Optimal. Leaf size=513 \[ -\frac {c (d+e x)^{n+1} (a e+c d x)}{2 a^2 \left (a+c x^2\right ) \left (a e^2+c d^2\right )}+\frac {e (d+e x)^{n+1} \, _2F_1\left (2,n+1;n+2;\frac {e x}{d}+1\right )}{a^2 d^2 (n+1)}-\frac {c (d+e x)^{n+1} \left (\sqrt {-a} \sqrt {c} d e n+a e^2 (1-n)+c d^2\right ) \, _2F_1\left (1,n+1;n+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{4 (-a)^{5/2} (n+1) \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (a e^2+c d^2\right )}+\frac {c (d+e x)^{n+1} \left (-\sqrt {-a} \sqrt {c} d e n+a e^2 (1-n)+c d^2\right ) \, _2F_1\left (1,n+1;n+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{4 (-a)^{5/2} (n+1) \left (\sqrt {-a} e+\sqrt {c} d\right ) \left (a e^2+c d^2\right )}-\frac {c (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{2 (-a)^{5/2} (n+1) \left (\sqrt {c} d-\sqrt {-a} e\right )}+\frac {c (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{2 (-a)^{5/2} (n+1) \left (\sqrt {-a} e+\sqrt {c} d\right )} \]
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Rubi [A] time = 0.70, antiderivative size = 513, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {961, 65, 741, 831, 68, 712} \[ -\frac {c (d+e x)^{n+1} (a e+c d x)}{2 a^2 \left (a+c x^2\right ) \left (a e^2+c d^2\right )}+\frac {e (d+e x)^{n+1} \, _2F_1\left (2,n+1;n+2;\frac {e x}{d}+1\right )}{a^2 d^2 (n+1)}-\frac {c (d+e x)^{n+1} \left (\sqrt {-a} \sqrt {c} d e n+a e^2 (1-n)+c d^2\right ) \, _2F_1\left (1,n+1;n+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{4 (-a)^{5/2} (n+1) \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (a e^2+c d^2\right )}+\frac {c (d+e x)^{n+1} \left (-\sqrt {-a} \sqrt {c} d e n+a e^2 (1-n)+c d^2\right ) \, _2F_1\left (1,n+1;n+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{4 (-a)^{5/2} (n+1) \left (\sqrt {-a} e+\sqrt {c} d\right ) \left (a e^2+c d^2\right )}-\frac {c (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{2 (-a)^{5/2} (n+1) \left (\sqrt {c} d-\sqrt {-a} e\right )}+\frac {c (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{2 (-a)^{5/2} (n+1) \left (\sqrt {-a} e+\sqrt {c} d\right )} \]
Antiderivative was successfully verified.
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Rule 65
Rule 68
Rule 712
Rule 741
Rule 831
Rule 961
Rubi steps
\begin {align*} \int \frac {(d+e x)^n}{x^2 \left (a+c x^2\right )^2} \, dx &=\int \left (\frac {(d+e x)^n}{a^2 x^2}-\frac {c (d+e x)^n}{a \left (a+c x^2\right )^2}-\frac {c (d+e x)^n}{a^2 \left (a+c x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {(d+e x)^n}{x^2} \, dx}{a^2}-\frac {c \int \frac {(d+e x)^n}{a+c x^2} \, dx}{a^2}-\frac {c \int \frac {(d+e x)^n}{\left (a+c x^2\right )^2} \, dx}{a}\\ &=-\frac {c (a e+c d x) (d+e x)^{1+n}}{2 a^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac {e (d+e x)^{1+n} \, _2F_1\left (2,1+n;2+n;1+\frac {e x}{d}\right )}{a^2 d^2 (1+n)}-\frac {c \int \left (\frac {\sqrt {-a} (d+e x)^n}{2 a \left (\sqrt {-a}-\sqrt {c} x\right )}+\frac {\sqrt {-a} (d+e x)^n}{2 a \left (\sqrt {-a}+\sqrt {c} x\right )}\right ) \, dx}{a^2}+\frac {c \int \frac {(d+e x)^n \left (-c d^2-a e^2 (1-n)+c d e n x\right )}{a+c x^2} \, dx}{2 a^2 \left (c d^2+a e^2\right )}\\ &=-\frac {c (a e+c d x) (d+e x)^{1+n}}{2 a^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac {e (d+e x)^{1+n} \, _2F_1\left (2,1+n;2+n;1+\frac {e x}{d}\right )}{a^2 d^2 (1+n)}+\frac {c \int \frac {(d+e x)^n}{\sqrt {-a}-\sqrt {c} x} \, dx}{2 (-a)^{5/2}}+\frac {c \int \frac {(d+e x)^n}{\sqrt {-a}+\sqrt {c} x} \, dx}{2 (-a)^{5/2}}+\frac {c \int \left (\frac {\left (\sqrt {-a} \left (-c d^2-a e^2 (1-n)\right )-a \sqrt {c} d e n\right ) (d+e x)^n}{2 a \left (\sqrt {-a}-\sqrt {c} x\right )}+\frac {\left (\sqrt {-a} \left (-c d^2-a e^2 (1-n)\right )+a \sqrt {c} d e n\right ) (d+e x)^n}{2 a \left (\sqrt {-a}+\sqrt {c} x\right )}\right ) \, dx}{2 a^2 \left (c d^2+a e^2\right )}\\ &=-\frac {c (a e+c d x) (d+e x)^{1+n}}{2 a^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac {c (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 )}{2 (-a)^{5/2} \left (\sqrt {c} d-\sqrt {-a} e\right ) (1+n)}+\frac {c (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 )}{2 (-a)^{5/2} \left (\sqrt {c} d+\sqrt {-a} e\right ) (1+n)}+\frac {e (d+e x)^{1+n} \, _2F_1\left (2,1+n;2+n;1+\frac {e x}{d}\right )}{a^2 d^2 (1+n)}+\frac {\left (c \left (c d^2+a e^2 (1-n)-\sqrt {-a} \sqrt {c} d e n\right )\right ) \int \frac {(d+e x)^n}{\sqrt {-a}-\sqrt {c} x} \, dx}{4 (-a)^{5/2} \left (c d^2+a e^2\right )}+\frac {\left (c \left (c d^2+a e^2 (1-n)+\sqrt {-a} \sqrt {c} d e n\right )\right ) \int \frac {(d+e x)^n}{\sqrt {-a}+\sqrt {c} x} \, dx}{4 (-a)^{5/2} \left (c d^2+a e^2\right )}\\ &=-\frac {c (a e+c d x) (d+e x)^{1+n}}{2 a^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac {c (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 )}{2 (-a)^{5/2} \left (\sqrt {c} d-\sqrt {-a} e\right ) (1+n)}-\frac {c \left (c d^2+a e^2 (1-n)+\sqrt {-a} \sqrt {c} d e 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 (-a)^{5/2} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (c d^2+a e^2\right ) (1+n)}+\frac {c (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 )}{2 (-a)^{5/2} \left (\sqrt {c} d+\sqrt {-a} e\right ) (1+n)}+\frac {c \left (c d^2+a e^2 (1-n)-\sqrt {-a} \sqrt {c} d e 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 (-a)^{5/2} \left (\sqrt {c} d+\sqrt {-a} e\right ) \left (c d^2+a e^2\right ) (1+n)}+\frac {e (d+e x)^{1+n} \, _2F_1\left (2,1+n;2+n;1+\frac {e x}{d}\right )}{a^2 d^2 (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.67, size = 437, normalized size = 0.85 \[ \frac {1}{4} (d+e x)^{n+1} \left (-\frac {2 c (a e+c d x)}{a^2 \left (a+c x^2\right ) \left (a e^2+c d^2\right )}+\frac {4 e \, _2F_1\left (2,n+1;n+2;\frac {e x}{d}+1\right )}{a^2 d^2 (n+1)}+\frac {a c \left (\frac {\left (\sqrt {-a} \sqrt {c} d e n-a e^2 (n-1)+c d^2\right ) \, _2F_1\left (1,n+1;n+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{\sqrt {c} d-\sqrt {-a} e}-\frac {\left (-\sqrt {-a} \sqrt {c} d e n-a e^2 (n-1)+c d^2\right ) \, _2F_1\left (1,n+1;n+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{\sqrt {-a} e+\sqrt {c} d}\right )}{(-a)^{7/2} (n+1) \left (a e^2+c d^2\right )}+\frac {2 c \, _2F_1\left (1,n+1;n+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{(-a)^{5/2} (n+1) \left (\sqrt {-a} e-\sqrt {c} d\right )}+\frac {2 c \, _2F_1\left (1,n+1;n+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{(-a)^{5/2} (n+1) \left (\sqrt {-a} e+\sqrt {c} d\right )}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 1.03, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e x + d\right )}^{n}}{c^{2} x^{6} + 2 \, a c x^{4} + a^{2} x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{n}}{{\left (c x^{2} + a\right )}^{2} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x +d \right )^{n}}{\left (c \,x^{2}+a \right )^{2} x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{n}}{{\left (c x^{2} + a\right )}^{2} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (d+e\,x\right )}^n}{x^2\,{\left (c\,x^2+a\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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