Optimal. Leaf size=149 \[ \frac {d e^{3/2} \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{b^{9/4}}+\frac {d e^{3/2} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{b^{9/4}}-\frac {2 d e \sqrt {e x}}{b^2 \sqrt [4]{a+b x^2}}+\frac {2 (e x)^{5/2} (b c-a d)}{5 a b e \left (a+b x^2\right )^{5/4}} \]
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Rubi [A] time = 0.09, antiderivative size = 149, 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 = {452, 288, 329, 240, 212, 208, 205} \begin {gather*} \frac {d e^{3/2} \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{b^{9/4}}+\frac {d e^{3/2} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{b^{9/4}}-\frac {2 d e \sqrt {e x}}{b^2 \sqrt [4]{a+b x^2}}+\frac {2 (e x)^{5/2} (b c-a d)}{5 a b e \left (a+b x^2\right )^{5/4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 208
Rule 212
Rule 240
Rule 288
Rule 329
Rule 452
Rubi steps
\begin {align*} \int \frac {(e x)^{3/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx &=\frac {2 (b c-a d) (e x)^{5/2}}{5 a b e \left (a+b x^2\right )^{5/4}}+\frac {d \int \frac {(e x)^{3/2}}{\left (a+b x^2\right )^{5/4}} \, dx}{b}\\ &=\frac {2 (b c-a d) (e x)^{5/2}}{5 a b e \left (a+b x^2\right )^{5/4}}-\frac {2 d e \sqrt {e x}}{b^2 \sqrt [4]{a+b x^2}}+\frac {\left (d e^2\right ) \int \frac {1}{\sqrt {e x} \sqrt [4]{a+b x^2}} \, dx}{b^2}\\ &=\frac {2 (b c-a d) (e x)^{5/2}}{5 a b e \left (a+b x^2\right )^{5/4}}-\frac {2 d e \sqrt {e x}}{b^2 \sqrt [4]{a+b x^2}}+\frac {(2 d e) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{b^2}\\ &=\frac {2 (b c-a d) (e x)^{5/2}}{5 a b e \left (a+b x^2\right )^{5/4}}-\frac {2 d e \sqrt {e x}}{b^2 \sqrt [4]{a+b x^2}}+\frac {(2 d e) \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 )}{b^2}\\ &=\frac {2 (b c-a d) (e x)^{5/2}}{5 a b e \left (a+b x^2\right )^{5/4}}-\frac {2 d e \sqrt {e x}}{b^2 \sqrt [4]{a+b x^2}}+\frac {\left (d e^2\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 )}{b^2}+\frac {\left (d e^2\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 )}{b^2}\\ &=\frac {2 (b c-a d) (e x)^{5/2}}{5 a b e \left (a+b x^2\right )^{5/4}}-\frac {2 d e \sqrt {e x}}{b^2 \sqrt [4]{a+b x^2}}+\frac {d e^{3/2} \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{b^{9/4}}+\frac {d e^{3/2} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{b^{9/4}}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 77, normalized size = 0.52 \begin {gather*} \frac {2 x (e x)^{3/2} \left (5 d x^2 \left (a+b x^2\right ) \sqrt [4]{\frac {b x^2}{a}+1} \, _2F_1\left (\frac {9}{4},\frac {9}{4};\frac {13}{4};-\frac {b x^2}{a}\right )+9 a c\right )}{45 a^2 \left (a+b x^2\right )^{5/4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 28.61, size = 206, normalized size = 1.38 \begin {gather*} \frac {e^{3/2} \left (a+b x^2\right )^{3/4} \left (-\frac {2 \left (5 a^2 d e^{7/2} \sqrt {e x}+6 a b d e^{3/2} (e x)^{5/2}-b^2 c e^{3/2} (e x)^{5/2}\right )}{5 a b^2 \left (a e^2+b e^2 x^2\right )^{5/4}}+\frac {d e^{3/2} \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a e^2+b e^2 x^2}}\right )}{b^{9/4}}+\frac {d e^{3/2} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a e^2+b e^2 x^2}}\right )}{b^{9/4}}\right )}{\left (a e^2+b e^2 x^2\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.53, size = 448, normalized size = 3.01 \begin {gather*} -\frac {4 \, {\left (5 \, a^{2} d e - {\left (b^{2} c - 6 \, a b d\right )} e x^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {e x} + 20 \, {\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )} \left (\frac {d^{4} e^{6}}{b^{9}}\right )^{\frac {1}{4}} \arctan \left (-\frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {e x} b^{7} d e \left (\frac {d^{4} e^{6}}{b^{9}}\right )^{\frac {3}{4}} - {\left (b^{8} x^{2} + a b^{7}\right )} \left (\frac {d^{4} e^{6}}{b^{9}}\right )^{\frac {3}{4}} \sqrt {\frac {\sqrt {b x^{2} + a} d^{2} e^{3} x + {\left (b^{5} x^{2} + a b^{4}\right )} \sqrt {\frac {d^{4} e^{6}}{b^{9}}}}{b x^{2} + a}}}{b d^{4} e^{6} x^{2} + a d^{4} e^{6}}\right ) - 5 \, {\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )} \left (\frac {d^{4} e^{6}}{b^{9}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {e x} d e + {\left (b^{3} x^{2} + a b^{2}\right )} \left (\frac {d^{4} e^{6}}{b^{9}}\right )^{\frac {1}{4}}}{b x^{2} + a}\right ) + 5 \, {\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )} \left (\frac {d^{4} e^{6}}{b^{9}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {e x} d e - {\left (b^{3} x^{2} + a b^{2}\right )} \left (\frac {d^{4} e^{6}}{b^{9}}\right )^{\frac {1}{4}}}{b x^{2} + a}\right )}{10 \, {\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}} \end {gather*}
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 \begin {gather*} \int \frac {{\left (d x^{2} + c\right )} \left (e x\right )^{\frac {3}{2}}}{{\left (b x^{2} + a\right )}^{\frac {9}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (e x \right )^{\frac {3}{2}} \left (d \,x^{2}+c \right )}{\left (b \,x^{2}+a \right )^{\frac {9}{4}}}\, dx \end {gather*}
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 \begin {gather*} \int \frac {{\left (d x^{2} + c\right )} \left (e x\right )^{\frac {3}{2}}}{{\left (b x^{2} + a\right )}^{\frac {9}{4}}}\,{d x} \end {gather*}
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 \begin {gather*} \int \frac {{\left (e\,x\right )}^{3/2}\,\left (d\,x^2+c\right )}{{\left (b\,x^2+a\right )}^{9/4}} \,d x \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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