Optimal. Leaf size=363 \[ \frac {e \sqrt {e x} \sqrt {c-d x^2}}{2 (b c-a d) \left (a-b x^2\right )}+\frac {\sqrt [4]{c} d^{3/4} e^{3/2} \sqrt {1-\frac {d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 b (b c-a d) \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c+a d) e^{3/2} \sqrt {1-\frac {d x^2}{c}} \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a b \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c+a d) e^{3/2} \sqrt {1-\frac {d x^2}{c}} \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a b \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}} \]
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Rubi [A]
time = 0.36, antiderivative size = 363, 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, 482, 537,
230, 227, 418, 1233, 1232} \begin {gather*} \frac {\sqrt [4]{c} d^{3/4} e^{3/2} \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 )}{2 b \sqrt {c-d x^2} (b c-a d)}-\frac {\sqrt [4]{c} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (a d+b 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 )}{4 a b \sqrt [4]{d} \sqrt {c-d x^2} (b c-a d)}-\frac {\sqrt [4]{c} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (a d+b 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 )}{4 a b \sqrt [4]{d} \sqrt {c-d x^2} (b c-a d)}+\frac {e \sqrt {e x} \sqrt {c-d x^2}}{2 \left (a-b x^2\right ) (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 227
Rule 230
Rule 418
Rule 477
Rule 482
Rule 537
Rule 1232
Rule 1233
Rubi steps
\begin {align*} \int \frac {(e x)^{3/2}}{\left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx &=\frac {2 \text {Subst}\left (\int \frac {x^4}{\left (a-\frac {b x^4}{e^2}\right )^2 \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{e}\\ &=\frac {e \sqrt {e x} \sqrt {c-d x^2}}{2 (b c-a d) \left (a-b x^2\right )}-\frac {e \text {Subst}\left (\int \frac {c+\frac {d x^4}{e^2}}{\left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 (b c-a d)}\\ &=\frac {e \sqrt {e x} \sqrt {c-d x^2}}{2 (b c-a d) \left (a-b x^2\right )}+\frac {(d e) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 b (b c-a d)}-\frac {((b c+a d) e) \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 b (b c-a d)}\\ &=\frac {e \sqrt {e x} \sqrt {c-d x^2}}{2 (b c-a d) \left (a-b x^2\right )}-\frac {((b c+a d) e) \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 b (b c-a d)}-\frac {((b c+a d) e) \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 b (b c-a d)}+\frac {\left (d e \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 b (b c-a d) \sqrt {c-d x^2}}\\ &=\frac {e \sqrt {e x} \sqrt {c-d x^2}}{2 (b c-a d) \left (a-b x^2\right )}+\frac {\sqrt [4]{c} d^{3/4} e^{3/2} \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 )}{2 b (b c-a d) \sqrt {c-d x^2}}-\frac {\left ((b c+a d) e \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 b (b c-a d) \sqrt {c-d x^2}}-\frac {\left ((b c+a d) e \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 b (b c-a d) \sqrt {c-d x^2}}\\ &=\frac {e \sqrt {e x} \sqrt {c-d x^2}}{2 (b c-a d) \left (a-b x^2\right )}+\frac {\sqrt [4]{c} d^{3/4} e^{3/2} \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 )}{2 b (b c-a d) \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c+a d) e^{3/2} \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 )}{4 a b \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c+a d) e^{3/2} \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 )}{4 a b \sqrt [4]{d} (b c-a 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 = 10.12, size = 169, normalized size = 0.47 \begin {gather*} -\frac {e \sqrt {e x} \left (5 a \left (c-d x^2\right )+5 c \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 x^2 \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 )\right )}{10 a (-b c+a d) \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. \(2245\) vs.
\(2(281)=562\).
time = 0.12, size = 2246, normalized size = 6.19
method | result | size |
elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (-d \,x^{2}+c \right ) e x}\, \left (-\frac {e \sqrt {-d e \,x^{3}+c e x}}{2 \left (a d -b c \right ) \left (-b \,x^{2}+a \right )}-\frac {e^{2} \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 \left (a d -b c \right ) b \sqrt {-d e \,x^{3}+c e x}}-\frac {e^{2} \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 ) a}{8 \left (a d -b c \right ) b \sqrt {a b}\, \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}-\frac {e^{2} \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}{8 \left (a d -b c \right ) \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 {e^{2} \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 ) a}{8 \left (a d -b c \right ) b \sqrt {a b}\, \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}+\frac {e^{2} \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}{8 \left (a d -b c \right ) \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 )}{e x \sqrt {-d \,x^{2}+c}}\) | \(813\) |
default | \(\text {Expression too large to display}\) | \(2246\) |
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 (e x\right )^{\frac {3}{2}}}{\left (- a + b x^{2}\right )^{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 {{\left (e\,x\right )}^{3/2}}{{\left (a-b\,x^2\right )}^2\,\sqrt {c-d\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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