Optimal. Leaf size=421 \[ \frac {2 d (e x)^{3/2} \sqrt {c-d x^2}}{5 b e}+\frac {2 c^{3/4} \sqrt [4]{d} (7 b c-5 a d) \sqrt {e} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{5 b^2 \sqrt {c-d x^2}}-\frac {2 c^{3/4} \sqrt [4]{d} (7 b c-5 a d) \sqrt {e} \sqrt {1-\frac {d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{5 b^2 \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c-a d)^2 \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{\sqrt {a} b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d)^2 \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{\sqrt {a} b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}} \]
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Rubi [A]
time = 0.57, antiderivative size = 421, 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, 488,
598, 313, 230, 227, 1214, 1213, 435, 504, 1233, 1232} \begin {gather*} -\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (b c-a d)^2 \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 )}{\sqrt {a} b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (b c-a d)^2 \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 )}{\sqrt {a} b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {2 c^{3/4} \sqrt [4]{d} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (7 b c-5 a d) F\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{5 b^2 \sqrt {c-d x^2}}+\frac {2 c^{3/4} \sqrt [4]{d} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (7 b c-5 a d) E\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{5 b^2 \sqrt {c-d x^2}}+\frac {2 d (e x)^{3/2} \sqrt {c-d x^2}}{5 b e} \end {gather*}
Antiderivative was successfully verified.
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Rule 227
Rule 230
Rule 313
Rule 435
Rule 477
Rule 488
Rule 504
Rule 598
Rule 1213
Rule 1214
Rule 1232
Rule 1233
Rubi steps
\begin {align*} \int \frac {\sqrt {e x} \left (c-d x^2\right )^{3/2}}{a-b x^2} \, dx &=\frac {2 \text {Subst}\left (\int \frac {x^2 \left (c-\frac {d x^4}{e^2}\right )^{3/2}}{a-\frac {b x^4}{e^2}} \, dx,x,\sqrt {e x}\right )}{e}\\ &=\frac {2 d (e x)^{3/2} \sqrt {c-d x^2}}{5 b e}-\frac {(2 e) \text {Subst}\left (\int \frac {x^2 \left (-\frac {c (5 b c-3 a d)}{e^2}+\frac {d (7 b c-5 a 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 )}{5 b}\\ &=\frac {2 d (e x)^{3/2} \sqrt {c-d x^2}}{5 b e}-\frac {(2 e) \text {Subst}\left (\int \left (-\frac {d (7 b c-5 a d) x^2}{b e^2 \sqrt {c-\frac {d x^4}{e^2}}}-\frac {5 \left (b^2 c^2-2 a b c d+a^2 d^2\right ) x^2}{b 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 )}{5 b}\\ &=\frac {2 d (e x)^{3/2} \sqrt {c-d x^2}}{5 b e}+\frac {(2 d (7 b c-5 a d)) \text {Subst}\left (\int \frac {x^2}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{5 b^2 e}+\frac {\left (2 (b c-a d)^2\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^2 e}\\ &=\frac {2 d (e x)^{3/2} \sqrt {c-d x^2}}{5 b e}-\frac {\left (2 \sqrt {c} \sqrt {d} (7 b c-5 a d)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{5 b^2}+\frac {\left (2 \sqrt {c} \sqrt {d} (7 b c-5 a 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 )}{5 b^2}+\frac {\left ((b c-a d)^2 e\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^{5/2}}-\frac {\left ((b c-a d)^2 e\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^{5/2}}\\ &=\frac {2 d (e x)^{3/2} \sqrt {c-d x^2}}{5 b e}-\frac {\left (2 \sqrt {c} \sqrt {d} (7 b c-5 a 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 )}{5 b^2 \sqrt {c-d x^2}}+\frac {\left (2 \sqrt {c} \sqrt {d} (7 b c-5 a 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 )}{5 b^2 \sqrt {c-d x^2}}+\frac {\left ((b c-a d)^2 e \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^{5/2} \sqrt {c-d x^2}}-\frac {\left ((b c-a d)^2 e \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^{5/2} \sqrt {c-d x^2}}\\ &=\frac {2 d (e x)^{3/2} \sqrt {c-d x^2}}{5 b e}-\frac {2 c^{3/4} \sqrt [4]{d} (7 b c-5 a d) \sqrt {e} \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 \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c-a d)^2 \sqrt {e} \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 {a} b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d)^2 \sqrt {e} \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 {a} b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\left (2 \sqrt {c} \sqrt {d} (7 b c-5 a 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 )}{5 b^2 \sqrt {c-d x^2}}\\ &=\frac {2 d (e x)^{3/2} \sqrt {c-d x^2}}{5 b e}+\frac {2 c^{3/4} \sqrt [4]{d} (7 b c-5 a d) \sqrt {e} \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 \sqrt {c-d x^2}}-\frac {2 c^{3/4} \sqrt [4]{d} (7 b c-5 a d) \sqrt {e} \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 \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c-a d)^2 \sqrt {e} \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 {a} b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d)^2 \sqrt {e} \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 {a} b^{5/2} \sqrt [4]{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 = 10.18, size = 155, normalized size = 0.37 \begin {gather*} \frac {2 x \sqrt {e x} \left (7 c (5 b c-3 a d) \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 d \left (7 a \left (c-d x^2\right )+(-7 b c+5 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 )\right )}{105 a b \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. \(1915\) vs.
\(2(315)=630\).
time = 0.15, size = 1916, normalized size = 4.55
method | result | size |
risch | \(\frac {2 d \sqrt {-d \,x^{2}+c}\, x^{2} e}{5 b \sqrt {e x}}-\frac {\left (\frac {\left (5 a d -7 b c \right ) \sqrt {c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \left (-\frac {2 \sqrt {c d}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {c d}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{b \sqrt {-d e \,x^{3}+c e x}}+\frac {5 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {\sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 b d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}+\frac {\sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 b d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}\right )}{b}\right ) e \sqrt {\left (-d \,x^{2}+c \right ) e x}}{5 b \sqrt {e x}\, \sqrt {-d \,x^{2}+c}}\) | \(534\) |
elliptic | \(\text {Expression too large to display}\) | \(1272\) |
default | \(\text {Expression too large to display}\) | \(1916\) |
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 {c \sqrt {e x} \sqrt {c - d x^{2}}}{- a + b x^{2}}\, dx - \int \left (- \frac {d x^{2} \sqrt {e x} \sqrt {c - d x^{2}}}{- a + b x^{2}}\right )\, 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 {\sqrt {e\,x}\,{\left (c-d\,x^2\right )}^{3/2}}{a-b\,x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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