Optimal. Leaf size=414 \[ -\frac {\sqrt {a} \sqrt [4]{c} e^{5/2} \sqrt {1-\frac {d x^2}{c}} (b c-a d) \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt {a} \sqrt [4]{c} e^{5/2} \sqrt {1-\frac {d x^2}{c}} (b c-a d) \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {2 c^{3/4} e^{5/2} \sqrt {1-\frac {d x^2}{c}} (2 b c-5 a d) F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{5 b^2 d^{3/4} \sqrt {c-d x^2}}-\frac {2 c^{3/4} e^{5/2} \sqrt {1-\frac {d x^2}{c}} (2 b c-5 a d) E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{5 b^2 d^{3/4} \sqrt {c-d x^2}}-\frac {2 e (e x)^{3/2} \sqrt {c-d x^2}}{5 b} \]
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Rubi [A] time = 0.82, antiderivative size = 414, 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 = {466, 478, 584, 307, 224, 221, 1200, 1199, 424, 490, 1219, 1218} \[ \frac {2 c^{3/4} e^{5/2} \sqrt {1-\frac {d x^2}{c}} (2 b c-5 a d) F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{5 b^2 d^{3/4} \sqrt {c-d x^2}}-\frac {2 c^{3/4} e^{5/2} \sqrt {1-\frac {d x^2}{c}} (2 b c-5 a d) E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{5 b^2 d^{3/4} \sqrt {c-d x^2}}-\frac {\sqrt {a} \sqrt [4]{c} e^{5/2} \sqrt {1-\frac {d x^2}{c}} (b c-a d) \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt {a} \sqrt [4]{c} e^{5/2} \sqrt {1-\frac {d x^2}{c}} (b c-a d) \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {2 e (e x)^{3/2} \sqrt {c-d x^2}}{5 b} \]
Antiderivative was successfully verified.
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Rule 221
Rule 224
Rule 307
Rule 424
Rule 466
Rule 478
Rule 490
Rule 584
Rule 1199
Rule 1200
Rule 1218
Rule 1219
Rubi steps
\begin {align*} \int \frac {(e x)^{5/2} \sqrt {c-d x^2}}{a-b x^2} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {x^6 \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)^{3/2} \sqrt {c-d x^2}}{5 b}+\frac {(2 e) \operatorname {Subst}\left (\int \frac {x^2 \left (3 a c+\frac {(2 b c-5 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 )}{5 b}\\ &=-\frac {2 e (e x)^{3/2} \sqrt {c-d x^2}}{5 b}+\frac {(2 e) \operatorname {Subst}\left (\int \left (-\frac {(2 b c-5 a d) x^2}{b \sqrt {c-\frac {d x^4}{e^2}}}-\frac {5 \left (-a b c+a^2 d\right ) 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 )}{5 b}\\ &=-\frac {2 e (e x)^{3/2} \sqrt {c-d x^2}}{5 b}-\frac {(2 (2 b c-5 a d) e) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{5 b^2}+\frac {(2 a (b c-a d) e) \operatorname {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^2}\\ &=-\frac {2 e (e x)^{3/2} \sqrt {c-d x^2}}{5 b}+\frac {\left (2 \sqrt {c} (2 b c-5 a d) e^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{5 b^2 \sqrt {d}}-\frac {\left (2 \sqrt {c} (2 b c-5 a d) e^2\right ) \operatorname {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 )}{5 b^2 \sqrt {d}}+\frac {\left (a (b c-a d) e^3\right ) \operatorname {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^{5/2}}-\frac {\left (a (b c-a d) e^3\right ) \operatorname {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^{5/2}}\\ &=-\frac {2 e (e x)^{3/2} \sqrt {c-d x^2}}{5 b}+\frac {\left (2 \sqrt {c} (2 b c-5 a d) e^2 \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 )}{5 b^2 \sqrt {d} \sqrt {c-d x^2}}-\frac {\left (2 \sqrt {c} (2 b c-5 a d) e^2 \sqrt {1-\frac {d x^2}{c}}\right ) \operatorname {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 )}{5 b^2 \sqrt {d} \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 (\sqrt {a} e-\sqrt {b} x^2\right ) \sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{b^{5/2} \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 (\sqrt {a} e+\sqrt {b} x^2\right ) \sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{b^{5/2} \sqrt {c-d x^2}}\\ &=-\frac {2 e (e x)^{3/2} \sqrt {c-d x^2}}{5 b}+\frac {2 c^{3/4} (2 b c-5 a d) e^{5/2} \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 )}{5 b^2 d^{3/4} \sqrt {c-d x^2}}-\frac {\sqrt {a} \sqrt [4]{c} (b c-a d) e^{5/2} \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^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt {a} \sqrt [4]{c} (b c-a d) e^{5/2} \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^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {\left (2 \sqrt {c} (2 b c-5 a d) e^2 \sqrt {1-\frac {d x^2}{c}}\right ) \operatorname {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 )}{5 b^2 \sqrt {d} \sqrt {c-d x^2}}\\ &=-\frac {2 e (e x)^{3/2} \sqrt {c-d x^2}}{5 b}-\frac {2 c^{3/4} (2 b c-5 a d) e^{5/2} \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 )}{5 b^2 d^{3/4} \sqrt {c-d x^2}}+\frac {2 c^{3/4} (2 b c-5 a d) e^{5/2} \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 )}{5 b^2 d^{3/4} \sqrt {c-d x^2}}-\frac {\sqrt {a} \sqrt [4]{c} (b c-a d) e^{5/2} \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^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt {a} \sqrt [4]{c} (b c-a d) e^{5/2} \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^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}\\ \end {align*}
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Mathematica [C] time = 0.15, size = 143, normalized size = 0.35 \[ \frac {2 e (e x)^{3/2} \left (x^2 \sqrt {1-\frac {d x^2}{c}} (2 b c-5 a d) F_1\left (\frac {7}{4};\frac {1}{2},1;\frac {11}{4};\frac {d x^2}{c},\frac {b x^2}{a}\right )+7 a c \sqrt {1-\frac {d x^2}{c}} F_1\left (\frac {3}{4};\frac {1}{2},1;\frac {7}{4};\frac {d x^2}{c},\frac {b x^2}{a}\right )-7 a \left (c-d x^2\right )\right )}{35 a b \sqrt {c-d x^2}} \]
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 \[ \int -\frac {\sqrt {-d x^{2} + c} \left (e x\right )^{\frac {5}{2}}}{b x^{2} - a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 1491, normalized size = 3.60 \[ \text {result too large to display} \]
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 \[ -\int \frac {\sqrt {-d x^{2} + c} \left (e x\right )^{\frac {5}{2}}}{b x^{2} - a}\,{d x} \]
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 \[ \int \frac {{\left (e\,x\right )}^{5/2}\,\sqrt {c-d\,x^2}}{a-b\,x^2} \,d x \]
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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