Optimal. Leaf size=152 \[ \frac {\left (a+c x^2\right )^p (d+e x)^{m+1} \left (1-\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{-p} \left (1-\frac {d+e x}{\frac {\sqrt {-a} e}{\sqrt {c}}+d}\right )^{-p} F_1\left (m+1;-p,-p;m+2;\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}},\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )}{e (m+1)} \]
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Rubi [A] time = 0.06, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {760, 133} \[ \frac {\left (a+c x^2\right )^p (d+e x)^{m+1} \left (1-\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{-p} \left (1-\frac {d+e x}{\frac {\sqrt {-a} e}{\sqrt {c}}+d}\right )^{-p} F_1\left (m+1;-p,-p;m+2;\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}},\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )}{e (m+1)} \]
Antiderivative was successfully verified.
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Rule 133
Rule 760
Rubi steps
\begin {align*} \int (d+e x)^m \left (a+c x^2\right )^p \, dx &=\frac {\left (\left (a+c x^2\right )^p \left (1-\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{-p} \left (1-\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{-p}\right ) \operatorname {Subst}\left (\int x^m \left (1-\frac {x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^p \left (1-\frac {x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^p \, dx,x,d+e x\right )}{e}\\ &=\frac {(d+e x)^{1+m} \left (a+c x^2\right )^p \left (1-\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{-p} \left (1-\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{-p} F_1\left (1+m;-p,-p;2+m;\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}},\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )}{e (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 157, normalized size = 1.03 \[ \frac {\left (a+c x^2\right )^p (d+e x)^{m+1} \left (\frac {e \left (\sqrt {-\frac {a}{c}}-x\right )}{e \sqrt {-\frac {a}{c}}+d}\right )^{-p} \left (\frac {e \left (\sqrt {-\frac {a}{c}}+x\right )}{e \sqrt {-\frac {a}{c}}-d}\right )^{-p} F_1\left (m+1;-p,-p;m+2;\frac {d+e x}{d-\sqrt {-\frac {a}{c}} e},\frac {d+e x}{d+\sqrt {-\frac {a}{c}} e}\right )}{e (m+1)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c x^{2} + a\right )}^{p} {\left (e x + d\right )}^{m}, x\right ) \]
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 \[ \int {\left (c x^{2} + a\right )}^{p} {\left (e x + d\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.72, size = 0, normalized size = 0.00 \[ \int \left (c \,x^{2}+a \right )^{p} \left (e x +d \right )^{m}\, dx \]
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 \[ \int {\left (c x^{2} + a\right )}^{p} {\left (e x + d\right )}^{m}\,{d x} \]
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 \[ \int {\left (c\,x^2+a\right )}^p\,{\left (d+e\,x\right )}^m \,d x \]
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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