Optimal. Leaf size=180 \[ -\frac {a (10 b c-9 a d) e^3 \sqrt {e x} \sqrt [4]{a+b x^2}}{12 b^3}+\frac {(10 b c-9 a d) e (e x)^{5/2} \sqrt [4]{a+b x^2}}{30 b^2}+\frac {d (e x)^{9/2} \sqrt [4]{a+b x^2}}{5 b e}-\frac {a^{3/2} (10 b c-9 a d) e^2 \left (1+\frac {a}{b x^2}\right )^{3/4} (e x)^{3/2} F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{12 b^{5/2} \left (a+b x^2\right )^{3/4}} \]
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Rubi [A]
time = 0.09, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {470, 327, 335,
243, 342, 281, 237} \begin {gather*} -\frac {a^{3/2} e^2 (e x)^{3/2} \left (\frac {a}{b x^2}+1\right )^{3/4} (10 b c-9 a d) F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{12 b^{5/2} \left (a+b x^2\right )^{3/4}}-\frac {a e^3 \sqrt {e x} \sqrt [4]{a+b x^2} (10 b c-9 a d)}{12 b^3}+\frac {e (e x)^{5/2} \sqrt [4]{a+b x^2} (10 b c-9 a d)}{30 b^2}+\frac {d (e x)^{9/2} \sqrt [4]{a+b x^2}}{5 b e} \end {gather*}
Antiderivative was successfully verified.
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Rule 237
Rule 243
Rule 281
Rule 327
Rule 335
Rule 342
Rule 470
Rubi steps
\begin {align*} \int \frac {(e x)^{7/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{3/4}} \, dx &=\frac {d (e x)^{9/2} \sqrt [4]{a+b x^2}}{5 b e}-\frac {\left (-5 b c+\frac {9 a d}{2}\right ) \int \frac {(e x)^{7/2}}{\left (a+b x^2\right )^{3/4}} \, dx}{5 b}\\ &=\frac {(10 b c-9 a d) e (e x)^{5/2} \sqrt [4]{a+b x^2}}{30 b^2}+\frac {d (e x)^{9/2} \sqrt [4]{a+b x^2}}{5 b e}-\frac {\left (a (10 b c-9 a d) e^2\right ) \int \frac {(e x)^{3/2}}{\left (a+b x^2\right )^{3/4}} \, dx}{12 b^2}\\ &=-\frac {a (10 b c-9 a d) e^3 \sqrt {e x} \sqrt [4]{a+b x^2}}{12 b^3}+\frac {(10 b c-9 a d) e (e x)^{5/2} \sqrt [4]{a+b x^2}}{30 b^2}+\frac {d (e x)^{9/2} \sqrt [4]{a+b x^2}}{5 b e}+\frac {\left (a^2 (10 b c-9 a d) e^4\right ) \int \frac {1}{\sqrt {e x} \left (a+b x^2\right )^{3/4}} \, dx}{24 b^3}\\ &=-\frac {a (10 b c-9 a d) e^3 \sqrt {e x} \sqrt [4]{a+b x^2}}{12 b^3}+\frac {(10 b c-9 a d) e (e x)^{5/2} \sqrt [4]{a+b x^2}}{30 b^2}+\frac {d (e x)^{9/2} \sqrt [4]{a+b x^2}}{5 b e}+\frac {\left (a^2 (10 b c-9 a d) e^3\right ) \text {Subst}\left (\int \frac {1}{\left (a+\frac {b x^4}{e^2}\right )^{3/4}} \, dx,x,\sqrt {e x}\right )}{12 b^3}\\ &=-\frac {a (10 b c-9 a d) e^3 \sqrt {e x} \sqrt [4]{a+b x^2}}{12 b^3}+\frac {(10 b c-9 a d) e (e x)^{5/2} \sqrt [4]{a+b x^2}}{30 b^2}+\frac {d (e x)^{9/2} \sqrt [4]{a+b x^2}}{5 b e}+\frac {\left (a^2 (10 b c-9 a d) e^3 \left (1+\frac {a}{b x^2}\right )^{3/4} (e x)^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {a e^2}{b x^4}\right )^{3/4} x^3} \, dx,x,\sqrt {e x}\right )}{12 b^3 \left (a+b x^2\right )^{3/4}}\\ &=-\frac {a (10 b c-9 a d) e^3 \sqrt {e x} \sqrt [4]{a+b x^2}}{12 b^3}+\frac {(10 b c-9 a d) e (e x)^{5/2} \sqrt [4]{a+b x^2}}{30 b^2}+\frac {d (e x)^{9/2} \sqrt [4]{a+b x^2}}{5 b e}-\frac {\left (a^2 (10 b c-9 a d) e^3 \left (1+\frac {a}{b x^2}\right )^{3/4} (e x)^{3/2}\right ) \text {Subst}\left (\int \frac {x}{\left (1+\frac {a e^2 x^4}{b}\right )^{3/4}} \, dx,x,\frac {1}{\sqrt {e x}}\right )}{12 b^3 \left (a+b x^2\right )^{3/4}}\\ &=-\frac {a (10 b c-9 a d) e^3 \sqrt {e x} \sqrt [4]{a+b x^2}}{12 b^3}+\frac {(10 b c-9 a d) e (e x)^{5/2} \sqrt [4]{a+b x^2}}{30 b^2}+\frac {d (e x)^{9/2} \sqrt [4]{a+b x^2}}{5 b e}-\frac {\left (a^2 (10 b c-9 a d) e^3 \left (1+\frac {a}{b x^2}\right )^{3/4} (e x)^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {a e^2 x^2}{b}\right )^{3/4}} \, dx,x,\frac {1}{e x}\right )}{24 b^3 \left (a+b x^2\right )^{3/4}}\\ &=-\frac {a (10 b c-9 a d) e^3 \sqrt {e x} \sqrt [4]{a+b x^2}}{12 b^3}+\frac {(10 b c-9 a d) e (e x)^{5/2} \sqrt [4]{a+b x^2}}{30 b^2}+\frac {d (e x)^{9/2} \sqrt [4]{a+b x^2}}{5 b e}-\frac {a^{3/2} (10 b c-9 a d) e^2 \left (1+\frac {a}{b x^2}\right )^{3/4} (e x)^{3/2} F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{12 b^{5/2} \left (a+b x^2\right )^{3/4}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.11, size = 123, normalized size = 0.68 \begin {gather*} \frac {e^3 \sqrt {e x} \left (\left (a+b x^2\right ) \left (45 a^2 d+4 b^2 x^2 \left (5 c+3 d x^2\right )-2 a b \left (25 c+9 d x^2\right )\right )+5 a^2 (10 b c-9 a d) \left (1+\frac {b x^2}{a}\right )^{3/4} \, _2F_1\left (\frac {1}{4},\frac {3}{4};\frac {5}{4};-\frac {b x^2}{a}\right )\right )}{60 b^3 \left (a+b x^2\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (e x \right )^{\frac {7}{2}} \left (d \,x^{2}+c \right )}{\left (b \,x^{2}+a \right )^{\frac {3}{4}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 57.56, size = 94, normalized size = 0.52 \begin {gather*} \frac {c e^{\frac {7}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{4}} \Gamma \left (\frac {13}{4}\right )} + \frac {d e^{\frac {7}{2}} x^{\frac {13}{2}} \Gamma \left (\frac {13}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {13}{4} \\ \frac {17}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{4}} \Gamma \left (\frac {17}{4}\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*} \text {could not integrate} \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 )}^{7/2}\,\left (d\,x^2+c\right )}{{\left (b\,x^2+a\right )}^{3/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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