Optimal. Leaf size=247 \[ \frac {(d e-c f) x \sqrt {a-b x^2}}{c (b c+a d) \sqrt {c+d x^2}}+\frac {\sqrt {a} \sqrt {b} (d e-c f) \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{c d (b c+a d) \sqrt {a-b x^2} \sqrt {1+\frac {d x^2}{c}}}+\frac {\sqrt {a} f \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} F\left (\sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.16, antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {541, 538, 438,
437, 435, 432, 430} \begin {gather*} \frac {\sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} (d e-c f) E\left (\text {ArcSin}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{c d \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1} (a d+b c)}+\frac {\sqrt {a} f \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} F\left (\text {ArcSin}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x^2}}+\frac {x \sqrt {a-b x^2} (d e-c f)}{c \sqrt {c+d x^2} (a d+b c)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 430
Rule 432
Rule 435
Rule 437
Rule 438
Rule 538
Rule 541
Rubi steps
\begin {align*} \int \frac {e+f x^2}{\sqrt {a-b x^2} \left (c+d x^2\right )^{3/2}} \, dx &=\frac {(d e-c f) x \sqrt {a-b x^2}}{c (b c+a d) \sqrt {c+d x^2}}-\frac {\int \frac {-c (b e+a f)-b (d e-c f) x^2}{\sqrt {a-b x^2} \sqrt {c+d x^2}} \, dx}{c (b c+a d)}\\ &=\frac {(d e-c f) x \sqrt {a-b x^2}}{c (b c+a d) \sqrt {c+d x^2}}+\frac {f \int \frac {1}{\sqrt {a-b x^2} \sqrt {c+d x^2}} \, dx}{d}+\frac {(b (d e-c f)) \int \frac {\sqrt {c+d x^2}}{\sqrt {a-b x^2}} \, dx}{c d (b c+a d)}\\ &=\frac {(d e-c f) x \sqrt {a-b x^2}}{c (b c+a d) \sqrt {c+d x^2}}+\frac {\left (b (d e-c f) \sqrt {1-\frac {b x^2}{a}}\right ) \int \frac {\sqrt {c+d x^2}}{\sqrt {1-\frac {b x^2}{a}}} \, dx}{c d (b c+a d) \sqrt {a-b x^2}}+\frac {\left (f \sqrt {1+\frac {d x^2}{c}}\right ) \int \frac {1}{\sqrt {a-b x^2} \sqrt {1+\frac {d x^2}{c}}} \, dx}{d \sqrt {c+d x^2}}\\ &=\frac {(d e-c f) x \sqrt {a-b x^2}}{c (b c+a d) \sqrt {c+d x^2}}+\frac {\left (b (d e-c f) \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}{c d (b c+a d) \sqrt {a-b x^2} \sqrt {1+\frac {d x^2}{c}}}+\frac {\left (f \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}}\right ) \int \frac {1}{\sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}}} \, dx}{d \sqrt {a-b x^2} \sqrt {c+d x^2}}\\ &=\frac {(d e-c f) x \sqrt {a-b x^2}}{c (b c+a d) \sqrt {c+d x^2}}+\frac {\sqrt {a} \sqrt {b} (d e-c f) \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 )}{c d (b c+a d) \sqrt {a-b x^2} \sqrt {1+\frac {d x^2}{c}}}+\frac {\sqrt {a} f \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} F\left (\sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 7.54, size = 220, normalized size = 0.89 \begin {gather*} \frac {\sqrt {-\frac {b}{a}} d (d e-c f) x \left (a-b x^2\right )+i b c (-d e+c f) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \sinh ^{-1}\left (\sqrt {-\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )-i c (b c+a d) f \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} F\left (i \sinh ^{-1}\left (\sqrt {-\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )}{\sqrt {-\frac {b}{a}} c d (b c+a d) \sqrt {a-b x^2} \sqrt {c+d x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.12, size = 349, normalized size = 1.41
method | result | size |
default | \(\frac {\left (\sqrt {\frac {b}{a}}\, b c d f \,x^{3}-\sqrt {\frac {b}{a}}\, b \,d^{2} e \,x^{3}+\sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticF \left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) a c d f +\sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticF \left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) b \,c^{2} f -\sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticE \left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) b \,c^{2} f +\sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticE \left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) b c d e -\sqrt {\frac {b}{a}}\, a c d f x +\sqrt {\frac {b}{a}}\, a \,d^{2} e x \right ) \sqrt {-b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}{c d \sqrt {\frac {b}{a}}\, \left (a d +b c \right ) \left (-b d \,x^{4}+a d \,x^{2}-c \,x^{2} b +a c \right )}\) | \(349\) |
elliptic | \(\frac {\sqrt {\left (-b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (-\frac {\left (-b d \,x^{2}+a d \right ) x \left (c f -d e \right )}{d c \left (a d +b c \right ) \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (-b d \,x^{2}+a d \right )}}+\frac {\left (\frac {f}{d}-\frac {c f -d e}{d c}+\frac {a \left (c f -d e \right )}{c \left (a d +b c \right )}\right ) \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \EllipticF \left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-c \,x^{2} b +a c}}+\frac {\left (c f -d e \right ) b \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\EllipticF \left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )-\EllipticE \left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )\right )}{\left (a d +b c \right ) \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-c \,x^{2} b +a c}\, d}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(376\) |
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 {e + f x^{2}}{\sqrt {a - b x^{2}} \left (c + d x^{2}\right )^{\frac {3}{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 {f\,x^2+e}{\sqrt {a-b\,x^2}\,{\left (d\,x^2+c\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________