Optimal. Leaf size=444 \[ -\frac {2 \sqrt {c-d x^2}}{5 a c e (e x)^{5/2}}-\frac {2 (5 b c+3 a d) \sqrt {c-d x^2}}{5 a^2 c^2 e^3 \sqrt {e x}}-\frac {2 \sqrt [4]{d} (5 b c+3 a d) \sqrt {1-\frac {d x^2}{c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{5 a^2 c^{5/4} e^{7/2} \sqrt {c-d x^2}}+\frac {2 \sqrt [4]{d} (5 b c+3 a d) \sqrt {1-\frac {d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{5 a^2 c^{5/4} e^{7/2} \sqrt {c-d x^2}}-\frac {b^{3/2} \sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{a^{5/2} \sqrt [4]{d} e^{7/2} \sqrt {c-d x^2}}+\frac {b^{3/2} \sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{a^{5/2} \sqrt [4]{d} e^{7/2} \sqrt {c-d x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.64, antiderivative size = 444, 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, 491,
597, 598, 313, 230, 227, 1214, 1213, 435, 504, 1233, 1232} \begin {gather*} -\frac {b^{3/2} \sqrt [4]{c} \sqrt {1-\frac {d x^2}{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 )}{a^{5/2} \sqrt [4]{d} e^{7/2} \sqrt {c-d x^2}}+\frac {b^{3/2} \sqrt [4]{c} \sqrt {1-\frac {d x^2}{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 )}{a^{5/2} \sqrt [4]{d} e^{7/2} \sqrt {c-d x^2}}+\frac {2 \sqrt [4]{d} \sqrt {1-\frac {d x^2}{c}} (3 a d+5 b c) 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 c^{5/4} e^{7/2} \sqrt {c-d x^2}}-\frac {2 \sqrt [4]{d} \sqrt {1-\frac {d x^2}{c}} (3 a d+5 b c) 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 c^{5/4} e^{7/2} \sqrt {c-d x^2}}-\frac {2 \sqrt {c-d x^2} (3 a d+5 b c)}{5 a^2 c^2 e^3 \sqrt {e x}}-\frac {2 \sqrt {c-d x^2}}{5 a c e (e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 227
Rule 230
Rule 313
Rule 435
Rule 477
Rule 491
Rule 504
Rule 597
Rule 598
Rule 1213
Rule 1214
Rule 1232
Rule 1233
Rubi steps
\begin {align*} \int \frac {1}{(e x)^{7/2} \left (a-b x^2\right ) \sqrt {c-d x^2}} \, dx &=\frac {2 \text {Subst}\left (\int \frac {1}{x^6 \left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{e}\\ &=-\frac {2 \sqrt {c-d x^2}}{5 a c e (e x)^{5/2}}+\frac {2 \text {Subst}\left (\int \frac {\frac {5 b c+3 a d}{e^2}-\frac {3 b 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 c e}\\ &=-\frac {2 \sqrt {c-d x^2}}{5 a c e (e x)^{5/2}}-\frac {2 (5 b c+3 a d) \sqrt {c-d x^2}}{5 a^2 c^2 e^3 \sqrt {e x}}-\frac {2 \text {Subst}\left (\int \frac {x^2 \left (-\frac {5 b^2 c^2-5 a b c d-3 a^2 d^2}{e^4}-\frac {b d (5 b c+3 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^2 e}\\ &=-\frac {2 \sqrt {c-d x^2}}{5 a c e (e x)^{5/2}}-\frac {2 (5 b c+3 a d) \sqrt {c-d x^2}}{5 a^2 c^2 e^3 \sqrt {e x}}-\frac {2 \text {Subst}\left (\int \left (\frac {d (5 b c+3 a d) x^2}{e^4 \sqrt {c-\frac {d x^4}{e^2}}}-\frac {5 b^2 c^2 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^2 e}\\ &=-\frac {2 \sqrt {c-d x^2}}{5 a c e (e x)^{5/2}}-\frac {2 (5 b c+3 a d) \sqrt {c-d x^2}}{5 a^2 c^2 e^3 \sqrt {e x}}+\frac {\left (2 b^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 d (5 b c+3 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 c^2 e^5}\\ &=-\frac {2 \sqrt {c-d x^2}}{5 a c e (e x)^{5/2}}-\frac {2 (5 b c+3 a d) \sqrt {c-d x^2}}{5 a^2 c^2 e^3 \sqrt {e x}}+\frac {\left (2 \sqrt {d} (5 b c+3 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 c^{3/2} e^4}-\frac {\left (2 \sqrt {d} (5 b c+3 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 c^{3/2} e^4}+\frac {b^{3/2} \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 e^3}-\frac {b^{3/2} \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 e^3}\\ &=-\frac {2 \sqrt {c-d x^2}}{5 a c e (e x)^{5/2}}-\frac {2 (5 b c+3 a d) \sqrt {c-d x^2}}{5 a^2 c^2 e^3 \sqrt {e x}}+\frac {\left (2 \sqrt {d} (5 b c+3 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 c^{3/2} e^4 \sqrt {c-d x^2}}-\frac {\left (2 \sqrt {d} (5 b c+3 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 c^{3/2} e^4 \sqrt {c-d x^2}}+\frac {\left (b^{3/2} \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 e^3 \sqrt {c-d x^2}}-\frac {\left (b^{3/2} \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 e^3 \sqrt {c-d x^2}}\\ &=-\frac {2 \sqrt {c-d x^2}}{5 a c e (e x)^{5/2}}-\frac {2 (5 b c+3 a d) \sqrt {c-d x^2}}{5 a^2 c^2 e^3 \sqrt {e x}}+\frac {2 \sqrt [4]{d} (5 b c+3 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 c^{5/4} e^{7/2} \sqrt {c-d x^2}}-\frac {b^{3/2} \sqrt [4]{c} \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 [4]{d} e^{7/2} \sqrt {c-d x^2}}+\frac {b^{3/2} \sqrt [4]{c} \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 [4]{d} e^{7/2} \sqrt {c-d x^2}}-\frac {\left (2 \sqrt {d} (5 b c+3 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 c^{3/2} e^4 \sqrt {c-d x^2}}\\ &=-\frac {2 \sqrt {c-d x^2}}{5 a c e (e x)^{5/2}}-\frac {2 (5 b c+3 a d) \sqrt {c-d x^2}}{5 a^2 c^2 e^3 \sqrt {e x}}-\frac {2 \sqrt [4]{d} (5 b c+3 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 c^{5/4} e^{7/2} \sqrt {c-d x^2}}+\frac {2 \sqrt [4]{d} (5 b c+3 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 c^{5/4} e^{7/2} \sqrt {c-d x^2}}-\frac {b^{3/2} \sqrt [4]{c} \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 [4]{d} e^{7/2} \sqrt {c-d x^2}}+\frac {b^{3/2} \sqrt [4]{c} \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 [4]{d} e^{7/2} \sqrt {c-d x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in
optimal.
time = 10.17, size = 188, normalized size = 0.42 \begin {gather*} \frac {x \left (-42 a \left (c-d x^2\right ) \left (5 b c x^2+a \left (c+3 d x^2\right )\right )+14 \left (5 b^2 c^2-5 a b c d-3 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 b d (5 b c+3 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 )}{105 a^3 c^2 (e x)^{7/2} \sqrt {c-d x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1097\) vs.
\(2(332)=664\).
time = 0.13, size = 1098, normalized size = 2.47 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {1}{- a \left (e x\right )^{\frac {7}{2}} \sqrt {c - d x^{2}} + b x^{2} \left (e x\right )^{\frac {7}{2}} \sqrt {c - d x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (e\,x\right )}^{7/2}\,\left (a-b\,x^2\right )\,\sqrt {c-d\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________