Optimal. Leaf size=154 \[ \frac {(d+e x)^{m+1} \left (1-\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{3/2} \left (1-\frac {d+e x}{\frac {\sqrt {-a} e}{\sqrt {c}}+d}\right )^{3/2} F_1\left (m+1;\frac {3}{2},\frac {3}{2};m+2;\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}},\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )}{e (m+1) \left (a+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.06, antiderivative size = 154, normalized size of antiderivative = 1.00, 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} \[ \frac {(d+e x)^{m+1} \left (1-\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{3/2} \left (1-\frac {d+e x}{\frac {\sqrt {-a} e}{\sqrt {c}}+d}\right )^{3/2} F_1\left (m+1;\frac {3}{2},\frac {3}{2};m+2;\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}},\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )}{e (m+1) \left (a+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 133
Rule 760
Rubi steps
\begin {align*} \int \frac {(d+e x)^m}{\left (a+c x^2\right )^{3/2}} \, dx &=\frac {\left (\left (1-\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{3/2} \left (1-\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{3/2}\right ) \operatorname {Subst}\left (\int \frac {x^m}{\left (1-\frac {x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{3/2} \left (1-\frac {x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{3/2}} \, dx,x,d+e x\right )}{e \left (a+c x^2\right )^{3/2}}\\ &=\frac {(d+e x)^{1+m} \left (1-\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{3/2} \left (1-\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{3/2} F_1\left (1+m;\frac {3}{2},\frac {3}{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) \left (a+c x^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 159, normalized size = 1.03 \[ \frac {(d+e x)^{m+1} \left (\frac {e \left (\sqrt {-\frac {a}{c}}-x\right )}{e \sqrt {-\frac {a}{c}}+d}\right )^{3/2} \left (\frac {e \left (\sqrt {-\frac {a}{c}}+x\right )}{e \sqrt {-\frac {a}{c}}-d}\right )^{3/2} F_1\left (m+1;\frac {3}{2},\frac {3}{2};m+2;\frac {d+e x}{d-\sqrt {-\frac {a}{c}} e},\frac {d+e x}{d+\sqrt {-\frac {a}{c}} e}\right )}{e (m+1) \left (a+c x^2\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + a} {\left (e x + d\right )}^{m}}{c^{2} x^{4} + 2 \, a c x^{2} + a^{2}}, x\right ) \]
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 \[ \int \frac {{\left (e x + d\right )}^{m}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {\left (e x +d \right )^{m}}{\left (c \,x^{2}+a \right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {{\left (e x + d\right )}^{m}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (d+e\,x\right )}^m}{{\left (c\,x^2+a\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (d + e x\right )^{m}}{\left (a + c x^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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