Optimal. Leaf size=184 \[ -\frac {e^{5/2} (4 b c-7 a d) \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{4 b^{11/4}}+\frac {e^{5/2} (4 b c-7 a d) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{4 b^{11/4}}-\frac {e (e x)^{3/2} \sqrt [4]{a+b x^2} (4 b c-7 a d)}{6 a b^2}+\frac {2 (e x)^{7/2} (b c-a d)}{3 a b e \left (a+b x^2\right )^{3/4}} \]
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Rubi [A] time = 0.12, antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {457, 321, 329, 331, 298, 205, 208} \[ -\frac {e^{5/2} (4 b c-7 a d) \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{4 b^{11/4}}+\frac {e^{5/2} (4 b c-7 a d) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{4 b^{11/4}}-\frac {e (e x)^{3/2} \sqrt [4]{a+b x^2} (4 b c-7 a d)}{6 a b^2}+\frac {2 (e x)^{7/2} (b c-a d)}{3 a b e \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 208
Rule 298
Rule 321
Rule 329
Rule 331
Rule 457
Rubi steps
\begin {align*} \int \frac {(e x)^{5/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{7/4}} \, dx &=\frac {2 (b c-a d) (e x)^{7/2}}{3 a b e \left (a+b x^2\right )^{3/4}}+\frac {\left (2 \left (-2 b c+\frac {7 a d}{2}\right )\right ) \int \frac {(e x)^{5/2}}{\left (a+b x^2\right )^{3/4}} \, dx}{3 a b}\\ &=\frac {2 (b c-a d) (e x)^{7/2}}{3 a b e \left (a+b x^2\right )^{3/4}}-\frac {(4 b c-7 a d) e (e x)^{3/2} \sqrt [4]{a+b x^2}}{6 a b^2}+\frac {\left ((4 b c-7 a d) e^2\right ) \int \frac {\sqrt {e x}}{\left (a+b x^2\right )^{3/4}} \, dx}{4 b^2}\\ &=\frac {2 (b c-a d) (e x)^{7/2}}{3 a b e \left (a+b x^2\right )^{3/4}}-\frac {(4 b c-7 a d) e (e x)^{3/2} \sqrt [4]{a+b x^2}}{6 a b^2}+\frac {((4 b c-7 a d) e) \operatorname {Subst}\left (\int \frac {x^2}{\left (a+\frac {b x^4}{e^2}\right )^{3/4}} \, dx,x,\sqrt {e x}\right )}{2 b^2}\\ &=\frac {2 (b c-a d) (e x)^{7/2}}{3 a b e \left (a+b x^2\right )^{3/4}}-\frac {(4 b c-7 a d) e (e x)^{3/2} \sqrt [4]{a+b x^2}}{6 a b^2}+\frac {((4 b c-7 a d) e) \operatorname {Subst}\left (\int \frac {x^2}{1-\frac {b x^4}{e^2}} \, dx,x,\frac {\sqrt {e x}}{\sqrt [4]{a+b x^2}}\right )}{2 b^2}\\ &=\frac {2 (b c-a d) (e x)^{7/2}}{3 a b e \left (a+b x^2\right )^{3/4}}-\frac {(4 b c-7 a d) e (e x)^{3/2} \sqrt [4]{a+b x^2}}{6 a b^2}+\frac {\left ((4 b c-7 a d) e^3\right ) \operatorname {Subst}\left (\int \frac {1}{e-\sqrt {b} x^2} \, dx,x,\frac {\sqrt {e x}}{\sqrt [4]{a+b x^2}}\right )}{4 b^{5/2}}-\frac {\left ((4 b c-7 a d) e^3\right ) \operatorname {Subst}\left (\int \frac {1}{e+\sqrt {b} x^2} \, dx,x,\frac {\sqrt {e x}}{\sqrt [4]{a+b x^2}}\right )}{4 b^{5/2}}\\ &=\frac {2 (b c-a d) (e x)^{7/2}}{3 a b e \left (a+b x^2\right )^{3/4}}-\frac {(4 b c-7 a d) e (e x)^{3/2} \sqrt [4]{a+b x^2}}{6 a b^2}-\frac {(4 b c-7 a d) e^{5/2} \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{4 b^{11/4}}+\frac {(4 b c-7 a d) e^{5/2} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{4 b^{11/4}}\\ \end {align*}
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Mathematica [C] time = 0.13, size = 77, normalized size = 0.42 \[ \frac {x (e x)^{5/2} \left (\left (\frac {b x^2}{a}+1\right )^{3/4} (4 b c-7 a d) \, _2F_1\left (\frac {7}{4},\frac {7}{4};\frac {11}{4};-\frac {b x^2}{a}\right )+7 a d\right )}{14 a b \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
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fricas [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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d x^{2} + c\right )} \left (e x\right )^{\frac {5}{2}}}{{\left (b x^{2} + a\right )}^{\frac {7}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.11, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x \right )^{\frac {5}{2}} \left (d \,x^{2}+c \right )}{\left (b \,x^{2}+a \right )^{\frac {7}{4}}}\, dx \]
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 \[ \int \frac {{\left (d x^{2} + c\right )} \left (e x\right )^{\frac {5}{2}}}{{\left (b x^{2} + a\right )}^{\frac {7}{4}}}\,{d x} \]
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 \[ \int \frac {{\left (e\,x\right )}^{5/2}\,\left (d\,x^2+c\right )}{{\left (b\,x^2+a\right )}^{7/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 166.39, size = 94, normalized size = 0.51 \[ \frac {c e^{\frac {5}{2}} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{4}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {7}{4}} \Gamma \left (\frac {11}{4}\right )} + \frac {d e^{\frac {5}{2}} x^{\frac {11}{2}} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{4}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {7}{4}} \Gamma \left (\frac {15}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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