Optimal. Leaf size=514 \[ \frac {d (3 b c+2 a d) \sqrt {e x}}{6 a c (b c-a d)^2 e \left (c-d x^2\right )^{3/2}}+\frac {b \sqrt {e x}}{2 a (b c-a d) e \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d \left (3 b^2 c^2+17 a b c d-5 a^2 d^2\right ) \sqrt {e x}}{6 a c^2 (b c-a d)^3 e \sqrt {c-d x^2}}+\frac {d^{3/4} \left (3 b^2 c^2+17 a b c d-5 a^2 d^2\right ) \sqrt {1-\frac {d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{6 a c^{7/4} (b c-a d)^3 \sqrt {e} \sqrt {c-d x^2}}+\frac {b^2 \sqrt [4]{c} (3 b c-13 a d) \sqrt {1-\frac {d x^2}{c}} \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a^2 \sqrt [4]{d} (b c-a d)^3 \sqrt {e} \sqrt {c-d x^2}}+\frac {b^2 \sqrt [4]{c} (3 b c-13 a d) \sqrt {1-\frac {d x^2}{c}} \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a^2 \sqrt [4]{d} (b c-a d)^3 \sqrt {e} \sqrt {c-d x^2}} \]
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Rubi [A]
time = 0.67, antiderivative size = 514, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 9, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {477, 425, 541,
537, 230, 227, 418, 1233, 1232} \begin {gather*} \frac {d^{3/4} \sqrt {1-\frac {d x^2}{c}} \left (-5 a^2 d^2+17 a b c d+3 b^2 c^2\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 c^{7/4} \sqrt {e} \sqrt {c-d x^2} (b c-a d)^3}+\frac {b^2 \sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} (3 b c-13 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 \sqrt [4]{d} \sqrt {e} \sqrt {c-d x^2} (b c-a d)^3}+\frac {b^2 \sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} (3 b c-13 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 \sqrt [4]{d} \sqrt {e} \sqrt {c-d x^2} (b c-a d)^3}+\frac {d \sqrt {e x} \left (-5 a^2 d^2+17 a b c d+3 b^2 c^2\right )}{6 a c^2 e \sqrt {c-d x^2} (b c-a d)^3}+\frac {b \sqrt {e x}}{2 a e \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2} (b c-a d)}+\frac {d \sqrt {e x} (2 a d+3 b c)}{6 a c e \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 425
Rule 477
Rule 537
Rule 541
Rule 1232
Rule 1233
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {e x} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx &=\frac {2 \text {Subst}\left (\int \frac {1}{\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 \sqrt {e x}}{2 a (b c-a d) e \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {e \text {Subst}\left (\int \frac {\frac {3 b c-4 a d}{e^2}-\frac {9 b d x^4}{e^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) \sqrt {e x}}{6 a c (b c-a d)^2 e \left (c-d x^2\right )^{3/2}}+\frac {b \sqrt {e x}}{2 a (b c-a d) e \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}-\frac {e^3 \text {Subst}\left (\int \frac {-\frac {2 \left (9 b^2 c^2-24 a b c d+10 a^2 d^2\right )}{e^4}+\frac {10 b d (3 b c+2 a d) x^4}{e^6}}{\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) \sqrt {e x}}{6 a c (b c-a d)^2 e \left (c-d x^2\right )^{3/2}}+\frac {b \sqrt {e x}}{2 a (b c-a d) e \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d \left (3 b^2 c^2+17 a b c d-5 a^2 d^2\right ) \sqrt {e x}}{6 a c^2 (b c-a d)^3 e \sqrt {c-d x^2}}+\frac {e^5 \text {Subst}\left (\int \frac {\frac {4 \left (9 b^3 c^3-36 a b^2 c^2 d+17 a^2 b c d^2-5 a^3 d^3\right )}{e^6}-\frac {4 b d \left (3 b^2 c^2+17 a b c d-5 a^2 d^2\right ) x^4}{e^8}}{\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) \sqrt {e x}}{6 a c (b c-a d)^2 e \left (c-d x^2\right )^{3/2}}+\frac {b \sqrt {e x}}{2 a (b c-a d) e \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d \left (3 b^2 c^2+17 a b c d-5 a^2 d^2\right ) \sqrt {e x}}{6 a c^2 (b c-a d)^3 e \sqrt {c-d x^2}}+\frac {\left (b^2 (3 b c-13 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 (b c-a d)^3 e}+\frac {\left (d \left (3 b^2 c^2+17 a b c d-5 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{6 a c^2 (b c-a d)^3 e}\\ &=\frac {d (3 b c+2 a d) \sqrt {e x}}{6 a c (b c-a d)^2 e \left (c-d x^2\right )^{3/2}}+\frac {b \sqrt {e x}}{2 a (b c-a d) e \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d \left (3 b^2 c^2+17 a b c d-5 a^2 d^2\right ) \sqrt {e x}}{6 a c^2 (b c-a d)^3 e \sqrt {c-d x^2}}+\frac {\left (b^2 (3 b c-13 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^2 (b c-a d)^3 e}+\frac {\left (b^2 (3 b c-13 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^2 (b c-a d)^3 e}+\frac {\left (d \left (3 b^2 c^2+17 a b c d-5 a^2 d^2\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 c^2 (b c-a d)^3 e \sqrt {c-d x^2}}\\ &=\frac {d (3 b c+2 a d) \sqrt {e x}}{6 a c (b c-a d)^2 e \left (c-d x^2\right )^{3/2}}+\frac {b \sqrt {e x}}{2 a (b c-a d) e \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d \left (3 b^2 c^2+17 a b c d-5 a^2 d^2\right ) \sqrt {e x}}{6 a c^2 (b c-a d)^3 e \sqrt {c-d x^2}}+\frac {d^{3/4} \left (3 b^2 c^2+17 a b c d-5 a^2 d^2\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 c^{7/4} (b c-a d)^3 \sqrt {e} \sqrt {c-d x^2}}+\frac {\left (b^2 (3 b c-13 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 c-a d)^3 e \sqrt {c-d x^2}}+\frac {\left (b^2 (3 b c-13 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 c-a d)^3 e \sqrt {c-d x^2}}\\ &=\frac {d (3 b c+2 a d) \sqrt {e x}}{6 a c (b c-a d)^2 e \left (c-d x^2\right )^{3/2}}+\frac {b \sqrt {e x}}{2 a (b c-a d) e \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d \left (3 b^2 c^2+17 a b c d-5 a^2 d^2\right ) \sqrt {e x}}{6 a c^2 (b c-a d)^3 e \sqrt {c-d x^2}}+\frac {d^{3/4} \left (3 b^2 c^2+17 a b c d-5 a^2 d^2\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 c^{7/4} (b c-a d)^3 \sqrt {e} \sqrt {c-d x^2}}+\frac {b^2 \sqrt [4]{c} (3 b c-13 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 \sqrt [4]{d} (b c-a d)^3 \sqrt {e} \sqrt {c-d x^2}}+\frac {b^2 \sqrt [4]{c} (3 b c-13 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 \sqrt [4]{d} (b c-a d)^3 \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 = 10.39, size = 328, normalized size = 0.64 \begin {gather*} -\frac {5 a x \left (3 b^3 c^2 \left (c-d x^2\right )^2+a^3 d^3 \left (-7 c+5 d x^2\right )+a b^2 c d^2 x^2 \left (-19 c+17 d x^2\right )+a^2 b d^2 \left (19 c^2-10 c d x^2-5 d^2 x^4\right )\right )-5 \left (-9 b^3 c^3+36 a b^2 c^2 d-17 a^2 b c d^2+5 a^3 d^3\right ) x \left (a-b x^2\right ) \left (c-d 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 )+b d \left (3 b^2 c^2+17 a b c d-5 a^2 d^2\right ) x^3 \left (-a+b x^2\right ) \left (c-d 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 )}{30 a^2 c^2 (b c-a d)^3 \sqrt {e x} \left (-a+b x^2\right ) \left (c-d x^2\right )^{3/2}} \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. \(4763\) vs.
\(2(420)=840\).
time = 0.13, size = 4764, normalized size = 9.27
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1311\) |
default | \(\text {Expression too large to display}\) | \(4764\) |
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}{\sqrt {e x} \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}{\sqrt {e\,x}\,{\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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