Optimal. Leaf size=444 \[ -\frac {c e^3 (e x)^{3/2}}{d (b c-a d) \sqrt {c-d x^2}}+\frac {c^{3/4} (3 b c-2 a d) e^{9/2} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b d^{7/4} (b c-a d) \sqrt {c-d x^2}}-\frac {c^{3/4} (3 b c-2 a d) e^{9/2} \sqrt {1-\frac {d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b d^{7/4} (b c-a d) \sqrt {c-d x^2}}-\frac {a^{3/2} \sqrt [4]{c} e^{9/2} \sqrt {1-\frac {d x^2}{c}} \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b^{3/2} \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}}+\frac {a^{3/2} \sqrt [4]{c} e^{9/2} \sqrt {1-\frac {d x^2}{c}} \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b^{3/2} \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}} \]
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Rubi [A]
time = 0.56, antiderivative size = 444, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 12, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {477, 481,
598, 313, 230, 227, 1214, 1213, 435, 504, 1233, 1232} \begin {gather*} -\frac {a^{3/2} \sqrt [4]{c} e^{9/2} \sqrt {1-\frac {d x^2}{c}} \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\text {ArcSin}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b^{3/2} \sqrt [4]{d} \sqrt {c-d x^2} (b c-a d)}+\frac {a^{3/2} \sqrt [4]{c} e^{9/2} \sqrt {1-\frac {d x^2}{c}} \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\text {ArcSin}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b^{3/2} \sqrt [4]{d} \sqrt {c-d x^2} (b c-a d)}-\frac {c^{3/4} e^{9/2} \sqrt {1-\frac {d x^2}{c}} (3 b c-2 a d) F\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b d^{7/4} \sqrt {c-d x^2} (b c-a d)}+\frac {c^{3/4} e^{9/2} \sqrt {1-\frac {d x^2}{c}} (3 b c-2 a d) E\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b d^{7/4} \sqrt {c-d x^2} (b c-a d)}-\frac {c e^3 (e x)^{3/2}}{d \sqrt {c-d x^2} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 227
Rule 230
Rule 313
Rule 435
Rule 477
Rule 481
Rule 504
Rule 598
Rule 1213
Rule 1214
Rule 1232
Rule 1233
Rubi steps
\begin {align*} \int \frac {(e x)^{9/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx &=\frac {2 \text {Subst}\left (\int \frac {x^{10}}{\left (a-\frac {b x^4}{e^2}\right ) \left (c-\frac {d x^4}{e^2}\right )^{3/2}} \, dx,x,\sqrt {e x}\right )}{e}\\ &=-\frac {c e^3 (e x)^{3/2}}{d (b c-a d) \sqrt {c-d x^2}}+\frac {e^3 \text {Subst}\left (\int \frac {x^2 \left (3 a c-\frac {(3 b c-2 a d) x^4}{e^2}\right )}{\left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{d (b c-a d)}\\ &=-\frac {c e^3 (e x)^{3/2}}{d (b c-a d) \sqrt {c-d x^2}}+\frac {e^3 \text {Subst}\left (\int \left (\frac {(3 b c-2 a d) x^2}{b \sqrt {c-\frac {d x^4}{e^2}}}+\frac {2 a^2 d x^2}{b \left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}}\right ) \, dx,x,\sqrt {e x}\right )}{d (b c-a d)}\\ &=-\frac {c e^3 (e x)^{3/2}}{d (b c-a d) \sqrt {c-d x^2}}+\frac {\left (2 a^2 e^3\right ) \text {Subst}\left (\int \frac {x^2}{\left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{b (b c-a d)}+\frac {\left ((3 b c-2 a d) e^3\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{b d (b c-a d)}\\ &=-\frac {c e^3 (e x)^{3/2}}{d (b c-a d) \sqrt {c-d x^2}}-\frac {\left (\sqrt {c} (3 b c-2 a d) e^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{b d^{3/2} (b c-a d)}+\frac {\left (\sqrt {c} (3 b c-2 a d) e^4\right ) \text {Subst}\left (\int \frac {1+\frac {\sqrt {d} x^2}{\sqrt {c} e}}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{b d^{3/2} (b c-a d)}+\frac {\left (a^2 e^5\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a} e-\sqrt {b} x^2\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{b^{3/2} (b c-a d)}-\frac {\left (a^2 e^5\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a} e+\sqrt {b} x^2\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{b^{3/2} (b c-a d)}\\ &=-\frac {c e^3 (e x)^{3/2}}{d (b c-a d) \sqrt {c-d x^2}}-\frac {\left (\sqrt {c} (3 b c-2 a d) e^4 \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{b d^{3/2} (b c-a d) \sqrt {c-d x^2}}+\frac {\left (\sqrt {c} (3 b c-2 a d) e^4 \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {1+\frac {\sqrt {d} x^2}{\sqrt {c} e}}{\sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{b d^{3/2} (b c-a d) \sqrt {c-d x^2}}+\frac {\left (a^2 e^5 \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a} e-\sqrt {b} x^2\right ) \sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{b^{3/2} (b c-a d) \sqrt {c-d x^2}}-\frac {\left (a^2 e^5 \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a} e+\sqrt {b} x^2\right ) \sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{b^{3/2} (b c-a d) \sqrt {c-d x^2}}\\ &=-\frac {c e^3 (e x)^{3/2}}{d (b c-a d) \sqrt {c-d x^2}}-\frac {c^{3/4} (3 b c-2 a d) e^{9/2} \sqrt {1-\frac {d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b d^{7/4} (b c-a d) \sqrt {c-d x^2}}-\frac {a^{3/2} \sqrt [4]{c} e^{9/2} \sqrt {1-\frac {d x^2}{c}} \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b^{3/2} \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}}+\frac {a^{3/2} \sqrt [4]{c} e^{9/2} \sqrt {1-\frac {d x^2}{c}} \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b^{3/2} \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}}+\frac {\left (\sqrt {c} (3 b c-2 a d) e^4 \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {\sqrt {d} x^2}{\sqrt {c} e}}}{\sqrt {1-\frac {\sqrt {d} x^2}{\sqrt {c} e}}} \, dx,x,\sqrt {e x}\right )}{b d^{3/2} (b c-a d) \sqrt {c-d x^2}}\\ &=-\frac {c e^3 (e x)^{3/2}}{d (b c-a d) \sqrt {c-d x^2}}+\frac {c^{3/4} (3 b c-2 a d) e^{9/2} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b d^{7/4} (b c-a d) \sqrt {c-d x^2}}-\frac {c^{3/4} (3 b c-2 a d) e^{9/2} \sqrt {1-\frac {d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b d^{7/4} (b c-a d) \sqrt {c-d x^2}}-\frac {a^{3/2} \sqrt [4]{c} e^{9/2} \sqrt {1-\frac {d x^2}{c}} \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b^{3/2} \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}}+\frac {a^{3/2} \sqrt [4]{c} e^{9/2} \sqrt {1-\frac {d x^2}{c}} \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b^{3/2} \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in
optimal.
time = 10.15, size = 148, normalized size = 0.33 \begin {gather*} -\frac {e^3 (e x)^{3/2} \left (-7 a c+7 a c \sqrt {1-\frac {d x^2}{c}} F_1\left (\frac {3}{4};\frac {1}{2},1;\frac {7}{4};\frac {d x^2}{c},\frac {b x^2}{a}\right )+(-3 b c+2 a d) x^2 \sqrt {1-\frac {d x^2}{c}} F_1\left (\frac {7}{4};\frac {1}{2},1;\frac {11}{4};\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{7 a d (-b c+a d) \sqrt {c-d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1029\) vs.
\(2(344)=688\).
time = 0.13, size = 1030, normalized size = 2.32 Too large to display
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(-1)] Timed out
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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {\left (e x\right )^{\frac {9}{2}}}{- a c \sqrt {c - d x^{2}} + a d x^{2} \sqrt {c - d x^{2}} + b c x^{2} \sqrt {c - d x^{2}} - b d x^{4} \sqrt {c - d x^{2}}}\, dx \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 = -0.00 \begin {gather*} \int \frac {{\left (e\,x\right )}^{9/2}}{\left (a-b\,x^2\right )\,{\left (c-d\,x^2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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