Optimal. Leaf size=200 \[ -\frac {2 \left (\sqrt {-a}-\sqrt {c} x\right ) \sqrt [4]{-\frac {\left (\sqrt {-a}+\sqrt {c} x\right ) \left (\sqrt {-a} e+\sqrt {c} d\right )}{\left (\sqrt {-a}-\sqrt {c} x\right ) \left (\sqrt {c} d-\sqrt {-a} e\right )}} \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {1}{2};\frac {2 \sqrt {-a} \sqrt {c} (d+e x)}{\left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {-a}-\sqrt {c} x\right )}\right )}{\sqrt [4]{a+c x^2} \sqrt {d+e x} \left (\sqrt {-a} e+\sqrt {c} d\right )} \]
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Rubi [A] time = 0.09, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {727} \[ -\frac {2 \left (\sqrt {-a}-\sqrt {c} x\right ) \sqrt [4]{-\frac {\left (\sqrt {-a}+\sqrt {c} x\right ) \left (\sqrt {-a} e+\sqrt {c} d\right )}{\left (\sqrt {-a}-\sqrt {c} x\right ) \left (\sqrt {c} d-\sqrt {-a} e\right )}} \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {1}{2};\frac {2 \sqrt {-a} \sqrt {c} (d+e x)}{\left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {-a}-\sqrt {c} x\right )}\right )}{\sqrt [4]{a+c x^2} \sqrt {d+e x} \left (\sqrt {-a} e+\sqrt {c} d\right )} \]
Antiderivative was successfully verified.
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Rule 727
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^{3/2} \sqrt [4]{a+c x^2}} \, dx &=-\frac {2 \left (\sqrt {-a}-\sqrt {c} x\right ) \sqrt [4]{-\frac {\left (\sqrt {c} d+\sqrt {-a} e\right ) \left (\sqrt {-a}+\sqrt {c} x\right )}{\left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {-a}-\sqrt {c} x\right )}} \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {1}{2};\frac {2 \sqrt {-a} \sqrt {c} (d+e x)}{\left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {-a}-\sqrt {c} x\right )}\right )}{\left (\sqrt {c} d+\sqrt {-a} e\right ) \sqrt {d+e x} \sqrt [4]{a+c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.51, size = 108, normalized size = 0.54 \[ \frac {\left (a+c x^2\right )^{3/4} (c d x-a e) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {3}{2};-\frac {(a e-c d x)^2}{a c (d+e x)^2}\right )}{a c (d+e x)^{5/2} \left (\frac {\left (a+c x^2\right ) \left (a e^2+c d^2\right )}{a c (d+e x)^2}\right )^{3/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c x^{2} + a\right )}^{\frac {3}{4}} \sqrt {e x + d}}{c e^{2} x^{4} + 2 \, c d e x^{3} + 2 \, a d e x + a d^{2} + {\left (c d^{2} + a e^{2}\right )} x^{2}}, x\right ) \]
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 \[ \int \frac {1}{{\left (c x^{2} + a\right )}^{\frac {1}{4}} {\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.90, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (e x +d \right )^{\frac {3}{2}} \left (c \,x^{2}+a \right )^{\frac {1}{4}}}\, dx \]
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 \[ \int \frac {1}{{\left (c x^{2} + a\right )}^{\frac {1}{4}} {\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (c\,x^2+a\right )}^{1/4}\,{\left (d+e\,x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\sqrt [4]{a + c x^{2}} \left (d + e x\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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