Optimal. Leaf size=391 \[ -\frac {(d e-c f) x \sqrt {e+f x^2}}{3 c (b c-a d) \left (c+d x^2\right )^{3/2}}-\frac {(b c (5 d e-c f)-2 a d (d e+c f)) \sqrt {e+f x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{3 c^{3/2} \sqrt {d} (b c-a d)^2 \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {e^{3/2} \sqrt {f} \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 c^2 (b c-a d) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {b e^{3/2} (b e-a f) \sqrt {c+d x^2} \Pi \left (1-\frac {b e}{a f};\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{a c (b c-a d)^2 \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}} \]
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Rubi [A]
time = 0.26, antiderivative size = 391, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {558, 553, 540,
539, 429, 422} \begin {gather*} -\frac {\sqrt {e+f x^2} (b c (5 d e-c f)-2 a d (c f+d e)) E\left (\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{3 c^{3/2} \sqrt {d} \sqrt {c+d x^2} (b c-a d)^2 \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {e^{3/2} \sqrt {f} \sqrt {c+d x^2} F\left (\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 c^2 \sqrt {e+f x^2} (b c-a d) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {b e^{3/2} \sqrt {c+d x^2} (b e-a f) \Pi \left (1-\frac {b e}{a f};\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{a c \sqrt {f} \sqrt {e+f x^2} (b c-a d)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac {x \sqrt {e+f x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 422
Rule 429
Rule 539
Rule 540
Rule 553
Rule 558
Rubi steps
\begin {align*} \int \frac {\left (e+f x^2\right )^{3/2}}{\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx &=-\frac {\int \frac {\sqrt {e+f x^2} \left (2 b c d e-a d^2 e-b c^2 f+d^2 (b e-a f) x^2\right )}{\left (c+d x^2\right )^{5/2}} \, dx}{(b c-a d)^2}+\frac {(b (b e-a f)) \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{(b c-a d)^2}\\ &=-\frac {(d e-c f) x \sqrt {e+f x^2}}{3 c (b c-a d) \left (c+d x^2\right )^{3/2}}+\frac {b e^{3/2} (b e-a f) \sqrt {c+d x^2} \Pi \left (1-\frac {b e}{a f};\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{a c (b c-a d)^2 \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {\int \frac {-d e (b c (5 d e-2 c f)-a d (2 d e+c f))-d f (b c (4 d e-c f)-a d (d e+2 c f)) x^2}{\left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}} \, dx}{3 c d (b c-a d)^2}\\ &=-\frac {(d e-c f) x \sqrt {e+f x^2}}{3 c (b c-a d) \left (c+d x^2\right )^{3/2}}+\frac {b e^{3/2} (b e-a f) \sqrt {c+d x^2} \Pi \left (1-\frac {b e}{a f};\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{a c (b c-a d)^2 \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {(e f) \int \frac {1}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{3 c (b c-a d)}-\frac {(b c (5 d e-c f)-2 a d (d e+c f)) \int \frac {\sqrt {e+f x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{3 c (b c-a d)^2}\\ &=-\frac {(d e-c f) x \sqrt {e+f x^2}}{3 c (b c-a d) \left (c+d x^2\right )^{3/2}}-\frac {(b c (5 d e-c f)-2 a d (d e+c f)) \sqrt {e+f x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{3 c^{3/2} \sqrt {d} (b c-a d)^2 \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {e^{3/2} \sqrt {f} \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 c^2 (b c-a d) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {b e^{3/2} (b e-a f) \sqrt {c+d x^2} \Pi \left (1-\frac {b e}{a f};\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{a c (b c-a d)^2 \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 7.22, size = 999, normalized size = 2.55 \begin {gather*} \frac {3 a^2 c d^2 \sqrt {\frac {d}{c}} e^2 x-6 a b c^3 \left (\frac {d}{c}\right )^{3/2} e^2 x+2 a b c^3 \sqrt {\frac {d}{c}} e f x+a^2 c^3 \left (\frac {d}{c}\right )^{3/2} e f x-5 a b c d^2 \sqrt {\frac {d}{c}} e^2 x^3+2 a^2 d^3 \sqrt {\frac {d}{c}} e^2 x^3+5 a^2 c d^2 \sqrt {\frac {d}{c}} e f x^3-5 a b c^3 \left (\frac {d}{c}\right )^{3/2} e f x^3+2 a b c^3 \sqrt {\frac {d}{c}} f^2 x^3+a^2 c^3 \left (\frac {d}{c}\right )^{3/2} f^2 x^3-5 a b c d^2 \sqrt {\frac {d}{c}} e f x^5+2 a^2 d^3 \sqrt {\frac {d}{c}} e f x^5+2 a^2 c d^2 \sqrt {\frac {d}{c}} f^2 x^5+a b c^3 \left (\frac {d}{c}\right )^{3/2} f^2 x^5+i a e (b c (-5 d e+c f)+2 a d (d e+c f)) \left (c+d x^2\right ) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} E\left (i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )-i a (-d e+c f) (5 b c e-2 a d e-3 a c f) \left (c+d x^2\right ) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} F\left (i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )-3 i b^2 c^3 e^2 \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \Pi \left (\frac {b c}{a d};i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )+6 i a b c^3 e f \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \Pi \left (\frac {b c}{a d};i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )-3 i a^2 c^3 f^2 \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \Pi \left (\frac {b c}{a d};i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )-3 i b^2 c^2 d e^2 x^2 \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \Pi \left (\frac {b c}{a d};i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )+6 i a b c^2 d e f x^2 \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \Pi \left (\frac {b c}{a d};i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )-3 i a^2 c^2 d f^2 x^2 \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \Pi \left (\frac {b c}{a d};i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )}{3 a c^2 \sqrt {\frac {d}{c}} (b c-a d)^2 \left (c+d x^2\right )^{3/2} \sqrt {e+f 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. \(1875\) vs.
\(2(456)=912\).
time = 0.16, size = 1876, normalized size = 4.80
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1645\) |
default | \(\text {Expression too large to display}\) | \(1876\) |
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 (f\,x^2+e\right )}^{3/2}}{\left (b\,x^2+a\right )\,{\left (d\,x^2+c\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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