Optimal. Leaf size=372 \[ -\frac {2 \sqrt [4]{c} e^{7/2} \sqrt {1-\frac {d x^2}{c}} \left (-21 a^2 d^2+14 a b c d+2 b^2 c^2\right ) F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{21 b^3 d^{5/4} \sqrt {c-d x^2}}+\frac {a \sqrt [4]{c} e^{7/2} \sqrt {1-\frac {d x^2}{c}} (b c-a d) \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b^3 \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {a \sqrt [4]{c} e^{7/2} \sqrt {1-\frac {d x^2}{c}} (b c-a d) \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b^3 \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {2 e^3 \sqrt {e x} \sqrt {c-d x^2} (2 b c-7 a d)}{21 b^2 d}-\frac {2 e (e x)^{5/2} \sqrt {c-d x^2}}{7 b} \]
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Rubi [A] time = 0.80, antiderivative size = 372, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {466, 478, 582, 523, 224, 221, 409, 1219, 1218} \[ -\frac {2 \sqrt [4]{c} e^{7/2} \sqrt {1-\frac {d x^2}{c}} \left (-21 a^2 d^2+14 a b c d+2 b^2 c^2\right ) F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{21 b^3 d^{5/4} \sqrt {c-d x^2}}+\frac {2 e^3 \sqrt {e x} \sqrt {c-d x^2} (2 b c-7 a d)}{21 b^2 d}+\frac {a \sqrt [4]{c} e^{7/2} \sqrt {1-\frac {d x^2}{c}} (b c-a d) \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b^3 \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {a \sqrt [4]{c} e^{7/2} \sqrt {1-\frac {d x^2}{c}} (b c-a d) \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b^3 \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {2 e (e x)^{5/2} \sqrt {c-d x^2}}{7 b} \]
Antiderivative was successfully verified.
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Rule 221
Rule 224
Rule 409
Rule 466
Rule 478
Rule 523
Rule 582
Rule 1218
Rule 1219
Rubi steps
\begin {align*} \int \frac {(e x)^{7/2} \sqrt {c-d x^2}}{a-b x^2} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {x^8 \sqrt {c-\frac {d x^4}{e^2}}}{a-\frac {b x^4}{e^2}} \, dx,x,\sqrt {e x}\right )}{e}\\ &=-\frac {2 e (e x)^{5/2} \sqrt {c-d x^2}}{7 b}+\frac {(2 e) \operatorname {Subst}\left (\int \frac {x^4 \left (5 a c+\frac {(2 b c-7 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 )}{7 b}\\ &=\frac {2 (2 b c-7 a d) e^3 \sqrt {e x} \sqrt {c-d x^2}}{21 b^2 d}-\frac {2 e (e x)^{5/2} \sqrt {c-d x^2}}{7 b}-\frac {\left (2 e^5\right ) \operatorname {Subst}\left (\int \frac {\frac {a c (2 b c-7 a d)}{e^2}-\frac {\left (2 b^2 c^2+14 a b c d-21 a^2 d^2\right ) x^4}{e^4}}{\left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{21 b^2 d}\\ &=\frac {2 (2 b c-7 a d) e^3 \sqrt {e x} \sqrt {c-d x^2}}{21 b^2 d}-\frac {2 e (e x)^{5/2} \sqrt {c-d x^2}}{7 b}+\frac {\left (2 a^2 (b c-a d) e^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{b^3}-\frac {\left (2 \left (2 b^2 c^2+14 a b c d-21 a^2 d^2\right ) e^3\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{21 b^3 d}\\ &=\frac {2 (2 b c-7 a d) e^3 \sqrt {e x} \sqrt {c-d x^2}}{21 b^2 d}-\frac {2 e (e x)^{5/2} \sqrt {c-d x^2}}{7 b}+\frac {\left (a (b c-a d) e^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {\sqrt {b} x^2}{\sqrt {a} e}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{b^3}+\frac {\left (a (b c-a d) e^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {\sqrt {b} x^2}{\sqrt {a} e}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{b^3}-\frac {\left (2 \left (2 b^2 c^2+14 a b c d-21 a^2 d^2\right ) e^3 \sqrt {1-\frac {d x^2}{c}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{21 b^3 d \sqrt {c-d x^2}}\\ &=\frac {2 (2 b c-7 a d) e^3 \sqrt {e x} \sqrt {c-d x^2}}{21 b^2 d}-\frac {2 e (e x)^{5/2} \sqrt {c-d x^2}}{7 b}-\frac {2 \sqrt [4]{c} \left (2 b^2 c^2+14 a b c d-21 a^2 d^2\right ) e^{7/2} \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 )}{21 b^3 d^{5/4} \sqrt {c-d x^2}}+\frac {\left (a (b c-a d) e^3 \sqrt {1-\frac {d x^2}{c}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {\sqrt {b} x^2}{\sqrt {a} e}\right ) \sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{b^3 \sqrt {c-d x^2}}+\frac {\left (a (b c-a d) e^3 \sqrt {1-\frac {d x^2}{c}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {\sqrt {b} x^2}{\sqrt {a} e}\right ) \sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{b^3 \sqrt {c-d x^2}}\\ &=\frac {2 (2 b c-7 a d) e^3 \sqrt {e x} \sqrt {c-d x^2}}{21 b^2 d}-\frac {2 e (e x)^{5/2} \sqrt {c-d x^2}}{7 b}-\frac {2 \sqrt [4]{c} \left (2 b^2 c^2+14 a b c d-21 a^2 d^2\right ) e^{7/2} \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 )}{21 b^3 d^{5/4} \sqrt {c-d x^2}}+\frac {a \sqrt [4]{c} (b c-a d) e^{7/2} \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 \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {a \sqrt [4]{c} (b c-a d) e^{7/2} \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 \sqrt [4]{d} \sqrt {c-d x^2}}\\ \end {align*}
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Mathematica [C] time = 0.22, size = 187, normalized size = 0.50 \[ \frac {2 e^3 \sqrt {e x} \left (x^2 \sqrt {1-\frac {d x^2}{c}} \left (-21 a^2 d^2+14 a b c d+2 b^2 c^2\right ) F_1\left (\frac {5}{4};\frac {1}{2},1;\frac {9}{4};\frac {d x^2}{c},\frac {b x^2}{a}\right )+5 a c \sqrt {1-\frac {d x^2}{c}} (7 a d-2 b c) F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};\frac {d x^2}{c},\frac {b x^2}{a}\right )-5 a \left (c-d x^2\right ) \left (7 a d-2 b c+3 b d x^2\right )\right )}{105 a b^2 d \sqrt {c-d x^2}} \]
Warning: Unable to verify antiderivative.
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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 \[ \int -\frac {\sqrt {-d x^{2} + c} \left (e x\right )^{\frac {7}{2}}}{b x^{2} - a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.13, size = 1479, normalized size = 3.98 \[ \text {result too large to display} \]
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 \[ -\int \frac {\sqrt {-d x^{2} + c} \left (e x\right )^{\frac {7}{2}}}{b x^{2} - a}\,{d x} \]
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 \[ \int \frac {{\left (e\,x\right )}^{7/2}\,\sqrt {c-d\,x^2}}{a-b\,x^2} \,d x \]
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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