Optimal. Leaf size=366 \[ \frac {(b c-a d) \sqrt {e x} \sqrt {c-d x^2}}{2 a b e \left (a-b x^2\right )}+\frac {\sqrt [4]{c} d^{3/4} (b c+3 a d) \sqrt {1-\frac {d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a b^2 \sqrt {e} \sqrt {c-d x^2}}+\frac {3 \sqrt [4]{c} (b c-a d) (b c+a d) \sqrt {1-\frac {d x^2}{c}} \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a^2 b^2 \sqrt [4]{d} \sqrt {e} \sqrt {c-d x^2}}+\frac {3 \sqrt [4]{c} (b c-a d) (b c+a d) \sqrt {1-\frac {d x^2}{c}} \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a^2 b^2 \sqrt [4]{d} \sqrt {e} \sqrt {c-d x^2}} \]
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Rubi [A]
time = 0.41, antiderivative size = 366, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 8, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {477, 424, 537,
230, 227, 418, 1233, 1232} \begin {gather*} \frac {3 \sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} (a d+b c) (b c-a d) \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 )}{4 a^2 b^2 \sqrt [4]{d} \sqrt {e} \sqrt {c-d x^2}}+\frac {3 \sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} (a d+b c) (b c-a d) \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 )}{4 a^2 b^2 \sqrt [4]{d} \sqrt {e} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} d^{3/4} \sqrt {1-\frac {d x^2}{c}} (3 a d+b c) F\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a b^2 \sqrt {e} \sqrt {c-d x^2}}+\frac {\sqrt {e x} \sqrt {c-d x^2} (b c-a d)}{2 a b e \left (a-b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 227
Rule 230
Rule 418
Rule 424
Rule 477
Rule 537
Rule 1232
Rule 1233
Rubi steps
\begin {align*} \int \frac {\left (c-d x^2\right )^{3/2}}{\sqrt {e x} \left (a-b x^2\right )^2} \, dx &=\frac {2 \text {Subst}\left (\int \frac {\left (c-\frac {d x^4}{e^2}\right )^{3/2}}{\left (a-\frac {b x^4}{e^2}\right )^2} \, dx,x,\sqrt {e x}\right )}{e}\\ &=\frac {(b c-a d) \sqrt {e x} \sqrt {c-d x^2}}{2 a b e \left (a-b x^2\right )}-\frac {e \text {Subst}\left (\int \frac {-\frac {c (3 b c+a d)}{e^2}+\frac {d (b c+3 a d) 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 )}{2 a b}\\ &=\frac {(b c-a d) \sqrt {e x} \sqrt {c-d x^2}}{2 a b e \left (a-b x^2\right )}+\frac {(3 (b c-a d) (b c+a d)) \text {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 )}{2 a b^2 e}+\frac {(d (b c+3 a d)) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 a b^2 e}\\ &=\frac {(b c-a d) \sqrt {e x} \sqrt {c-d x^2}}{2 a b e \left (a-b x^2\right )}+\frac {(3 (b c-a d) (b c+a d)) \text {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 )}{4 a^2 b^2 e}+\frac {(3 (b c-a d) (b c+a d)) \text {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 )}{4 a^2 b^2 e}+\frac {\left (d (b c+3 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 )}{2 a b^2 e \sqrt {c-d x^2}}\\ &=\frac {(b c-a d) \sqrt {e x} \sqrt {c-d x^2}}{2 a b e \left (a-b x^2\right )}+\frac {\sqrt [4]{c} d^{3/4} (b c+3 a 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 )}{2 a b^2 \sqrt {e} \sqrt {c-d x^2}}+\frac {\left (3 (b c-a d) (b c+a d) \sqrt {1-\frac {d x^2}{c}}\right ) \text {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 )}{4 a^2 b^2 e \sqrt {c-d x^2}}+\frac {\left (3 (b c-a d) (b c+a d) \sqrt {1-\frac {d x^2}{c}}\right ) \text {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 )}{4 a^2 b^2 e \sqrt {c-d x^2}}\\ &=\frac {(b c-a d) \sqrt {e x} \sqrt {c-d x^2}}{2 a b e \left (a-b x^2\right )}+\frac {\sqrt [4]{c} d^{3/4} (b c+3 a 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 )}{2 a b^2 \sqrt {e} \sqrt {c-d x^2}}+\frac {3 \sqrt [4]{c} (b c-a d) (b c+a d) \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 )}{4 a^2 b^2 \sqrt [4]{d} \sqrt {e} \sqrt {c-d x^2}}+\frac {3 \sqrt [4]{c} (b c-a d) (b c+a d) \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 )}{4 a^2 b^2 \sqrt [4]{d} \sqrt {e} \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 = 10.15, size = 187, normalized size = 0.51 \begin {gather*} \frac {5 a (-b c+a d) x \left (c-d x^2\right )+5 c (3 b c+a d) x \left (-a+b x^2\right ) \sqrt {1-\frac {d x^2}{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 )+d (b c+3 a d) x^3 \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} F_1\left (\frac {5}{4};\frac {1}{2},1;\frac {9}{4};\frac {d x^2}{c},\frac {b x^2}{a}\right )}{10 a^2 b \sqrt {e x} \left (-a+b x^2\right ) \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. \(2518\) vs.
\(2(284)=568\).
time = 0.12, size = 2519, normalized size = 6.88
method | result | size |
elliptic | \(\frac {\sqrt {\left (-d \,x^{2}+c \right ) e x}\, \left (-\frac {\left (a d -b c \right ) \sqrt {-d e \,x^{3}+c e x}}{2 a e b \left (-b \,x^{2}+a \right )}+\frac {3 d \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}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{4 \sqrt {-d e \,x^{3}+c e x}\, b^{2}}+\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}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) c}{4 \sqrt {-d e \,x^{3}+c e x}\, a b}+\frac {3 a d \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 )}{8 b^{2} \sqrt {a b}\, \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}-\frac {3 \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 ) c^{2}}{8 a \sqrt {a b}\, d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}-\frac {3 a d \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 )}{8 b^{2} \sqrt {a b}\, \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}+\frac {3 \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 ) c^{2}}{8 a \sqrt {a b}\, 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}}\) | \(851\) |
default | \(\text {Expression too large to display}\) | \(2519\) |
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 {\left (c - d x^{2}\right )^{\frac {3}{2}}}{\sqrt {e x} \left (- a + b x^{2}\right )^{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 {{\left (c-d\,x^2\right )}^{3/2}}{\sqrt {e\,x}\,{\left (a-b\,x^2\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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