Optimal. Leaf size=202 \[ -\frac {\left (\sqrt {-a} A \sqrt {c}+a B\right ) (d+e x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{2 a \sqrt {c} (m+2) \left (\sqrt {c} d-\sqrt {-a} e\right )}-\frac {\left (\frac {\sqrt {-a} B}{\sqrt {c}}+A\right ) (d+e x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{2 \sqrt {-a} (m+2) \left (\sqrt {-a} e+\sqrt {c} d\right )} \]
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Rubi [A] time = 0.19, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {831, 68} \[ -\frac {\left (\sqrt {-a} A \sqrt {c}+a B\right ) (d+e x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{2 a \sqrt {c} (m+2) \left (\sqrt {c} d-\sqrt {-a} e\right )}-\frac {\left (\frac {\sqrt {-a} B}{\sqrt {c}}+A\right ) (d+e x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{2 \sqrt {-a} (m+2) \left (\sqrt {-a} e+\sqrt {c} d\right )} \]
Antiderivative was successfully verified.
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Rule 68
Rule 831
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^{1+m}}{a+c x^2} \, dx &=\int \left (\frac {\left (\sqrt {-a} A-\frac {a B}{\sqrt {c}}\right ) (d+e x)^{1+m}}{2 a \left (\sqrt {-a}-\sqrt {c} x\right )}+\frac {\left (\sqrt {-a} A+\frac {a B}{\sqrt {c}}\right ) (d+e x)^{1+m}}{2 a \left (\sqrt {-a}+\sqrt {c} x\right )}\right ) \, dx\\ &=\frac {1}{2} \left (\frac {a A}{(-a)^{3/2}}-\frac {B}{\sqrt {c}}\right ) \int \frac {(d+e x)^{1+m}}{\sqrt {-a}-\sqrt {c} x} \, dx+\frac {1}{2} \left (\frac {a A}{(-a)^{3/2}}+\frac {B}{\sqrt {c}}\right ) \int \frac {(d+e x)^{1+m}}{\sqrt {-a}+\sqrt {c} x} \, dx\\ &=-\frac {\left (\frac {a A}{(-a)^{3/2}}+\frac {B}{\sqrt {c}}\right ) (d+e x)^{2+m} \, _2F_1\left (1,2+m;3+m;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{2 \left (\sqrt {c} d-\sqrt {-a} e\right ) (2+m)}+\frac {\left (\frac {a A}{(-a)^{3/2}}-\frac {B}{\sqrt {c}}\right ) (d+e x)^{2+m} \, _2F_1\left (1,2+m;3+m;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{2 \left (\sqrt {c} d+\sqrt {-a} e\right ) (2+m)}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 182, normalized size = 0.90 \[ \frac {(d+e x)^{m+2} \left (\frac {\left (\sqrt {-a} A \sqrt {c}+a B\right ) \, _2F_1\left (1,m+2;m+3;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{\sqrt {-a} e-\sqrt {c} d}+\frac {\left (\sqrt {-a} A \sqrt {c}-a B\right ) \, _2F_1\left (1,m+2;m+3;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{\sqrt {-a} e+\sqrt {c} d}\right )}{2 a \sqrt {c} (m+2)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B x + A\right )} {\left (e x + d\right )}^{m + 1}}{c x^{2} + a}, x\right ) \]
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 \[ \int \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{m + 1}}{c x^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.87, size = 0, normalized size = 0.00 \[ \int \frac {\left (B x +A \right ) \left (e x +d \right )^{m +1}}{c \,x^{2}+a}\, dx \]
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 \[ \int \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{m + 1}}{c x^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^{m+1}}{c\,x^2+a} \,d x \]
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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