Optimal. Leaf size=814 \[ -\frac {d^2 x}{3 c (b c-a d) (d e-c f) \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}-\frac {d^2 (b c (5 d e-9 c f)-2 a d (d e-3 c f)) x}{3 c^2 (b c-a d)^2 (d e-c f)^2 \sqrt {c+d x^2} \sqrt {e+f x^2}}+\frac {b^2 f^{3/2} \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{(b c-a d)^2 \sqrt {e} (b e-a f) (d e-c f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {d \sqrt {f} \left (b c \left (5 d^2 e^2-7 c d e f-6 c^2 f^2\right )-a d \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right )\right ) \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 c^2 (b c-a d)^2 \sqrt {e} (d e-c f)^3 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {b^2 \sqrt {e} \sqrt {f} (2 b d e-b c f-a d f) \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{c (b c-a d)^2 (b e-a f)^2 (d e-c f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {d^2 \sqrt {e} \sqrt {f} (b c (7 d e-15 c f)-a d (d e-9 c f)) \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 c^2 (b c-a d)^2 (d e-c f)^3 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {b^4 e^{3/2} \sqrt {c+d x^2} \Pi \left (1-\frac {b e}{a f};\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{a c (b c-a d)^2 \sqrt {f} (b e-a f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}} \]
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Rubi [A]
time = 0.63, antiderivative size = 814, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {560, 553, 539,
429, 422, 541} \begin {gather*} \frac {e^{3/2} \sqrt {d x^2+c} \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 ) b^4}{a c (b c-a d)^2 \sqrt {f} (b e-a f)^2 \sqrt {\frac {e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt {f x^2+e}}+\frac {f^{3/2} \sqrt {d x^2+c} E\left (\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right ) b^2}{(b c-a d)^2 \sqrt {e} (b e-a f) (d e-c f) \sqrt {\frac {e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt {f x^2+e}}-\frac {\sqrt {e} \sqrt {f} (2 b d e-b c f-a d f) \sqrt {d x^2+c} F\left (\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right ) b^2}{c (b c-a d)^2 (b e-a f)^2 (d e-c f) \sqrt {\frac {e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt {f x^2+e}}-\frac {d \sqrt {f} \left (b c \left (5 d^2 e^2-7 c d f e-6 c^2 f^2\right )-a d \left (2 d^2 e^2-7 c d f e-3 c^2 f^2\right )\right ) \sqrt {d x^2+c} E\left (\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 c^2 (b c-a d)^2 \sqrt {e} (d e-c f)^3 \sqrt {\frac {e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt {f x^2+e}}+\frac {d^2 \sqrt {e} \sqrt {f} (b c (7 d e-15 c f)-a d (d e-9 c f)) \sqrt {d x^2+c} F\left (\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 c^2 (b c-a d)^2 (d e-c f)^3 \sqrt {\frac {e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt {f x^2+e}}-\frac {d^2 (b c (5 d e-9 c f)-2 a d (d e-3 c f)) x}{3 c^2 (b c-a d)^2 (d e-c f)^2 \sqrt {d x^2+c} \sqrt {f x^2+e}}-\frac {d^2 x}{3 c (b c-a d) (d e-c f) \left (d x^2+c\right )^{3/2} \sqrt {f x^2+e}} \end {gather*}
Antiderivative was successfully verified.
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Rule 422
Rule 429
Rule 539
Rule 541
Rule 553
Rule 560
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^{3/2}} \, dx &=\frac {b^2 \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx}{(b c-a d)^2}-\frac {d \int \frac {2 b c-a d+b d x^2}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^{3/2}} \, dx}{(b c-a d)^2}\\ &=-\frac {d^2 x}{3 c (b c-a d) (d e-c f) \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}+\frac {b^4 \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{(b c-a d)^2 (b e-a f)^2}-\frac {\left (b^2 f\right ) \int \frac {2 b e-a f+b f x^2}{\sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx}{(b c-a d)^2 (b e-a f)^2}+\frac {d \int \frac {-b c (5 d e-6 c f)+a d (2 d e-3 c f)-3 d (b c-a d) f x^2}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}} \, dx}{3 c (b c-a d)^2 (d e-c f)}\\ &=-\frac {d^2 x}{3 c (b c-a d) (d e-c f) \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}-\frac {d^2 (b c (5 d e-9 c f)-2 a d (d e-3 c f)) x}{3 c^2 (b c-a d)^2 (d e-c f)^2 \sqrt {c+d x^2} \sqrt {e+f x^2}}+\frac {b^4 e^{3/2} \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} (b e-a f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {d \int \frac {-c f (2 b c (d e-3 c f)+a d (d e+3 c f))+d f (b c (5 d e-9 c f)-2 a d (d e-3 c f)) x^2}{\sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx}{3 c^2 (b c-a d)^2 (d e-c f)^2}+\frac {\left (b^2 f^2\right ) \int \frac {\sqrt {c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{(b c-a d)^2 (b e-a f) (d e-c f)}-\frac {\left (b^2 f (2 b d e-b c f-a d f)\right ) \int \frac {1}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{(b c-a d)^2 (b e-a f)^2 (d e-c f)}\\ &=-\frac {d^2 x}{3 c (b c-a d) (d e-c f) \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}-\frac {d^2 (b c (5 d e-9 c f)-2 a d (d e-3 c f)) x}{3 c^2 (b c-a d)^2 (d e-c f)^2 \sqrt {c+d x^2} \sqrt {e+f x^2}}+\frac {b^2 f^{3/2} \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{(b c-a d)^2 \sqrt {e} (b e-a f) (d e-c f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {b^2 \sqrt {e} \sqrt {f} (2 b d e-b c f-a d 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 )}{c (b c-a d)^2 (b e-a f)^2 (d e-c f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {b^4 e^{3/2} \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} (b e-a f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {\left (d^2 f (b c (7 d e-15 c f)-a d (d e-9 c f))\right ) \int \frac {1}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{3 c (b c-a d)^2 (d e-c f)^3}-\frac {\left (d f \left (b c \left (5 d^2 e^2-7 c d e f-6 c^2 f^2\right )-a d \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right )\right )\right ) \int \frac {\sqrt {c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{3 c^2 (b c-a d)^2 (d e-c f)^3}\\ &=-\frac {d^2 x}{3 c (b c-a d) (d e-c f) \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}-\frac {d^2 (b c (5 d e-9 c f)-2 a d (d e-3 c f)) x}{3 c^2 (b c-a d)^2 (d e-c f)^2 \sqrt {c+d x^2} \sqrt {e+f x^2}}+\frac {b^2 f^{3/2} \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{(b c-a d)^2 \sqrt {e} (b e-a f) (d e-c f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {d \sqrt {f} \left (b c \left (5 d^2 e^2-7 c d e f-6 c^2 f^2\right )-a d \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right )\right ) \sqrt {c+d x^2} E\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)^2 \sqrt {e} (d e-c f)^3 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {b^2 \sqrt {e} \sqrt {f} (2 b d e-b c f-a d 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 )}{c (b c-a d)^2 (b e-a f)^2 (d e-c f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {d^2 \sqrt {e} \sqrt {f} (b c (7 d e-15 c f)-a d (d e-9 c 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)^2 (d e-c f)^3 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {b^4 e^{3/2} \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} (b e-a f)^2 \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 = 10.04, size = 1645, normalized size = 2.02 \begin {gather*} \frac {-i a d e \left (2 a b d (d e-3 c f) (d e+c f)^2+a^2 d^2 f \left (-2 d^2 e^2+7 c d e f+3 c^2 f^2\right )+b^2 c \left (-5 d^3 e^3+10 c d^2 e^2 f+3 c^3 f^3\right )\right ) \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 )+\frac {\sqrt {\frac {d}{c}} \left (6 a b^2 c^2 d^5 e^4 x-3 a^2 b c d^6 e^4 x-11 a b^2 c^3 d^4 e^3 f x+2 a^2 b c^2 d^5 e^3 f x+3 a^3 c d^6 e^3 f x+11 a^2 b c^3 d^4 e^2 f^2 x-8 a^3 c^2 d^5 e^2 f^2 x-3 a b^2 c^6 d f^4 x+6 a^2 b c^5 d^2 f^4 x-3 a^3 c^4 d^3 f^4 x+5 a b^2 c d^6 e^4 x^3-2 a^2 b d^7 e^4 x^3-4 a b^2 c^2 d^5 e^3 f x^3-a^2 b c d^6 e^3 f x^3+2 a^3 d^7 e^3 f x^3-11 a b^2 c^3 d^4 e^2 f^2 x^3+12 a^2 b c^2 d^5 e^2 f^2 x^3-4 a^3 c d^6 e^2 f^2 x^3+11 a^2 b c^3 d^4 e f^3 x^3-8 a^3 c^2 d^5 e f^3 x^3-6 a b^2 c^5 d^2 f^4 x^3+12 a^2 b c^4 d^3 f^4 x^3-6 a^3 c^3 d^4 f^4 x^3+5 a b^2 c d^6 e^3 f x^5-2 a^2 b d^7 e^3 f x^5-10 a b^2 c^2 d^5 e^2 f^2 x^5+2 a^2 b c d^6 e^2 f^2 x^5+2 a^3 d^7 e^2 f^2 x^5+10 a^2 b c^2 d^5 e f^3 x^5-7 a^3 c d^6 e f^3 x^5-3 a b^2 c^4 d^3 f^4 x^5+6 a^2 b c^3 d^4 f^4 x^5-3 a^3 c^2 d^5 f^4 x^5-i a c d^2 \sqrt {\frac {d}{c}} e (b e-a f) (-d e+c f) (2 a d (d e-3 c f)+b c (-5 d e+9 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^3 c^4 d^3 \sqrt {\frac {d}{c}} e^4 \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 )-9 i b^3 c^7 \left (\frac {d}{c}\right )^{5/2} e^3 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 )+9 i b^3 c^7 \left (\frac {d}{c}\right )^{3/2} e^2 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^3 c^7 \sqrt {\frac {d}{c}} e f^3 \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^3 c^3 d^4 \sqrt {\frac {d}{c}} e^4 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 )-9 i b^3 c^4 d^3 \sqrt {\frac {d}{c}} e^3 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 )+9 i b^3 c^7 \left (\frac {d}{c}\right )^{5/2} e^2 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 i b^3 c^7 \left (\frac {d}{c}\right )^{3/2} e f^3 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 )\right )}{d}}{3 a c^2 \sqrt {\frac {d}{c}} (b c-a d)^2 e (b e-a f) (-d e+c f)^3 \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. \(4114\) vs.
\(2(923)=1846\).
time = 0.19, size = 4115, normalized size = 5.06
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(2127\) |
default | \(\text {Expression too large to display}\) | \(4115\) |
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 (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac {5}{2}} \left (e + f x^{2}\right )^{\frac {3}{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 (b\,x^2+a\right )\,{\left (d\,x^2+c\right )}^{5/2}\,{\left (f\,x^2+e\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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