Optimal. Leaf size=88 \[ \frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {-a+b x^2} \sqrt {1+\frac {d x^2}{c}}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {438, 437, 435}
\begin {gather*} \frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} E\left (\text {ArcSin}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {b x^2-a} \sqrt {\frac {d x^2}{c}+1}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 435
Rule 437
Rule 438
Rubi steps
\begin {align*} \int \frac {\sqrt {c+d x^2}}{\sqrt {-a+b x^2}} \, dx &=\frac {\sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {c+d x^2}}{\sqrt {1-\frac {b x^2}{a}}} \, dx}{\sqrt {-a+b x^2}}\\ &=\frac {\left (\sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2}\right ) \int \frac {\sqrt {1+\frac {d x^2}{c}}}{\sqrt {1-\frac {b x^2}{a}}} \, dx}{\sqrt {-a+b x^2} \sqrt {1+\frac {d x^2}{c}}}\\ &=\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {-a+b x^2} \sqrt {1+\frac {d x^2}{c}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.75, size = 88, normalized size = 1.00 \begin {gather*} \frac {\sqrt {\frac {a-b x^2}{a}} \sqrt {c+d x^2} E\left (\sin ^{-1}\left (\sqrt {\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )}{\sqrt {\frac {b}{a}} \sqrt {-a+b x^2} \sqrt {\frac {c+d x^2}{c}}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(166\) vs.
\(2(73)=146\).
time = 0.08, size = 167, normalized size = 1.90
method | result | size |
default | \(\frac {\left (-a d \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right )-c \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right ) b +a d \EllipticE \left (x \sqrt {-\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right )\right ) \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}-a}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {-b \,x^{2}+a}{a}}}{\left (-b d \,x^{4}+a d \,x^{2}-c \,x^{2} b +a c \right ) \sqrt {-\frac {d}{c}}\, b}\) | \(167\) |
elliptic | \(\frac {\sqrt {-\left (-b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (\frac {c \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1-\frac {b \,x^{2}}{a}}\, \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {b d \,x^{4}-a d \,x^{2}+c \,x^{2} b -a c}}+\frac {d a \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1-\frac {b \,x^{2}}{a}}\, \left (\EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )-\EllipticE \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {b d \,x^{4}-a d \,x^{2}+c \,x^{2} b -a c}\, b}\right )}{\sqrt {b \,x^{2}-a}\, \sqrt {d \,x^{2}+c}}\) | \(264\) |
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 {\sqrt {c + d x^{2}}}{\sqrt {- a + b 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 = -0.01 \begin {gather*} \int \frac {\sqrt {d\,x^2+c}}{\sqrt {b\,x^2-a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________