Optimal. Leaf size=379 \[ -\frac {2 \sqrt {c-d x^2}}{a c e \sqrt {e x}}-\frac {2 \sqrt [4]{d} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{a \sqrt [4]{c} e^{3/2} \sqrt {c-d x^2}}+\frac {2 \sqrt [4]{d} \sqrt {1-\frac {d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{a \sqrt [4]{c} e^{3/2} \sqrt {c-d x^2}}-\frac {\sqrt {b} \sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{a^{3/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt {b} \sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{a^{3/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}} \]
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Rubi [A]
time = 0.47, antiderivative size = 379, 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, 491,
598, 313, 230, 227, 1214, 1213, 435, 504, 1233, 1232} \begin {gather*} -\frac {\sqrt {b} \sqrt [4]{c} \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 )}{a^{3/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt {b} \sqrt [4]{c} \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 )}{a^{3/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {2 \sqrt [4]{d} \sqrt {1-\frac {d x^2}{c}} F\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{a \sqrt [4]{c} e^{3/2} \sqrt {c-d x^2}}-\frac {2 \sqrt [4]{d} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{a \sqrt [4]{c} e^{3/2} \sqrt {c-d x^2}}-\frac {2 \sqrt {c-d x^2}}{a c e \sqrt {e x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 227
Rule 230
Rule 313
Rule 435
Rule 477
Rule 491
Rule 504
Rule 598
Rule 1213
Rule 1214
Rule 1232
Rule 1233
Rubi steps
\begin {align*} \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right ) \sqrt {c-d x^2}} \, dx &=\frac {2 \text {Subst}\left (\int \frac {1}{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 )}{e}\\ &=-\frac {2 \sqrt {c-d x^2}}{a c e \sqrt {e x}}+\frac {2 \text {Subst}\left (\int \frac {x^2 \left (\frac {b c-a d}{e^2}+\frac {b d x^4}{e^4}\right )}{\left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{a c e}\\ &=-\frac {2 \sqrt {c-d x^2}}{a c e \sqrt {e x}}+\frac {2 \text {Subst}\left (\int \left (-\frac {d x^2}{e^2 \sqrt {c-\frac {d x^4}{e^2}}}+\frac {b c x^2}{e^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 )}{a c e}\\ &=-\frac {2 \sqrt {c-d x^2}}{a c e \sqrt {e x}}+\frac {(2 b) \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 )}{a e^3}-\frac {(2 d) \text {Subst}\left (\int \frac {x^2}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{a c e^3}\\ &=-\frac {2 \sqrt {c-d x^2}}{a c e \sqrt {e x}}+\frac {\left (2 \sqrt {d}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{a \sqrt {c} e^2}-\frac {\left (2 \sqrt {d}\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 )}{a \sqrt {c} e^2}+\frac {\sqrt {b} \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 )}{a e}-\frac {\sqrt {b} \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 )}{a e}\\ &=-\frac {2 \sqrt {c-d x^2}}{a c e \sqrt {e x}}+\frac {\left (2 \sqrt {d} \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 )}{a \sqrt {c} e^2 \sqrt {c-d x^2}}-\frac {\left (2 \sqrt {d} \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 )}{a \sqrt {c} e^2 \sqrt {c-d x^2}}+\frac {\left (\sqrt {b} \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 )}{a e \sqrt {c-d x^2}}-\frac {\left (\sqrt {b} \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 )}{a e \sqrt {c-d x^2}}\\ &=-\frac {2 \sqrt {c-d x^2}}{a c e \sqrt {e x}}+\frac {2 \sqrt [4]{d} \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 )}{a \sqrt [4]{c} e^{3/2} \sqrt {c-d x^2}}-\frac {\sqrt {b} \sqrt [4]{c} \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 )}{a^{3/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt {b} \sqrt [4]{c} \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 )}{a^{3/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}-\frac {\left (2 \sqrt {d} \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 )}{a \sqrt {c} e^2 \sqrt {c-d x^2}}\\ &=-\frac {2 \sqrt {c-d x^2}}{a c e \sqrt {e x}}-\frac {2 \sqrt [4]{d} \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 )}{a \sqrt [4]{c} e^{3/2} \sqrt {c-d x^2}}+\frac {2 \sqrt [4]{d} \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 )}{a \sqrt [4]{c} e^{3/2} \sqrt {c-d x^2}}-\frac {\sqrt {b} \sqrt [4]{c} \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 )}{a^{3/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt {b} \sqrt [4]{c} \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 )}{a^{3/2} \sqrt [4]{d} e^{3/2} \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 = 10.11, size = 146, normalized size = 0.39 \begin {gather*} \frac {x \left (-42 a \left (c-d x^2\right )+14 (b c-a d) x^2 \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 )+6 b d x^4 \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 )}{21 a^2 c (e x)^{3/2} \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. \(825\) vs.
\(2(279)=558\).
time = 0.12, size = 826, normalized size = 2.18
method | result | size |
elliptic | \(\frac {\sqrt {\left (-d \,x^{2}+c \right ) e x}\, \left (-\frac {2 \left (-d e \,x^{2}+c e \right )}{e^{2} c a \sqrt {x \left (-d e \,x^{2}+c e \right )}}+\frac {2 \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{e a \sqrt {-d e \,x^{3}+c e x}}-\frac {\sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{e a \sqrt {-d e \,x^{3}+c e x}}-\frac {\sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 e a d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}-\frac {\sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 e a d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}\right )}{\sqrt {e x}\, \sqrt {-d \,x^{2}+c}}\) | \(530\) |
default | \(-\frac {\left (4 \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}}}\, \EllipticE \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) a c d -4 \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}}}\, \EllipticE \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) b \,c^{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}}}\, \EllipticF \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) a c d +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}}}\, \EllipticF \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) b \,c^{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}}}\, \sqrt {a b}\, \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 {c d}\, c -\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}\, \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 {c d}\, c -\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}}}\, \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 ) b \,c^{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}}}\, \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 ) b \,c^{2}+4 a \,d^{2} x^{2}-4 b c d \,x^{2}-4 a c d +4 b \,c^{2}\right ) d b}{2 \sqrt {-d \,x^{2}+c}\, c \left (\sqrt {c d}\, b -\sqrt {a b}\, d \right ) \left (\sqrt {c d}\, b +\sqrt {a b}\, d \right ) a e \sqrt {e x}}\) | \(826\) |
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 {1}{- a \left (e x\right )^{\frac {3}{2}} \sqrt {c - d x^{2}} + b x^{2} \left (e x\right )^{\frac {3}{2}} \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 {1}{{\left (e\,x\right )}^{3/2}\,\left (a-b\,x^2\right )\,\sqrt {c-d\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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