Optimal. Leaf size=485 \[ \frac {3 d e (e x)^{3/2}}{2 (b c-a d)^2 \sqrt {c-d x^2}}+\frac {e (e x)^{3/2}}{2 (b c-a d) \left (a-b x^2\right ) \sqrt {c-d x^2}}-\frac {3 c^{3/4} \sqrt [4]{d} e^{5/2} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 (b c-a d)^2 \sqrt {c-d x^2}}+\frac {3 c^{3/4} \sqrt [4]{d} e^{5/2} \sqrt {1-\frac {d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 (b c-a d)^2 \sqrt {c-d x^2}}+\frac {3 \sqrt [4]{c} (b c+a d) e^{5/2} \sqrt {1-\frac {d x^2}{c}} \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 \sqrt {a} \sqrt {b} \sqrt [4]{d} (b c-a d)^2 \sqrt {c-d x^2}}-\frac {3 \sqrt [4]{c} (b c+a d) e^{5/2} \sqrt {1-\frac {d x^2}{c}} \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 \sqrt {a} \sqrt {b} \sqrt [4]{d} (b c-a d)^2 \sqrt {c-d x^2}} \]
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Rubi [A]
time = 0.67, antiderivative size = 485, 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, 482,
593, 598, 313, 230, 227, 1214, 1213, 435, 504, 1233, 1232} \begin {gather*} \frac {3 c^{3/4} \sqrt [4]{d} e^{5/2} \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 \sqrt {c-d x^2} (b c-a d)^2}-\frac {3 c^{3/4} \sqrt [4]{d} e^{5/2} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 \sqrt {c-d x^2} (b c-a d)^2}+\frac {3 \sqrt [4]{c} e^{5/2} \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 \sqrt {a} \sqrt {b} \sqrt [4]{d} \sqrt {c-d x^2} (b c-a d)^2}-\frac {3 \sqrt [4]{c} e^{5/2} \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 \sqrt {a} \sqrt {b} \sqrt [4]{d} \sqrt {c-d x^2} (b c-a d)^2}+\frac {3 d e (e x)^{3/2}}{2 \sqrt {c-d x^2} (b c-a d)^2}+\frac {e (e x)^{3/2}}{2 \left (a-b x^2\right ) \sqrt {c-d x^2} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 227
Rule 230
Rule 313
Rule 435
Rule 477
Rule 482
Rule 504
Rule 593
Rule 598
Rule 1213
Rule 1214
Rule 1232
Rule 1233
Rubi steps
\begin {align*} \int \frac {(e x)^{5/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx &=\frac {2 \text {Subst}\left (\int \frac {x^6}{\left (a-\frac {b x^4}{e^2}\right )^2 \left (c-\frac {d x^4}{e^2}\right )^{3/2}} \, dx,x,\sqrt {e x}\right )}{e}\\ &=\frac {e (e x)^{3/2}}{2 (b c-a d) \left (a-b x^2\right ) \sqrt {c-d x^2}}-\frac {e \text {Subst}\left (\int \frac {x^2 \left (3 c+\frac {3 d x^4}{e^2}\right )}{\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 )}{2 (b c-a d)}\\ &=\frac {3 d e (e x)^{3/2}}{2 (b c-a d)^2 \sqrt {c-d x^2}}+\frac {e (e x)^{3/2}}{2 (b c-a d) \left (a-b x^2\right ) \sqrt {c-d x^2}}+\frac {e^3 \text {Subst}\left (\int \frac {x^2 \left (-\frac {6 c (b c+2 a d)}{e^2}+\frac {6 b c d x^4}{e^4}\right )}{\left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{4 c (b c-a d)^2}\\ &=\frac {3 d e (e x)^{3/2}}{2 (b c-a d)^2 \sqrt {c-d x^2}}+\frac {e (e x)^{3/2}}{2 (b c-a d) \left (a-b x^2\right ) \sqrt {c-d x^2}}+\frac {e^3 \text {Subst}\left (\int \left (-\frac {6 c d x^2}{e^2 \sqrt {c-\frac {d x^4}{e^2}}}-\frac {6 \left (b c^2+a c d\right ) x^2}{e^2 \left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}}\right ) \, dx,x,\sqrt {e x}\right )}{4 c (b c-a d)^2}\\ &=\frac {3 d e (e x)^{3/2}}{2 (b c-a d)^2 \sqrt {c-d x^2}}+\frac {e (e x)^{3/2}}{2 (b c-a d) \left (a-b x^2\right ) \sqrt {c-d x^2}}-\frac {(3 d e) \text {Subst}\left (\int \frac {x^2}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 (b c-a d)^2}-\frac {(3 (b c+a d) e) \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 )}{2 (b c-a d)^2}\\ &=\frac {3 d e (e x)^{3/2}}{2 (b c-a d)^2 \sqrt {c-d x^2}}+\frac {e (e x)^{3/2}}{2 (b c-a d) \left (a-b x^2\right ) \sqrt {c-d x^2}}+\frac {\left (3 \sqrt {c} \sqrt {d} e^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 (b c-a d)^2}-\frac {\left (3 \sqrt {c} \sqrt {d} e^2\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 )}{2 (b c-a d)^2}-\frac {\left (3 (b c+a d) e^3\right ) \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 )}{4 \sqrt {b} (b c-a d)^2}+\frac {\left (3 (b c+a d) e^3\right ) \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 )}{4 \sqrt {b} (b c-a d)^2}\\ &=\frac {3 d e (e x)^{3/2}}{2 (b c-a d)^2 \sqrt {c-d x^2}}+\frac {e (e x)^{3/2}}{2 (b c-a d) \left (a-b x^2\right ) \sqrt {c-d x^2}}+\frac {\left (3 \sqrt {c} \sqrt {d} e^2 \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 c-a d)^2 \sqrt {c-d x^2}}-\frac {\left (3 \sqrt {c} \sqrt {d} e^2 \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 )}{2 (b c-a d)^2 \sqrt {c-d x^2}}-\frac {\left (3 (b c+a d) e^3 \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 )}{4 \sqrt {b} (b c-a d)^2 \sqrt {c-d x^2}}+\frac {\left (3 (b c+a d) e^3 \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 )}{4 \sqrt {b} (b c-a d)^2 \sqrt {c-d x^2}}\\ &=\frac {3 d e (e x)^{3/2}}{2 (b c-a d)^2 \sqrt {c-d x^2}}+\frac {e (e x)^{3/2}}{2 (b c-a d) \left (a-b x^2\right ) \sqrt {c-d x^2}}+\frac {3 c^{3/4} \sqrt [4]{d} e^{5/2} \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 c-a d)^2 \sqrt {c-d x^2}}+\frac {3 \sqrt [4]{c} (b c+a d) e^{5/2} \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 \sqrt {a} \sqrt {b} \sqrt [4]{d} (b c-a d)^2 \sqrt {c-d x^2}}-\frac {3 \sqrt [4]{c} (b c+a d) e^{5/2} \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 \sqrt {a} \sqrt {b} \sqrt [4]{d} (b c-a d)^2 \sqrt {c-d x^2}}-\frac {\left (3 \sqrt {c} \sqrt {d} e^2 \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 )}{2 (b c-a d)^2 \sqrt {c-d x^2}}\\ &=\frac {3 d e (e x)^{3/2}}{2 (b c-a d)^2 \sqrt {c-d x^2}}+\frac {e (e x)^{3/2}}{2 (b c-a d) \left (a-b x^2\right ) \sqrt {c-d x^2}}-\frac {3 c^{3/4} \sqrt [4]{d} e^{5/2} \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 )}{2 (b c-a d)^2 \sqrt {c-d x^2}}+\frac {3 c^{3/4} \sqrt [4]{d} e^{5/2} \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 c-a d)^2 \sqrt {c-d x^2}}+\frac {3 \sqrt [4]{c} (b c+a d) e^{5/2} \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 \sqrt {a} \sqrt {b} \sqrt [4]{d} (b c-a d)^2 \sqrt {c-d x^2}}-\frac {3 \sqrt [4]{c} (b c+a d) e^{5/2} \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 \sqrt {a} \sqrt {b} \sqrt [4]{d} (b c-a d)^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 = 10.20, size = 185, normalized size = 0.38 \begin {gather*} \frac {e (e x)^{3/2} \left (-7 a \left (2 a d+b \left (c-3 d x^2\right )\right )+7 (b c+2 a d) \left (a-b x^2\right ) \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 )+3 b d x^2 \left (-a+b x^2\right ) \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 )}{14 a (b c-a d)^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. \(2548\) vs.
\(2(369)=738\).
time = 0.13, size = 2549, normalized size = 5.26
method | result | size |
elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (-d \,x^{2}+c \right ) e x}\, \left (\frac {b \,e^{2} x \sqrt {-d e \,x^{3}+c e x}}{2 \left (a d -b c \right )^{2} \left (-b \,x^{2}+a \right )}+\frac {d \,e^{3} x^{2}}{\left (a d -b c \right )^{2} \sqrt {-\left (x^{2}-\frac {c}{d}\right ) d e x}}+\frac {3 e^{3} c \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{2 \left (a d -b c \right )^{2} \sqrt {-d e \,x^{3}+c e x}}-\frac {3 e^{3} c \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 )^{2} \sqrt {-d e \,x^{3}+c e x}}+\frac {3 e^{3} \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 )^{2} b \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}+\frac {3 e^{3} \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 )^{2} d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}+\frac {3 e^{3} \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 )^{2} b \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}+\frac {3 e^{3} \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 )^{2} 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}}\) | \(917\) |
default | \(\text {Expression too large to display}\) | \(2549\) |
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(-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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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 )}^{5/2}}{{\left (a-b\,x^2\right )}^2\,{\left (c-d\,x^2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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