332 \[ \frac{(d+e x)^{n+1} \left (3 \sqrt{-a} c d^2+a \sqrt{c} d e n+\sqrt{-a} a e^2 (n+3)\right ) \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{4 c^2 (n+1) \left (\sqrt{c} d-\sqrt{-a} e\right ) \left (a e^2+c d^2\right )}-\frac{(d+e x)^{n+1} \left (3 \sqrt{-a} c d^2-a \sqrt{c} d e n+\sqrt{-a} a e^2 (n+3)\right ) \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{4 c^2 (n+1) \left (\sqrt{-a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right )}+\frac{a (d+e x)^{n+1} (a e+c d x)}{2 c^2 \left (a+c x^2\right ) \left (a e^2+c d^2\right )}+\frac{(d+e x)^{n+1}}{c^2 e (n+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.450803, antiderivative size = 332, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {1649, 1629, 68} \[ \frac{(d+e x)^{n+1} \left (3 \sqrt{-a} c d^2+a \sqrt{c} d e n+\sqrt{-a} a e^2 (n+3)\right ) \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{4 c^2 (n+1) \left (\sqrt{c} d-\sqrt{-a} e\right ) \left (a e^2+c d^2\right )}-\frac{(d+e x)^{n+1} \left (3 \sqrt{-a} c d^2-a \sqrt{c} d e n+\sqrt{-a} a e^2 (n+3)\right ) \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{4 c^2 (n+1) \left (\sqrt{-a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right )}+\frac{a (d+e x)^{n+1} (a e+c d x)}{2 c^2 \left (a+c x^2\right ) \left (a e^2+c d^2\right )}+\frac{(d+e x)^{n+1}}{c^2 e (n+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1649
Rule 1629
Rule 68
Rubi steps
\begin{align*} \int \frac{x^4 (d+e x)^n}{\left (a+c x^2\right )^2} \, dx &=\frac{a (a e+c d x) (d+e x)^{1+n}}{2 c^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac{\int \frac{(d+e x)^n \left (\frac{a^2 \left (c d^2+a e^2 (1+n)\right )}{c^2}+\frac{a^2 d e n x}{c}-2 a \left (d^2+\frac{a e^2}{c}\right ) x^2\right )}{a+c x^2} \, dx}{2 a \left (c d^2+a e^2\right )}\\ &=\frac{a (a e+c d x) (d+e x)^{1+n}}{2 c^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac{\int \left (-\frac{2 a \left (c d^2+a e^2\right ) (d+e x)^n}{c^2}+\frac{\left (-\frac{a^3 d e n}{c^{3/2}}+\sqrt{-a} \left (\frac{3 a^2 d^2}{c}+\frac{3 a^3 e^2}{c^2}+\frac{a^3 e^2 n}{c^2}\right )\right ) (d+e x)^n}{2 a \left (\sqrt{-a}-\sqrt{c} x\right )}+\frac{\left (\frac{a^3 d e n}{c^{3/2}}+\sqrt{-a} \left (\frac{3 a^2 d^2}{c}+\frac{3 a^3 e^2}{c^2}+\frac{a^3 e^2 n}{c^2}\right )\right ) (d+e x)^n}{2 a \left (\sqrt{-a}+\sqrt{c} x\right )}\right ) \, dx}{2 a \left (c d^2+a e^2\right )}\\ &=\frac{(d+e x)^{1+n}}{c^2 e (1+n)}+\frac{a (a e+c d x) (d+e x)^{1+n}}{2 c^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac{\left (3 \sqrt{-a} c d^2-a \sqrt{c} d e n+\sqrt{-a} a e^2 (3+n)\right ) \int \frac{(d+e x)^n}{\sqrt{-a}-\sqrt{c} x} \, dx}{4 c^2 \left (c d^2+a e^2\right )}-\frac{\left (3 \sqrt{-a} c d^2+a \sqrt{c} d e n+\sqrt{-a} a e^2 (3+n)\right ) \int \frac{(d+e x)^n}{\sqrt{-a}+\sqrt{c} x} \, dx}{4 c^2 \left (c d^2+a e^2\right )}\\ &=\frac{(d+e x)^{1+n}}{c^2 e (1+n)}+\frac{a (a e+c d x) (d+e x)^{1+n}}{2 c^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac{\left (3 \sqrt{-a} c d^2+a \sqrt{c} d e n+\sqrt{-a} a e^2 (3+n)\right ) (d+e x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{4 c^2 \left (\sqrt{c} d-\sqrt{-a} e\right ) \left (c d^2+a e^2\right ) (1+n)}-\frac{\left (3 \sqrt{-a} c d^2-a \sqrt{c} d e n+\sqrt{-a} a e^2 (3+n)\right ) (d+e x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{4 c^2 \left (\sqrt{c} d+\sqrt{-a} e\right ) \left (c d^2+a e^2\right ) (1+n)}\\ \end{align*}
Mathematica [A] time = 0.788456, size = 413, normalized size = 1.24 \[ \frac{(d+e x)^{n+1} \left (\frac{a \left (\frac{\left (\sqrt{-a} \sqrt{c} d e n-a e^2 (n-1)+c d^2\right ) \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{\sqrt{c} d-\sqrt{-a} e}-\frac{\left (-\sqrt{-a} \sqrt{c} d e n-a e^2 (n-1)+c d^2\right ) \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{\sqrt{-a} e+\sqrt{c} d}\right )}{\sqrt{-a} (n+1) \left (a e^2+c d^2\right )}+\frac{2 a (a e+c d x)}{\left (a+c x^2\right ) \left (a e^2+c d^2\right )}+\frac{4 \sqrt{-a} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{(n+1) \left (\sqrt{c} d-\sqrt{-a} e\right )}-\frac{4 \sqrt{-a} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{(n+1) \left (\sqrt{-a} e+\sqrt{c} d\right )}+\frac{4}{e n+e}\right )}{4 c^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.716, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex+d \right ) ^{n}{x}^{4}}{ \left ( c{x}^{2}+a \right ) ^{2}}}\, 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*} \int \frac{{\left (e x + d\right )}^{n} x^{4}}{{\left (c x^{2} + a\right )}^{2}}\,{d x} \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{{\left (e x + d\right )}^{n} x^{4}}{c^{2} x^{4} + 2 \, a c x^{2} + a^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{{\left (e x + d\right )}^{n} x^{4}}{{\left (c x^{2} + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]