Optimal. Leaf size=194 \[ \frac {\sqrt {-a} (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{2 c (n+1) \left (\sqrt {c} d-\sqrt {-a} e\right )}-\frac {\sqrt {-a} (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{2 c (n+1) \left (\sqrt {-a} e+\sqrt {c} d\right )}+\frac {(d+e x)^{n+1}}{c e (n+1)} \]
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Rubi [A] time = 0.23, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1629, 712, 68} \[ \frac {\sqrt {-a} (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{2 c (n+1) \left (\sqrt {c} d-\sqrt {-a} e\right )}-\frac {\sqrt {-a} (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{2 c (n+1) \left (\sqrt {-a} e+\sqrt {c} d\right )}+\frac {(d+e x)^{n+1}}{c e (n+1)} \]
Antiderivative was successfully verified.
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Rule 68
Rule 712
Rule 1629
Rubi steps
\begin {align*} \int \frac {x^2 (d+e x)^n}{a+c x^2} \, dx &=\int \left (\frac {(d+e x)^n}{c}-\frac {a (d+e x)^n}{c \left (a+c x^2\right )}\right ) \, dx\\ &=\frac {(d+e x)^{1+n}}{c e (1+n)}-\frac {a \int \frac {(d+e x)^n}{a+c x^2} \, dx}{c}\\ &=\frac {(d+e x)^{1+n}}{c e (1+n)}-\frac {a \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}{c}\\ &=\frac {(d+e x)^{1+n}}{c e (1+n)}-\frac {\sqrt {-a} \int \frac {(d+e x)^n}{\sqrt {-a}-\sqrt {c} x} \, dx}{2 c}-\frac {\sqrt {-a} \int \frac {(d+e x)^n}{\sqrt {-a}+\sqrt {c} x} \, dx}{2 c}\\ &=\frac {(d+e x)^{1+n}}{c e (1+n)}+\frac {\sqrt {-a} (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 c \left (\sqrt {c} d-\sqrt {-a} e\right ) (1+n)}-\frac {\sqrt {-a} (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 c \left (\sqrt {c} d+\sqrt {-a} e\right ) (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 170, normalized size = 0.88 \[ \frac {(d+e x)^{n+1} \left (2 \left (a e^2+c d^2\right )+e \left (\sqrt {-a} \sqrt {c} d-a e\right ) \, _2F_1\left (1,n+1;n+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )-e \left (\sqrt {-a} \sqrt {c} d+a e\right ) \, _2F_1\left (1,n+1;n+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )\right )}{2 c e (n+1) \left (a e^2+c d^2\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e x + d\right )}^{n} x^{2}}{c x^{2} + a}, x\right ) \]
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 \[ \int \frac {{\left (e x + d\right )}^{n} x^{2}}{c x^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \left (e x +d \right )^{n}}{c \,x^{2}+a}\, dx \]
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 \[ \int \frac {{\left (e x + d\right )}^{n} x^{2}}{c x^{2} + a}\,{d x} \]
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 \[ \int \frac {x^2\,{\left (d+e\,x\right )}^n}{c\,x^2+a} \,d x \]
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 \[ \int \frac {x^{2} \left (d + e x\right )^{n}}{a + c x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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