Optimal. Leaf size=154 \[ \frac{(d+e x)^{m+1} \sqrt{1-\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}}} \sqrt{1-\frac{d+e x}{\frac{\sqrt{-a} e}{\sqrt{c}}+d}} F_1\left (m+1;\frac{1}{2},\frac{1}{2};m+2;\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}},\frac{d+e x}{d+\frac{\sqrt{-a} e}{\sqrt{c}}}\right )}{e (m+1) \sqrt{a+c x^2}} \]
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Rubi [A] time = 0.0615328, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {760, 133} \[ \frac{(d+e x)^{m+1} \sqrt{1-\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}}} \sqrt{1-\frac{d+e x}{\frac{\sqrt{-a} e}{\sqrt{c}}+d}} F_1\left (m+1;\frac{1}{2},\frac{1}{2};m+2;\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}},\frac{d+e x}{d+\frac{\sqrt{-a} e}{\sqrt{c}}}\right )}{e (m+1) \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
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Rule 760
Rule 133
Rubi steps
\begin{align*} \int \frac{(d+e x)^m}{\sqrt{a+c x^2}} \, dx &=\frac{\left (\sqrt{1-\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}}} \sqrt{1-\frac{d+e x}{d+\frac{\sqrt{-a} e}{\sqrt{c}}}}\right ) \operatorname{Subst}\left (\int \frac{x^m}{\sqrt{1-\frac{x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}}} \sqrt{1-\frac{x}{d+\frac{\sqrt{-a} e}{\sqrt{c}}}}} \, dx,x,d+e x\right )}{e \sqrt{a+c x^2}}\\ &=\frac{(d+e x)^{1+m} \sqrt{1-\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}}} \sqrt{1-\frac{d+e x}{d+\frac{\sqrt{-a} e}{\sqrt{c}}}} F_1\left (1+m;\frac{1}{2},\frac{1}{2};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) \sqrt{a+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.112678, size = 159, normalized size = 1.03 \[ \frac{(d+e x)^{m+1} \sqrt{\frac{e \left (\sqrt{-\frac{a}{c}}-x\right )}{e \sqrt{-\frac{a}{c}}+d}} \sqrt{\frac{e \left (\sqrt{-\frac{a}{c}}+x\right )}{e \sqrt{-\frac{a}{c}}-d}} F_1\left (m+1;\frac{1}{2},\frac{1}{2};m+2;\frac{d+e x}{d-\sqrt{-\frac{a}{c}} e},\frac{d+e x}{d+\sqrt{-\frac{a}{c}} e}\right )}{e (m+1) \sqrt{a+c x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \text{hanged} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{m}}{\sqrt{c x^{2} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (e x + d\right )}^{m}}{\sqrt{c x^{2} + a}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d + e x\right )^{m}}{\sqrt{a + c x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{m}}{\sqrt{c x^{2} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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