Optimal. Leaf size=355 \[ -\frac {7 \sqrt {c-d x^2}}{6 a^2 e (e x)^{3/2}}+\frac {\sqrt {c-d x^2}}{2 a e (e x)^{3/2} \left (a-b x^2\right )}+\frac {7 \sqrt [4]{c} d^{3/4} \sqrt {1-\frac {d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{6 a^2 e^{5/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (7 b c-5 a d) \sqrt {1-\frac {d x^2}{c}} \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a^3 \sqrt [4]{d} e^{5/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (7 b c-5 a d) \sqrt {1-\frac {d x^2}{c}} \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a^3 \sqrt [4]{d} e^{5/2} \sqrt {c-d x^2}} \]
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Rubi [A]
time = 0.44, antiderivative size = 355, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 9, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {477, 480, 597,
537, 230, 227, 418, 1233, 1232} \begin {gather*} \frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} (7 b c-5 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^3 \sqrt [4]{d} e^{5/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} (7 b c-5 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^3 \sqrt [4]{d} e^{5/2} \sqrt {c-d x^2}}+\frac {7 \sqrt [4]{c} d^{3/4} \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 )}{6 a^2 e^{5/2} \sqrt {c-d x^2}}-\frac {7 \sqrt {c-d x^2}}{6 a^2 e (e x)^{3/2}}+\frac {\sqrt {c-d x^2}}{2 a e (e x)^{3/2} \left (a-b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 227
Rule 230
Rule 418
Rule 477
Rule 480
Rule 537
Rule 597
Rule 1232
Rule 1233
Rubi steps
\begin {align*} \int \frac {\sqrt {c-d x^2}}{(e x)^{5/2} \left (a-b x^2\right )^2} \, dx &=\frac {2 \text {Subst}\left (\int \frac {\sqrt {c-\frac {d x^4}{e^2}}}{x^4 \left (a-\frac {b x^4}{e^2}\right )^2} \, dx,x,\sqrt {e x}\right )}{e}\\ &=\frac {\sqrt {c-d x^2}}{2 a e (e x)^{3/2} \left (a-b x^2\right )}-\frac {\text {Subst}\left (\int \frac {-7 c+\frac {5 d x^4}{e^2}}{x^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 e}\\ &=-\frac {7 \sqrt {c-d x^2}}{6 a^2 e (e x)^{3/2}}+\frac {\sqrt {c-d x^2}}{2 a e (e x)^{3/2} \left (a-b x^2\right )}+\frac {\text {Subst}\left (\int \frac {\frac {c (21 b c-8 a d)}{e^2}-\frac {7 b c 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 )}{6 a^2 c e}\\ &=-\frac {7 \sqrt {c-d x^2}}{6 a^2 e (e x)^{3/2}}+\frac {\sqrt {c-d x^2}}{2 a e (e x)^{3/2} \left (a-b x^2\right )}+\frac {(7 d) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{6 a^2 e^3}+\frac {(7 b c-5 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^2 e^3}\\ &=-\frac {7 \sqrt {c-d x^2}}{6 a^2 e (e x)^{3/2}}+\frac {\sqrt {c-d x^2}}{2 a e (e x)^{3/2} \left (a-b x^2\right )}+\frac {(7 b c-5 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^3 e^3}+\frac {(7 b c-5 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^3 e^3}+\frac {\left (7 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 )}{6 a^2 e^3 \sqrt {c-d x^2}}\\ &=-\frac {7 \sqrt {c-d x^2}}{6 a^2 e (e x)^{3/2}}+\frac {\sqrt {c-d x^2}}{2 a e (e x)^{3/2} \left (a-b x^2\right )}+\frac {7 \sqrt [4]{c} d^{3/4} \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 )}{6 a^2 e^{5/2} \sqrt {c-d x^2}}+\frac {\left ((7 b c-5 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^3 e^3 \sqrt {c-d x^2}}+\frac {\left ((7 b c-5 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^3 e^3 \sqrt {c-d x^2}}\\ &=-\frac {7 \sqrt {c-d x^2}}{6 a^2 e (e x)^{3/2}}+\frac {\sqrt {c-d x^2}}{2 a e (e x)^{3/2} \left (a-b x^2\right )}+\frac {7 \sqrt [4]{c} d^{3/4} \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 )}{6 a^2 e^{5/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (7 b c-5 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^3 \sqrt [4]{d} e^{5/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (7 b c-5 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^3 \sqrt [4]{d} e^{5/2} \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 = 10.15, size = 181, normalized size = 0.51 \begin {gather*} \frac {x \left (5 a \left (4 a-7 b x^2\right ) \left (c-d x^2\right )+5 (-21 b c+8 a d) x^2 \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 )+7 b d x^4 \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 )}{30 a^3 (e x)^{5/2} \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. \(2303\) vs.
\(2(267)=534\).
time = 0.13, size = 2304, normalized size = 6.49
method | result | size |
elliptic | \(\frac {\sqrt {\left (-d \,x^{2}+c \right ) e x}\, \left (\frac {b \sqrt {-d e \,x^{3}+c e x}}{2 e^{3} a^{2} \left (-b \,x^{2}+a \right )}-\frac {2 \sqrt {-d e \,x^{3}+c e x}}{3 e^{3} a^{2} x^{2}}+\frac {7 \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 )}{12 a^{2} e^{2} \sqrt {-d e \,x^{3}+c e x}}+\frac {5 \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 a \,e^{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 {7 \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 ) b c}{8 a^{2} e^{2} \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 {5 \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 a \,e^{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 {7 \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 ) b c}{8 a^{2} e^{2} \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}}\) | \(784\) |
default | \(\text {Expression too large to display}\) | \(2304\) |
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 {\sqrt {c - d x^{2}}}{\left (e x\right )^{\frac {5}{2}} \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 {\sqrt {c-d\,x^2}}{{\left (e\,x\right )}^{5/2}\,{\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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