Optimal. Leaf size=606 \[ \frac {d (3 b c+2 a d)}{6 a c (b c-a d)^2 e (e x)^{3/2} \left (c-d x^2\right )^{3/2}}+\frac {b}{2 a (b c-a d) e (e x)^{3/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d \left (b^2 c^2+7 a b c d-3 a^2 d^2\right )}{2 a c^2 (b c-a d)^3 e (e x)^{3/2} \sqrt {c-d x^2}}-\frac {\left (7 b^3 c^3-12 a b^2 c^2 d+35 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {c-d x^2}}{6 a^2 c^3 (b c-a d)^3 e (e x)^{3/2}}+\frac {d^{3/4} \left (7 b^3 c^3-12 a b^2 c^2 d+35 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {1-\frac {d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{6 a^2 c^{11/4} (b c-a d)^3 e^{5/2} \sqrt {c-d x^2}}+\frac {b^3 \sqrt [4]{c} (7 b c-17 a d) \sqrt {1-\frac {d x^2}{c}} \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a^3 \sqrt [4]{d} (b c-a d)^3 e^{5/2} \sqrt {c-d x^2}}+\frac {b^3 \sqrt [4]{c} (7 b c-17 a d) \sqrt {1-\frac {d x^2}{c}} \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a^3 \sqrt [4]{d} (b c-a d)^3 e^{5/2} \sqrt {c-d x^2}} \]
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Rubi [A]
time = 0.99, antiderivative size = 606, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 10, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {477, 483,
593, 597, 537, 230, 227, 418, 1233, 1232} \begin {gather*} \frac {b^3 \sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} (7 b c-17 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} (b c-a d)^3}+\frac {b^3 \sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} (7 b c-17 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} (b c-a d)^3}+\frac {d \left (-3 a^2 d^2+7 a b c d+b^2 c^2\right )}{2 a c^2 e (e x)^{3/2} \sqrt {c-d x^2} (b c-a d)^3}+\frac {d^{3/4} \sqrt {1-\frac {d x^2}{c}} \left (-15 a^3 d^3+35 a^2 b c d^2-12 a b^2 c^2 d+7 b^3 c^3\right ) 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 c^{11/4} e^{5/2} \sqrt {c-d x^2} (b c-a d)^3}-\frac {\sqrt {c-d x^2} \left (-15 a^3 d^3+35 a^2 b c d^2-12 a b^2 c^2 d+7 b^3 c^3\right )}{6 a^2 c^3 e (e x)^{3/2} (b c-a d)^3}+\frac {b}{2 a e (e x)^{3/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2} (b c-a d)}+\frac {d (2 a d+3 b c)}{6 a c e (e x)^{3/2} \left (c-d x^2\right )^{3/2} (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 227
Rule 230
Rule 418
Rule 477
Rule 483
Rule 537
Rule 593
Rule 597
Rule 1232
Rule 1233
Rubi steps
\begin {align*} \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx &=\frac {2 \text {Subst}\left (\int \frac {1}{x^4 \left (a-\frac {b x^4}{e^2}\right )^2 \left (c-\frac {d x^4}{e^2}\right )^{5/2}} \, dx,x,\sqrt {e x}\right )}{e}\\ &=\frac {b}{2 a (b c-a d) e (e x)^{3/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {e \text {Subst}\left (\int \frac {\frac {7 b c-4 a d}{e^2}-\frac {13 b d x^4}{e^4}}{x^4 \left (a-\frac {b x^4}{e^2}\right ) \left (c-\frac {d x^4}{e^2}\right )^{5/2}} \, dx,x,\sqrt {e x}\right )}{2 a (b c-a d)}\\ &=\frac {d (3 b c+2 a d)}{6 a c (b c-a d)^2 e (e x)^{3/2} \left (c-d x^2\right )^{3/2}}+\frac {b}{2 a (b c-a d) e (e x)^{3/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}-\frac {e^3 \text {Subst}\left (\int \frac {-\frac {6 \left (7 b^2 c^2-8 a b c d+6 a^2 d^2\right )}{e^4}+\frac {18 b d (3 b c+2 a d) x^4}{e^6}}{x^4 \left (a-\frac {b x^4}{e^2}\right ) \left (c-\frac {d x^4}{e^2}\right )^{3/2}} \, dx,x,\sqrt {e x}\right )}{12 a c (b c-a d)^2}\\ &=\frac {d (3 b c+2 a d)}{6 a c (b c-a d)^2 e (e x)^{3/2} \left (c-d x^2\right )^{3/2}}+\frac {b}{2 a (b c-a d) e (e x)^{3/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d \left (b^2 c^2+7 a b c d-3 a^2 d^2\right )}{2 a c^2 (b c-a d)^3 e (e x)^{3/2} \sqrt {c-d x^2}}+\frac {e^5 \text {Subst}\left (\int \frac {\frac {12 \left (7 b^3 c^3-12 a b^2 c^2 d+35 a^2 b c d^2-15 a^3 d^3\right )}{e^6}-\frac {60 b d \left (b^2 c^2+7 a b c d-3 a^2 d^2\right ) x^4}{e^8}}{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 )}{24 a c^2 (b c-a d)^3}\\ &=\frac {d (3 b c+2 a d)}{6 a c (b c-a d)^2 e (e x)^{3/2} \left (c-d x^2\right )^{3/2}}+\frac {b}{2 a (b c-a d) e (e x)^{3/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d \left (b^2 c^2+7 a b c d-3 a^2 d^2\right )}{2 a c^2 (b c-a d)^3 e (e x)^{3/2} \sqrt {c-d x^2}}-\frac {\left (7 b^3 c^3-12 a b^2 c^2 d+35 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {c-d x^2}}{6 a^2 c^3 (b c-a d)^3 e (e x)^{3/2}}-\frac {e^5 \text {Subst}\left (\int \frac {-\frac {12 \left (21 b^4 c^4-44 a b^3 c^3 d-12 a^2 b^2 c^2 d^2+35 a^3 b c d^3-15 a^4 d^4\right )}{e^8}+\frac {12 b d \left (7 b^3 c^3-12 a b^2 c^2 d+35 a^2 b c d^2-15 a^3 d^3\right ) x^4}{e^{10}}}{\left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{72 a^2 c^3 (b c-a d)^3}\\ &=\frac {d (3 b c+2 a d)}{6 a c (b c-a d)^2 e (e x)^{3/2} \left (c-d x^2\right )^{3/2}}+\frac {b}{2 a (b c-a d) e (e x)^{3/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d \left (b^2 c^2+7 a b c d-3 a^2 d^2\right )}{2 a c^2 (b c-a d)^3 e (e x)^{3/2} \sqrt {c-d x^2}}-\frac {\left (7 b^3 c^3-12 a b^2 c^2 d+35 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {c-d x^2}}{6 a^2 c^3 (b c-a d)^3 e (e x)^{3/2}}+\frac {\left (b^3 (7 b c-17 a d)\right ) \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 (b c-a d)^3 e^3}+\frac {\left (d \left (7 b^3 c^3-12 a b^2 c^2 d+35 a^2 b c d^2-15 a^3 d^3\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{6 a^2 c^3 (b c-a d)^3 e^3}\\ &=\frac {d (3 b c+2 a d)}{6 a c (b c-a d)^2 e (e x)^{3/2} \left (c-d x^2\right )^{3/2}}+\frac {b}{2 a (b c-a d) e (e x)^{3/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d \left (b^2 c^2+7 a b c d-3 a^2 d^2\right )}{2 a c^2 (b c-a d)^3 e (e x)^{3/2} \sqrt {c-d x^2}}-\frac {\left (7 b^3 c^3-12 a b^2 c^2 d+35 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {c-d x^2}}{6 a^2 c^3 (b c-a d)^3 e (e x)^{3/2}}+\frac {\left (b^3 (7 b c-17 a d)\right ) \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 (b c-a d)^3 e^3}+\frac {\left (b^3 (7 b c-17 a d)\right ) \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 (b c-a d)^3 e^3}+\frac {\left (d \left (7 b^3 c^3-12 a b^2 c^2 d+35 a^2 b c d^2-15 a^3 d^3\right ) \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 c^3 (b c-a d)^3 e^3 \sqrt {c-d x^2}}\\ &=\frac {d (3 b c+2 a d)}{6 a c (b c-a d)^2 e (e x)^{3/2} \left (c-d x^2\right )^{3/2}}+\frac {b}{2 a (b c-a d) e (e x)^{3/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d \left (b^2 c^2+7 a b c d-3 a^2 d^2\right )}{2 a c^2 (b c-a d)^3 e (e x)^{3/2} \sqrt {c-d x^2}}-\frac {\left (7 b^3 c^3-12 a b^2 c^2 d+35 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {c-d x^2}}{6 a^2 c^3 (b c-a d)^3 e (e x)^{3/2}}+\frac {d^{3/4} \left (7 b^3 c^3-12 a b^2 c^2 d+35 a^2 b c d^2-15 a^3 d^3\right ) \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 c^{11/4} (b c-a d)^3 e^{5/2} \sqrt {c-d x^2}}+\frac {\left (b^3 (7 b c-17 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 (b c-a d)^3 e^3 \sqrt {c-d x^2}}+\frac {\left (b^3 (7 b c-17 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 (b c-a d)^3 e^3 \sqrt {c-d x^2}}\\ &=\frac {d (3 b c+2 a d)}{6 a c (b c-a d)^2 e (e x)^{3/2} \left (c-d x^2\right )^{3/2}}+\frac {b}{2 a (b c-a d) e (e x)^{3/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d \left (b^2 c^2+7 a b c d-3 a^2 d^2\right )}{2 a c^2 (b c-a d)^3 e (e x)^{3/2} \sqrt {c-d x^2}}-\frac {\left (7 b^3 c^3-12 a b^2 c^2 d+35 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {c-d x^2}}{6 a^2 c^3 (b c-a d)^3 e (e x)^{3/2}}+\frac {d^{3/4} \left (7 b^3 c^3-12 a b^2 c^2 d+35 a^2 b c d^2-15 a^3 d^3\right ) \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 c^{11/4} (b c-a d)^3 e^{5/2} \sqrt {c-d x^2}}+\frac {b^3 \sqrt [4]{c} (7 b c-17 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} (b c-a d)^3 e^{5/2} \sqrt {c-d x^2}}+\frac {b^3 \sqrt [4]{c} (7 b c-17 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} (b c-a d)^3 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 = 11.01, size = 427, normalized size = 0.70 \begin {gather*} \frac {x \left (-\frac {5 a \left (7 b^4 c^3 x^2 \left (c-d x^2\right )^2-4 a b^3 c^2 \left (c-d x^2\right )^2 \left (c+3 d x^2\right )+a^4 d^3 \left (4 c^2-21 c d x^2+15 d^2 x^4\right )-a^3 b d^2 \left (12 c^3-45 c^2 d x^2+14 c d^2 x^4+15 d^3 x^6\right )+a^2 b^2 c d \left (12 c^3-12 c^2 d x^2-37 c d^2 x^4+35 d^3 x^6\right )\right )}{(-b c+a d)^3 \left (a-b x^2\right ) \left (c-d x^2\right )}+\frac {5 \left (21 b^4 c^4-44 a b^3 c^3 d-12 a^2 b^2 c^2 d^2+35 a^3 b c d^3-15 a^4 d^4\right ) x^2 \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 )}{(b c-a d)^3}-\frac {b d \left (7 b^3 c^3-12 a b^2 c^2 d+35 a^2 b c d^2-15 a^3 d^3\right ) x^4 \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 )}{(b c-a d)^3}\right )}{30 a^3 c^3 (e x)^{5/2} \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. \(5235\) vs.
\(2(506)=1012\).
time = 0.15, size = 5236, normalized size = 8.64
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1464\) |
default | \(\text {Expression too large to display}\) | \(5236\) |
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 {1}{\left (e x\right )^{\frac {5}{2}} \left (- a + b x^{2}\right )^{2} \left (c - d x^{2}\right )^{\frac {5}{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 {1}{{\left (e\,x\right )}^{5/2}\,{\left (a-b\,x^2\right )}^2\,{\left (c-d\,x^2\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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