Optimal. Leaf size=221 \[ \frac {e^{7/2} (4 b c-9 a d) \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{4 b^{13/4}}+\frac {e^{7/2} (4 b c-9 a d) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{4 b^{13/4}}-\frac {e^3 \sqrt {e x} (4 b c-9 a d)}{2 b^3 \sqrt [4]{a+b x^2}}-\frac {e (e x)^{5/2} (4 b c-9 a d)}{10 a b^2 \sqrt [4]{a+b x^2}}+\frac {2 (e x)^{9/2} (b c-a d)}{5 a b e \left (a+b x^2\right )^{5/4}} \]
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Rubi [A] time = 0.13, antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {457, 285, 288, 329, 240, 212, 208, 205} \[ -\frac {e^3 \sqrt {e x} (4 b c-9 a d)}{2 b^3 \sqrt [4]{a+b x^2}}+\frac {e^{7/2} (4 b c-9 a d) \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{4 b^{13/4}}+\frac {e^{7/2} (4 b c-9 a d) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{4 b^{13/4}}-\frac {e (e x)^{5/2} (4 b c-9 a d)}{10 a b^2 \sqrt [4]{a+b x^2}}+\frac {2 (e x)^{9/2} (b c-a d)}{5 a b e \left (a+b x^2\right )^{5/4}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 208
Rule 212
Rule 240
Rule 285
Rule 288
Rule 329
Rule 457
Rubi steps
\begin {align*} \int \frac {(e x)^{7/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx &=\frac {2 (b c-a d) (e x)^{9/2}}{5 a b e \left (a+b x^2\right )^{5/4}}+\frac {\left (2 \left (-2 b c+\frac {9 a d}{2}\right )\right ) \int \frac {(e x)^{7/2}}{\left (a+b x^2\right )^{5/4}} \, dx}{5 a b}\\ &=\frac {2 (b c-a d) (e x)^{9/2}}{5 a b e \left (a+b x^2\right )^{5/4}}-\frac {(4 b c-9 a d) e (e x)^{5/2}}{10 a b^2 \sqrt [4]{a+b x^2}}+\frac {\left ((4 b c-9 a d) e^2\right ) \int \frac {(e x)^{3/2}}{\left (a+b x^2\right )^{5/4}} \, dx}{4 b^2}\\ &=\frac {2 (b c-a d) (e x)^{9/2}}{5 a b e \left (a+b x^2\right )^{5/4}}-\frac {(4 b c-9 a d) e^3 \sqrt {e x}}{2 b^3 \sqrt [4]{a+b x^2}}-\frac {(4 b c-9 a d) e (e x)^{5/2}}{10 a b^2 \sqrt [4]{a+b x^2}}+\frac {\left ((4 b c-9 a d) e^4\right ) \int \frac {1}{\sqrt {e x} \sqrt [4]{a+b x^2}} \, dx}{4 b^3}\\ &=\frac {2 (b c-a d) (e x)^{9/2}}{5 a b e \left (a+b x^2\right )^{5/4}}-\frac {(4 b c-9 a d) e^3 \sqrt {e x}}{2 b^3 \sqrt [4]{a+b x^2}}-\frac {(4 b c-9 a d) e (e x)^{5/2}}{10 a b^2 \sqrt [4]{a+b x^2}}+\frac {\left ((4 b c-9 a d) e^3\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 b^3}\\ &=\frac {2 (b c-a d) (e x)^{9/2}}{5 a b e \left (a+b x^2\right )^{5/4}}-\frac {(4 b c-9 a d) e^3 \sqrt {e x}}{2 b^3 \sqrt [4]{a+b x^2}}-\frac {(4 b c-9 a d) e (e x)^{5/2}}{10 a b^2 \sqrt [4]{a+b x^2}}+\frac {\left ((4 b c-9 a d) e^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {b x^4}{e^2}} \, dx,x,\frac {\sqrt {e x}}{\sqrt [4]{a+b x^2}}\right )}{2 b^3}\\ &=\frac {2 (b c-a d) (e x)^{9/2}}{5 a b e \left (a+b x^2\right )^{5/4}}-\frac {(4 b c-9 a d) e^3 \sqrt {e x}}{2 b^3 \sqrt [4]{a+b x^2}}-\frac {(4 b c-9 a d) e (e x)^{5/2}}{10 a b^2 \sqrt [4]{a+b x^2}}+\frac {\left ((4 b c-9 a d) e^4\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^3}+\frac {\left ((4 b c-9 a d) e^4\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^3}\\ &=\frac {2 (b c-a d) (e x)^{9/2}}{5 a b e \left (a+b x^2\right )^{5/4}}-\frac {(4 b c-9 a d) e^3 \sqrt {e x}}{2 b^3 \sqrt [4]{a+b x^2}}-\frac {(4 b c-9 a d) e (e x)^{5/2}}{10 a b^2 \sqrt [4]{a+b x^2}}+\frac {(4 b c-9 a d) e^{7/2} \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{4 b^{13/4}}+\frac {(4 b c-9 a d) e^{7/2} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{4 b^{13/4}}\\ \end {align*}
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Mathematica [C] time = 0.12, size = 91, normalized size = 0.41 \[ \frac {e^3 x^4 \sqrt {e x} \left (9 a^2 d+\left (a+b x^2\right ) \sqrt [4]{\frac {b x^2}{a}+1} (4 b c-9 a d) \, _2F_1\left (\frac {9}{4},\frac {9}{4};\frac {13}{4};-\frac {b x^2}{a}\right )\right )}{18 a^2 b \left (a+b x^2\right )^{5/4}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.05, size = 997, normalized size = 4.51 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (d x^{2} + c\right )} \left (e x\right )^{\frac {7}{2}}}{{\left (b x^{2} + a\right )}^{\frac {9}{4}}}\,{d x} \]
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 \[ \int \frac {\left (e x \right )^{\frac {7}{2}} \left (d \,x^{2}+c \right )}{\left (b \,x^{2}+a \right )^{\frac {9}{4}}}\, dx \]
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 \[ \int \frac {{\left (d x^{2} + c\right )} \left (e x\right )^{\frac {7}{2}}}{{\left (b x^{2} + a\right )}^{\frac {9}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (e\,x\right )}^{7/2}\,\left (d\,x^2+c\right )}{{\left (b\,x^2+a\right )}^{9/4}} \,d x \]
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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