Optimal. Leaf size=191 \[ \frac {x \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} F_1\left (\frac {1}{2};-p,2;\frac {3}{2};-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^2}+\frac {e^2 x^3 \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} F_1\left (\frac {3}{2};-p,2;\frac {5}{2};-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{3 d^4}-\frac {c d e \left (a+c x^2\right )^{p+1} \, _2F_1\left (2,p+1;p+2;\frac {e^2 \left (c x^2+a\right )}{c d^2+a e^2}\right )}{(p+1) \left (a e^2+c d^2\right )^2} \]
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Rubi [A] time = 0.18, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {757, 430, 429, 444, 68, 511, 510} \[ \frac {x \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} F_1\left (\frac {1}{2};-p,2;\frac {3}{2};-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^2}+\frac {e^2 x^3 \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} F_1\left (\frac {3}{2};-p,2;\frac {5}{2};-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{3 d^4}-\frac {c d e \left (a+c x^2\right )^{p+1} \, _2F_1\left (2,p+1;p+2;\frac {e^2 \left (c x^2+a\right )}{c d^2+a e^2}\right )}{(p+1) \left (a e^2+c d^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 68
Rule 429
Rule 430
Rule 444
Rule 510
Rule 511
Rule 757
Rubi steps
\begin {align*} \int \frac {\left (a+c x^2\right )^p}{(d+e x)^2} \, dx &=\int \left (\frac {d^2 \left (a+c x^2\right )^p}{\left (d^2-e^2 x^2\right )^2}-\frac {2 d e x \left (a+c x^2\right )^p}{\left (d^2-e^2 x^2\right )^2}+\frac {e^2 x^2 \left (a+c x^2\right )^p}{\left (-d^2+e^2 x^2\right )^2}\right ) \, dx\\ &=d^2 \int \frac {\left (a+c x^2\right )^p}{\left (d^2-e^2 x^2\right )^2} \, dx-(2 d e) \int \frac {x \left (a+c x^2\right )^p}{\left (d^2-e^2 x^2\right )^2} \, dx+e^2 \int \frac {x^2 \left (a+c x^2\right )^p}{\left (-d^2+e^2 x^2\right )^2} \, dx\\ &=-\left ((d e) \operatorname {Subst}\left (\int \frac {(a+c x)^p}{\left (d^2-e^2 x\right )^2} \, dx,x,x^2\right )\right )+\left (d^2 \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p}\right ) \int \frac {\left (1+\frac {c x^2}{a}\right )^p}{\left (d^2-e^2 x^2\right )^2} \, dx+\left (e^2 \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p}\right ) \int \frac {x^2 \left (1+\frac {c x^2}{a}\right )^p}{\left (-d^2+e^2 x^2\right )^2} \, dx\\ &=\frac {x \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} F_1\left (\frac {1}{2};-p,2;\frac {3}{2};-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^2}+\frac {e^2 x^3 \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} F_1\left (\frac {3}{2};-p,2;\frac {5}{2};-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{3 d^4}-\frac {c d e \left (a+c x^2\right )^{1+p} \, _2F_1\left (2,1+p;2+p;\frac {e^2 \left (a+c x^2\right )}{c d^2+a e^2}\right )}{\left (c d^2+a e^2\right )^2 (1+p)}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 141, normalized size = 0.74 \[ \frac {\left (a+c x^2\right )^p \left (\frac {e \left (x-\sqrt {-\frac {a}{c}}\right )}{d+e x}\right )^{-p} \left (\frac {e \left (\sqrt {-\frac {a}{c}}+x\right )}{d+e x}\right )^{-p} F_1\left (1-2 p;-p,-p;2-2 p;\frac {d-\sqrt {-\frac {a}{c}} e}{d+e x},\frac {d+\sqrt {-\frac {a}{c}} e}{d+e x}\right )}{e (2 p-1) (d+e x)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c x^{2} + a\right )}^{p}}{e^{2} x^{2} + 2 \, d e x + d^{2}}, x\right ) \]
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 \[ \int \frac {{\left (c x^{2} + a\right )}^{p}}{{\left (e x + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.75, size = 0, normalized size = 0.00 \[ \int \frac {\left (c \,x^{2}+a \right )^{p}}{\left (e x +d \right )^{2}}\, dx \]
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 \[ \int \frac {{\left (c x^{2} + a\right )}^{p}}{{\left (e x + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c\,x^2+a\right )}^p}{{\left (d+e\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + c x^{2}\right )^{p}}{\left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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