Optimal. Leaf size=459 \[ -\frac {2 c \sqrt {c-d x^2}}{5 a e (e x)^{5/2}}-\frac {2 (5 b c-7 a d) \sqrt {c-d x^2}}{5 a^2 e^3 \sqrt {e x}}-\frac {2 c^{3/4} \sqrt [4]{d} (5 b c-7 a d) \sqrt {1-\frac {d x^2}{c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{5 a^2 e^{7/2} \sqrt {c-d x^2}}+\frac {2 c^{3/4} \sqrt [4]{d} (5 b c-7 a d) \sqrt {1-\frac {d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{5 a^2 e^{7/2} \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c-a d)^2 \sqrt {1-\frac {d x^2}{c}} \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{a^{5/2} \sqrt {b} \sqrt [4]{d} e^{7/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d)^2 \sqrt {1-\frac {d x^2}{c}} \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{a^{5/2} \sqrt {b} \sqrt [4]{d} e^{7/2} \sqrt {c-d x^2}} \]
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Rubi [A]
time = 0.73, antiderivative size = 459, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 13, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.433, Rules used = {477, 485,
597, 598, 313, 230, 227, 1214, 1213, 435, 504, 1233, 1232} \begin {gather*} -\frac {\sqrt [4]{c} \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 )}{a^{5/2} \sqrt {b} \sqrt [4]{d} e^{7/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \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 )}{a^{5/2} \sqrt {b} \sqrt [4]{d} e^{7/2} \sqrt {c-d x^2}}+\frac {2 c^{3/4} \sqrt [4]{d} \sqrt {1-\frac {d x^2}{c}} (5 b c-7 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 a^2 e^{7/2} \sqrt {c-d x^2}}-\frac {2 c^{3/4} \sqrt [4]{d} \sqrt {1-\frac {d x^2}{c}} (5 b c-7 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 a^2 e^{7/2} \sqrt {c-d x^2}}-\frac {2 \sqrt {c-d x^2} (5 b c-7 a d)}{5 a^2 e^3 \sqrt {e x}}-\frac {2 c \sqrt {c-d x^2}}{5 a e (e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 227
Rule 230
Rule 313
Rule 435
Rule 477
Rule 485
Rule 504
Rule 597
Rule 598
Rule 1213
Rule 1214
Rule 1232
Rule 1233
Rubi steps
\begin {align*} \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{7/2} \left (a-b x^2\right )} \, dx &=\frac {2 \text {Subst}\left (\int \frac {\left (c-\frac {d x^4}{e^2}\right )^{3/2}}{x^6 \left (a-\frac {b x^4}{e^2}\right )} \, dx,x,\sqrt {e x}\right )}{e}\\ &=-\frac {2 c \sqrt {c-d x^2}}{5 a e (e x)^{5/2}}+\frac {2 \text {Subst}\left (\int \frac {\frac {c (5 b c-7 a d)}{e^2}-\frac {d (3 b c-5 a d) x^4}{e^4}}{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 )}{5 a e}\\ &=-\frac {2 c \sqrt {c-d x^2}}{5 a e (e x)^{5/2}}-\frac {2 (5 b c-7 a d) \sqrt {c-d x^2}}{5 a^2 e^3 \sqrt {e x}}-\frac {2 \text {Subst}\left (\int \frac {x^2 \left (-\frac {c \left (5 b^2 c^2-15 a b c d+12 a^2 d^2\right )}{e^4}-\frac {b c d (5 b c-7 a d) x^4}{e^6}\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 a^2 c e}\\ &=-\frac {2 c \sqrt {c-d x^2}}{5 a e (e x)^{5/2}}-\frac {2 (5 b c-7 a d) \sqrt {c-d x^2}}{5 a^2 e^3 \sqrt {e x}}-\frac {2 \text {Subst}\left (\int \left (\frac {c d (5 b c-7 a d) x^2}{e^4 \sqrt {c-\frac {d x^4}{e^2}}}-\frac {5 \left (b^2 c^3-2 a b c^2 d+a^2 c d^2\right ) x^2}{e^4 \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 a^2 c e}\\ &=-\frac {2 c \sqrt {c-d x^2}}{5 a e (e x)^{5/2}}-\frac {2 (5 b c-7 a d) \sqrt {c-d x^2}}{5 a^2 e^3 \sqrt {e x}}-\frac {(2 d (5 b c-7 a d)) \text {Subst}\left (\int \frac {x^2}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{5 a^2 e^5}+\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 )}{a^2 e^5}\\ &=-\frac {2 c \sqrt {c-d x^2}}{5 a e (e x)^{5/2}}-\frac {2 (5 b c-7 a d) \sqrt {c-d x^2}}{5 a^2 e^3 \sqrt {e x}}+\frac {\left (2 \sqrt {c} \sqrt {d} (5 b c-7 a d)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{5 a^2 e^4}-\frac {\left (2 \sqrt {c} \sqrt {d} (5 b c-7 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 a^2 e^4}+\frac {(b c-a d)^2 \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^2 \sqrt {b} e^3}-\frac {(b c-a d)^2 \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^2 \sqrt {b} e^3}\\ &=-\frac {2 c \sqrt {c-d x^2}}{5 a e (e x)^{5/2}}-\frac {2 (5 b c-7 a d) \sqrt {c-d x^2}}{5 a^2 e^3 \sqrt {e x}}+\frac {\left (2 \sqrt {c} \sqrt {d} (5 b c-7 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 a^2 e^4 \sqrt {c-d x^2}}-\frac {\left (2 \sqrt {c} \sqrt {d} (5 b c-7 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 a^2 e^4 \sqrt {c-d x^2}}+\frac {\left ((b c-a d)^2 \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^2 \sqrt {b} e^3 \sqrt {c-d x^2}}-\frac {\left ((b c-a d)^2 \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^2 \sqrt {b} e^3 \sqrt {c-d x^2}}\\ &=-\frac {2 c \sqrt {c-d x^2}}{5 a e (e x)^{5/2}}-\frac {2 (5 b c-7 a d) \sqrt {c-d x^2}}{5 a^2 e^3 \sqrt {e x}}+\frac {2 c^{3/4} \sqrt [4]{d} (5 b c-7 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 )}{5 a^2 e^{7/2} \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c-a d)^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 )}{a^{5/2} \sqrt {b} \sqrt [4]{d} e^{7/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d)^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 )}{a^{5/2} \sqrt {b} \sqrt [4]{d} e^{7/2} \sqrt {c-d x^2}}-\frac {\left (2 \sqrt {c} \sqrt {d} (5 b c-7 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 a^2 e^4 \sqrt {c-d x^2}}\\ &=-\frac {2 c \sqrt {c-d x^2}}{5 a e (e x)^{5/2}}-\frac {2 (5 b c-7 a d) \sqrt {c-d x^2}}{5 a^2 e^3 \sqrt {e x}}-\frac {2 c^{3/4} \sqrt [4]{d} (5 b c-7 a 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 )}{5 a^2 e^{7/2} \sqrt {c-d x^2}}+\frac {2 c^{3/4} \sqrt [4]{d} (5 b c-7 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 )}{5 a^2 e^{7/2} \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c-a d)^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 )}{a^{5/2} \sqrt {b} \sqrt [4]{d} e^{7/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d)^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 )}{a^{5/2} \sqrt {b} \sqrt [4]{d} e^{7/2} \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 = 10.19, size = 187, normalized size = 0.41 \begin {gather*} \frac {x \left (14 \left (5 b^2 c^2-15 a b c d+12 a^2 d^2\right ) x^4 \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 \left (7 a \left (c-d x^2\right ) \left (a c+5 b c x^2-7 a d x^2\right )+b d (-5 b c+7 a d) x^6 \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^3 (e x)^{7/2} \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. \(2016\) vs.
\(2(347)=694\).
time = 0.14, size = 2017, normalized size = 4.39
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1328\) |
default | \(\text {Expression too large to display}\) | \(2017\) |
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 {c - d x^{2}}}{- a \left (e x\right )^{\frac {7}{2}} + b x^{2} \left (e x\right )^{\frac {7}{2}}}\, dx - \int \left (- \frac {d x^{2} \sqrt {c - d x^{2}}}{- a \left (e x\right )^{\frac {7}{2}} + b x^{2} \left (e x\right )^{\frac {7}{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 {{\left (c-d\,x^2\right )}^{3/2}}{{\left (e\,x\right )}^{7/2}\,\left (a-b\,x^2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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