Optimal. Leaf size=485 \[ -\frac {b f x \sqrt {c+d x^2}}{2 a (b c-a d) (b e-a f) \sqrt {e+f x^2}}+\frac {b^2 x \sqrt {c+d x^2} \sqrt {e+f x^2}}{2 a (b c-a d) (b e-a f) \left (a+b x^2\right )}+\frac {b \sqrt {e} \sqrt {f} \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{2 a (b c-a d) (b e-a f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {d \sqrt {e} \sqrt {f} \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{2 c (b c-a d) (b e-a f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {\sqrt {-c} \left (b^2 c e+3 a^2 d f-2 a b (d e+c f)\right ) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \Pi \left (\frac {b c}{a d};\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {-c}}\right )|\frac {c f}{d e}\right )}{2 a^2 \sqrt {d} (b c-a d) (b e-a f) \sqrt {c+d x^2} \sqrt {e+f x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.23, antiderivative size = 485, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {563, 552, 551,
545, 429, 506, 422} \begin {gather*} \frac {\sqrt {-c} \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {f x^2}{e}+1} \left (3 a^2 d f-2 a b (c f+d e)+b^2 c e\right ) \Pi \left (\frac {b c}{a d};\text {ArcSin}\left (\frac {\sqrt {d} x}{\sqrt {-c}}\right )|\frac {c f}{d e}\right )}{2 a^2 \sqrt {d} \sqrt {c+d x^2} \sqrt {e+f x^2} (b c-a d) (b e-a f)}-\frac {d \sqrt {e} \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 )}{2 c \sqrt {e+f x^2} (b c-a d) (b e-a f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {b \sqrt {e} \sqrt {f} \sqrt {c+d x^2} E\left (\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{2 a \sqrt {e+f x^2} (b c-a d) (b e-a f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {b^2 x \sqrt {c+d x^2} \sqrt {e+f x^2}}{2 a \left (a+b x^2\right ) (b c-a d) (b e-a f)}-\frac {b f x \sqrt {c+d x^2}}{2 a \sqrt {e+f x^2} (b c-a d) (b e-a f)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 422
Rule 429
Rule 506
Rule 545
Rule 551
Rule 552
Rule 563
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^2\right )^2 \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx &=\frac {b^2 x \sqrt {c+d x^2} \sqrt {e+f x^2}}{2 a (b c-a d) (b e-a f) \left (a+b x^2\right )}-\frac {(d f) \int \frac {a+b x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{2 a (b c-a d) (b e-a f)}+\frac {\left (b^2 c e+3 a^2 d f-2 a b (d e+c f)\right ) \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{2 a (b c-a d) (b e-a f)}\\ &=\frac {b^2 x \sqrt {c+d x^2} \sqrt {e+f x^2}}{2 a (b c-a d) (b e-a f) \left (a+b x^2\right )}-\frac {(d f) \int \frac {1}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{2 (b c-a d) (b e-a f)}-\frac {(b d f) \int \frac {x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{2 a (b c-a d) (b e-a f)}+\frac {\left (\left (b^2 c e+3 a^2 d f-2 a b (d e+c f)\right ) \sqrt {1+\frac {d x^2}{c}}\right ) \int \frac {1}{\left (a+b x^2\right ) \sqrt {1+\frac {d x^2}{c}} \sqrt {e+f x^2}} \, dx}{2 a (b c-a d) (b e-a f) \sqrt {c+d x^2}}\\ &=-\frac {b f x \sqrt {c+d x^2}}{2 a (b c-a d) (b e-a f) \sqrt {e+f x^2}}+\frac {b^2 x \sqrt {c+d x^2} \sqrt {e+f x^2}}{2 a (b c-a d) (b e-a f) \left (a+b x^2\right )}-\frac {d \sqrt {e} \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 )}{2 c (b c-a d) (b e-a f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {(b e f) \int \frac {\sqrt {c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{2 a (b c-a d) (b e-a f)}+\frac {\left (\left (b^2 c e+3 a^2 d f-2 a b (d e+c f)\right ) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}}\right ) \int \frac {1}{\left (a+b x^2\right ) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}}} \, dx}{2 a (b c-a d) (b e-a f) \sqrt {c+d x^2} \sqrt {e+f x^2}}\\ &=-\frac {b f x \sqrt {c+d x^2}}{2 a (b c-a d) (b e-a f) \sqrt {e+f x^2}}+\frac {b^2 x \sqrt {c+d x^2} \sqrt {e+f x^2}}{2 a (b c-a d) (b e-a f) \left (a+b x^2\right )}+\frac {b \sqrt {e} \sqrt {f} \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{2 a (b c-a d) (b e-a 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 {e} \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 )}{2 c (b c-a d) (b e-a f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {\sqrt {-c} \left (b^2 c e+3 a^2 d f-2 a b (d e+c f)\right ) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \Pi \left (\frac {b c}{a d};\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {-c}}\right )|\frac {c f}{d e}\right )}{2 a^2 \sqrt {d} (b c-a d) (b e-a f) \sqrt {c+d x^2} \sqrt {e+f x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 4.47, size = 587, normalized size = 1.21 \begin {gather*} \frac {\frac {b^2 c e x}{a+b x^2}+\frac {b^2 d e x^3}{a+b x^2}+\frac {b^2 c f x^3}{a+b x^2}+\frac {b^2 d f x^5}{a+b x^2}+i b c \sqrt {\frac {d}{c}} e \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 c \sqrt {\frac {d}{c}} (b e-a f) \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 )-\frac {i b^2 c e \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 )}{a \sqrt {\frac {d}{c}}}+2 i b c \sqrt {\frac {d}{c}} e \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 )+\frac {2 i b c 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 )}{\sqrt {\frac {d}{c}}}-3 i a c \sqrt {\frac {d}{c}} 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 )}{2 a (-b c+a d) (-b e+a f) \sqrt {c+d x^2} \sqrt {e+f 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. \(1077\) vs.
\(2(502)=1004\).
time = 0.15, size = 1078, normalized size = 2.22 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}{\left (a + b x^{2}\right )^{2} \sqrt {c + d x^{2}} \sqrt {e + f 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 (b\,x^2+a\right )}^2\,\sqrt {d\,x^2+c}\,\sqrt {f\,x^2+e}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________