Optimal. Leaf size=368 \[ \frac {\sqrt {c} \left (-3 a^2 c^2+7 a b c+2 b^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{15 a^2 d^{5/2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}-\frac {x \left (-3 a^2 c^2+7 a b c+2 b^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{15 a^2 d^2}-\frac {c^{3/2} (b-3 a c) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{15 a d^{5/2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}+\frac {x (b-3 a c) \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{15 a d^2}+\frac {x^3 \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{5 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.72, antiderivative size = 478, normalized size of antiderivative = 1.30, number of steps used = 8, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {6722, 1975, 478, 582, 531, 418, 492, 411} \[ -\frac {x \left (-3 a^2 c^2+7 a b c+2 b^2\right ) \sqrt {a c+a d x^2+b} \sqrt {a+\frac {b}{c+d x^2}}}{15 a^2 d^2 \sqrt {a \left (c+d x^2\right )+b}}+\frac {\sqrt {c} \left (-3 a^2 c^2+7 a b c+2 b^2\right ) \sqrt {a c+a d x^2+b} \sqrt {a+\frac {b}{c+d x^2}} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{15 a^2 d^{5/2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}} \sqrt {a \left (c+d x^2\right )+b}}-\frac {c^{3/2} (b-3 a c) \sqrt {a c+a d x^2+b} \sqrt {a+\frac {b}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{15 a d^{5/2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}} \sqrt {a \left (c+d x^2\right )+b}}+\frac {x (b-3 a c) \left (c+d x^2\right ) \sqrt {a c+a d x^2+b} \sqrt {a+\frac {b}{c+d x^2}}}{15 a d^2 \sqrt {a \left (c+d x^2\right )+b}}+\frac {x^3 \left (c+d x^2\right ) \sqrt {a c+a d x^2+b} \sqrt {a+\frac {b}{c+d x^2}}}{5 d \sqrt {a \left (c+d x^2\right )+b}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 411
Rule 418
Rule 478
Rule 492
Rule 531
Rule 582
Rule 1975
Rule 6722
Rubi steps
\begin {align*} \int x^4 \sqrt {a+\frac {b}{c+d x^2}} \, dx &=\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {x^4 \sqrt {b+a \left (c+d x^2\right )}}{\sqrt {c+d x^2}} \, dx}{\sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {x^4 \sqrt {b+a c+a d x^2}}{\sqrt {c+d x^2}} \, dx}{\sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {x^3 \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 d \sqrt {b+a \left (c+d x^2\right )}}-\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {x^2 \left (3 c (b+a c)-(b-3 a c) d x^2\right )}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{5 d \sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {(b-3 a c) x \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{15 a d^2 \sqrt {b+a \left (c+d x^2\right )}}+\frac {x^3 \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 d \sqrt {b+a \left (c+d x^2\right )}}+\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {-c (b-3 a c) (b+a c) d-\left (2 b^2+7 a b c-3 a^2 c^2\right ) d^2 x^2}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{15 a d^3 \sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {(b-3 a c) x \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{15 a d^2 \sqrt {b+a \left (c+d x^2\right )}}+\frac {x^3 \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 d \sqrt {b+a \left (c+d x^2\right )}}-\frac {\left (c (b-3 a c) (b+a c) \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {1}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{15 a d^2 \sqrt {b+a \left (c+d x^2\right )}}-\frac {\left (\left (2 b^2+7 a b c-3 a^2 c^2\right ) \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {x^2}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{15 a d \sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {\left (2 b^2+7 a b c-3 a^2 c^2\right ) x \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{15 a^2 d^2 \sqrt {b+a \left (c+d x^2\right )}}+\frac {(b-3 a c) x \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{15 a d^2 \sqrt {b+a \left (c+d x^2\right )}}+\frac {x^3 \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 d \sqrt {b+a \left (c+d x^2\right )}}-\frac {c^{3/2} (b-3 a c) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{15 a d^{5/2} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}} \sqrt {b+a \left (c+d x^2\right )}}+\frac {\left (c \left (2 b^2+7 a b c-3 a^2 c^2\right ) \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {\sqrt {b+a c+a d x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{15 a^2 d^2 \sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {\left (2 b^2+7 a b c-3 a^2 c^2\right ) x \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{15 a^2 d^2 \sqrt {b+a \left (c+d x^2\right )}}+\frac {(b-3 a c) x \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{15 a d^2 \sqrt {b+a \left (c+d x^2\right )}}+\frac {x^3 \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 d \sqrt {b+a \left (c+d x^2\right )}}+\frac {\sqrt {c} \left (2 b^2+7 a b c-3 a^2 c^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{15 a^2 d^{5/2} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}} \sqrt {b+a \left (c+d x^2\right )}}-\frac {c^{3/2} (b-3 a c) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{15 a d^{5/2} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}} \sqrt {b+a \left (c+d x^2\right )}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.93, size = 293, normalized size = 0.80 \[ \frac {\sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (x \left (c+d x^2\right ) \sqrt {\frac {a d}{a c+b}} \left (-3 a^2 \left (c^2-d^2 x^4\right )-2 a b \left (c-2 d x^2\right )+b^2\right )+i c \left (-3 a^2 c^2+7 a b c+2 b^2\right ) \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {a c+a d x^2+b}{a c+b}} E\left (i \sinh ^{-1}\left (\sqrt {\frac {a d}{b+a c}} x\right )|\frac {b}{a c}+1\right )-i b c (9 a c+b) \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {a c+a d x^2+b}{a c+b}} F\left (i \sinh ^{-1}\left (\sqrt {\frac {a d}{b+a c}} x\right )|\frac {b}{a c}+1\right )\right )}{15 a d^2 \sqrt {\frac {a d}{a c+b}} \left (a \left (c+d x^2\right )+b\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.80, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{4} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a + \frac {b}{d x^{2} + c}} x^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 662, normalized size = 1.80 \[ \frac {\left (3 \sqrt {-\frac {a d}{a c +b}}\, a^{2} d^{3} x^{7}+3 \sqrt {-\frac {a d}{a c +b}}\, a^{2} c \,d^{2} x^{5}+4 \sqrt {-\frac {a d}{a c +b}}\, a b \,d^{2} x^{5}-3 \sqrt {-\frac {a d}{a c +b}}\, a^{2} c^{2} d \,x^{3}+2 \sqrt {-\frac {a d}{a c +b}}\, a b c d \,x^{3}-3 \sqrt {-\frac {a d}{a c +b}}\, a^{2} c^{3} x +3 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, a^{2} c^{3} \EllipticE \left (\sqrt {-\frac {a d}{a c +b}}\, x , \sqrt {\frac {a c +b}{a c}}\right )+\sqrt {-\frac {a d}{a c +b}}\, b^{2} d \,x^{3}-2 \sqrt {-\frac {a d}{a c +b}}\, a b \,c^{2} x -7 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, a b \,c^{2} \EllipticE \left (\sqrt {-\frac {a d}{a c +b}}\, x , \sqrt {\frac {a c +b}{a c}}\right )+9 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, a b \,c^{2} \EllipticF \left (\sqrt {-\frac {a d}{a c +b}}\, x , \sqrt {\frac {a c +b}{a c}}\right )+\sqrt {-\frac {a d}{a c +b}}\, b^{2} c x -2 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, b^{2} c \EllipticE \left (\sqrt {-\frac {a d}{a c +b}}\, x , \sqrt {\frac {a c +b}{a c}}\right )+\sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, b^{2} c \EllipticF \left (\sqrt {-\frac {a d}{a c +b}}\, x , \sqrt {\frac {a c +b}{a c}}\right )\right ) \left (d \,x^{2}+c \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{15 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {-\frac {a d}{a c +b}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, a \,d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a + \frac {b}{d x^{2} + c}} x^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^4\,\sqrt {a+\frac {b}{d\,x^2+c}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{4} \sqrt {\frac {a c + a d x^{2} + b}{c + d x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________