Optimal. Leaf size=414 \[ -\frac {c^{3/4} e^{5/2} \sqrt {1-\frac {d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{d^{3/4} \sqrt {c-d x^2} (b c-a d)}+\frac {c^{3/4} e^{5/2} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{d^{3/4} \sqrt {c-d x^2} (b c-a d)}-\frac {\sqrt {a} \sqrt [4]{c} e^{5/2} \sqrt {1-\frac {d x^2}{c}} \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{\sqrt {b} \sqrt [4]{d} \sqrt {c-d x^2} (b c-a d)}+\frac {\sqrt {a} \sqrt [4]{c} e^{5/2} \sqrt {1-\frac {d x^2}{c}} \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{\sqrt {b} \sqrt [4]{d} \sqrt {c-d x^2} (b c-a d)}-\frac {e (e x)^{3/2}}{\sqrt {c-d x^2} (b c-a d)} \]
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Rubi [A] time = 0.69, 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, 471, 584, 307, 224, 221, 1200, 1199, 424, 490, 1219, 1218} \[ -\frac {c^{3/4} e^{5/2} \sqrt {1-\frac {d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{d^{3/4} \sqrt {c-d x^2} (b c-a d)}+\frac {c^{3/4} e^{5/2} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{d^{3/4} \sqrt {c-d x^2} (b c-a d)}-\frac {\sqrt {a} \sqrt [4]{c} e^{5/2} \sqrt {1-\frac {d x^2}{c}} \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{\sqrt {b} \sqrt [4]{d} \sqrt {c-d x^2} (b c-a d)}+\frac {\sqrt {a} \sqrt [4]{c} e^{5/2} \sqrt {1-\frac {d x^2}{c}} \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{\sqrt {b} \sqrt [4]{d} \sqrt {c-d x^2} (b c-a d)}-\frac {e (e x)^{3/2}}{\sqrt {c-d x^2} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 221
Rule 224
Rule 307
Rule 424
Rule 466
Rule 471
Rule 490
Rule 584
Rule 1199
Rule 1200
Rule 1218
Rule 1219
Rubi steps
\begin {align*} \int \frac {(e x)^{5/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {x^6}{\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 {e (e x)^{3/2}}{(b c-a d) \sqrt {c-d x^2}}+\frac {e \operatorname {Subst}\left (\int \frac {x^2 \left (3 a-\frac {b 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 )}{b c-a d}\\ &=-\frac {e (e x)^{3/2}}{(b c-a d) \sqrt {c-d x^2}}+\frac {e \operatorname {Subst}\left (\int \left (\frac {x^2}{\sqrt {c-\frac {d x^4}{e^2}}}+\frac {2 a x^2}{\left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}}\right ) \, dx,x,\sqrt {e x}\right )}{b c-a d}\\ &=-\frac {e (e x)^{3/2}}{(b c-a d) \sqrt {c-d x^2}}+\frac {e \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{b c-a d}+\frac {(2 a 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 c-a d}\\ &=-\frac {e (e x)^{3/2}}{(b c-a d) \sqrt {c-d x^2}}-\frac {\left (\sqrt {c} e^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{\sqrt {d} (b c-a d)}+\frac {\left (\sqrt {c} 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 )}{\sqrt {d} (b c-a d)}+\frac {\left (a 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 )}{\sqrt {b} (b c-a d)}-\frac {\left (a 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 )}{\sqrt {b} (b c-a d)}\\ &=-\frac {e (e x)^{3/2}}{(b c-a d) \sqrt {c-d x^2}}-\frac {\left (\sqrt {c} 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 )}{\sqrt {d} (b c-a d) \sqrt {c-d x^2}}+\frac {\left (\sqrt {c} 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 )}{\sqrt {d} (b c-a d) \sqrt {c-d x^2}}+\frac {\left (a 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 )}{\sqrt {b} (b c-a d) \sqrt {c-d x^2}}-\frac {\left (a 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 )}{\sqrt {b} (b c-a d) \sqrt {c-d x^2}}\\ &=-\frac {e (e x)^{3/2}}{(b c-a d) \sqrt {c-d x^2}}-\frac {c^{3/4} 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 )}{d^{3/4} (b c-a d) \sqrt {c-d x^2}}-\frac {\sqrt {a} \sqrt [4]{c} 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 )}{\sqrt {b} \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}}+\frac {\sqrt {a} \sqrt [4]{c} 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 )}{\sqrt {b} \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}}+\frac {\left (\sqrt {c} 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 )}{\sqrt {d} (b c-a d) \sqrt {c-d x^2}}\\ &=-\frac {e (e x)^{3/2}}{(b c-a d) \sqrt {c-d x^2}}+\frac {c^{3/4} 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 )}{d^{3/4} (b c-a d) \sqrt {c-d x^2}}-\frac {c^{3/4} 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 )}{d^{3/4} (b c-a d) \sqrt {c-d x^2}}-\frac {\sqrt {a} \sqrt [4]{c} 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 )}{\sqrt {b} \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}}+\frac {\sqrt {a} \sqrt [4]{c} 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 )}{\sqrt {b} \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}}\\ \end {align*}
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Mathematica [C] time = 0.10, size = 133, normalized size = 0.32 \[ -\frac {e (e x)^{3/2} \left (b x^2 \sqrt {1-\frac {d x^2}{c}} F_1\left (\frac {7}{4};\frac {1}{2},1;\frac {11}{4};\frac {d x^2}{c},\frac {b x^2}{a}\right )-7 a \sqrt {1-\frac {d x^2}{c}} F_1\left (\frac {3}{4};\frac {1}{2},1;\frac {7}{4};\frac {d x^2}{c},\frac {b x^2}{a}\right )+7 a\right )}{7 a \sqrt {c-d x^2} (b c-a d)} \]
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 \[ \int -\frac {\left (e x\right )^{\frac {5}{2}}}{{\left (b x^{2} - a\right )} {\left (-d x^{2} + c\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 841, normalized size = 2.03 \[ -\frac {\left (-2 a b \,d^{2} x^{2}+2 b^{2} c d \,x^{2}-2 \sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, a b c d \EllipticE \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )+\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, a b c d \EllipticF \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )-\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, a b c d \EllipticPi \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {c d}\, b -\sqrt {a b}\, d}, \frac {\sqrt {2}}{2}\right )-\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, a b c d \EllipticPi \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {c d}\, b +\sqrt {a b}\, d}, \frac {\sqrt {2}}{2}\right )+2 \sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, b^{2} c^{2} \EllipticE \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )-\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, b^{2} c^{2} \EllipticF \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )-\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \sqrt {a b}\, \sqrt {c d}\, a d \EllipticPi \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {c d}\, b -\sqrt {a b}\, d}, \frac {\sqrt {2}}{2}\right )+\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \sqrt {a b}\, \sqrt {c d}\, a d \EllipticPi \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {c d}\, b +\sqrt {a b}\, d}, \frac {\sqrt {2}}{2}\right )\right ) \sqrt {-d \,x^{2}+c}\, \sqrt {e x}\, e^{2}}{2 \left (\sqrt {c d}\, b -\sqrt {a b}\, d \right ) \left (\sqrt {c d}\, b +\sqrt {a b}\, d \right ) \left (a d -b c \right ) \left (d \,x^{2}-c \right ) x} \]
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 \[ -\int \frac {\left (e x\right )^{\frac {5}{2}}}{{\left (b x^{2} - a\right )} {\left (-d x^{2} + c\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (e\,x\right )}^{5/2}}{\left (a-b\,x^2\right )\,{\left (c-d\,x^2\right )}^{3/2}} \,d x \]
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 \[ - \int \frac {\left (e x\right )^{\frac {5}{2}}}{- a c \sqrt {c - d x^{2}} + a d x^{2} \sqrt {c - d x^{2}} + b c x^{2} \sqrt {c - d x^{2}} - b d x^{4} \sqrt {c - d x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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