Optimal. Leaf size=443 \[ \frac {c^{3/2} (3 a c+4 b) \left (a c+a d x^2+b\right ) F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{15 a^2 d^{5/2} \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}-\frac {x (3 a c+4 b) \left (a c+a d x^2+b\right )}{15 a^2 d^2 \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}-\frac {\sqrt {c} \left (3 a^2 c^2+13 a b c+8 b^2\right ) \left (a c+a d x^2+b\right ) E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{15 a^3 d^{5/2} \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}+\frac {x \left (3 a^2 c^2+13 a b c+8 b^2\right ) \left (a c+a d x^2+b\right )}{15 a^3 d^2 \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}+\frac {x^3 \left (a c+a d x^2+b\right )}{5 a d \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}} \]
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Rubi [A] time = 0.71, antiderivative size = 498, normalized size of antiderivative = 1.12, 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} \[ \frac {x \left (3 a^2 c^2+13 a b c+8 b^2\right ) \sqrt {a c+a d x^2+b} \sqrt {a \left (c+d x^2\right )+b}}{15 a^3 d^2 \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}-\frac {\sqrt {c} \left (3 a^2 c^2+13 a b c+8 b^2\right ) \sqrt {a c+a d x^2+b} \sqrt {a \left (c+d x^2\right )+b} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{15 a^3 d^{5/2} \left (c+d x^2\right ) \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}} \sqrt {a+\frac {b}{c+d x^2}}}+\frac {c^{3/2} (3 a c+4 b) \sqrt {a c+a d x^2+b} \sqrt {a \left (c+d x^2\right )+b} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{15 a^2 d^{5/2} \left (c+d x^2\right ) \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}} \sqrt {a+\frac {b}{c+d x^2}}}-\frac {x (3 a c+4 b) \sqrt {a c+a d x^2+b} \sqrt {a \left (c+d x^2\right )+b}}{15 a^2 d^2 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {x^3 \sqrt {a c+a d x^2+b} \sqrt {a \left (c+d x^2\right )+b}}{5 a d \sqrt {a+\frac {b}{c+d x^2}}} \]
Antiderivative was successfully verified.
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Rule 411
Rule 418
Rule 478
Rule 492
Rule 531
Rule 582
Rule 1975
Rule 6722
Rubi steps
\begin {align*} \int \frac {x^4}{\sqrt {a+\frac {b}{c+d x^2}}} \, dx &=\frac {\sqrt {b+a \left (c+d x^2\right )} \int \frac {x^4 \sqrt {c+d x^2}}{\sqrt {b+a \left (c+d x^2\right )}} \, dx}{\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=\frac {\sqrt {b+a \left (c+d x^2\right )} \int \frac {x^4 \sqrt {c+d x^2}}{\sqrt {b+a c+a d x^2}} \, dx}{\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=\frac {x^3 \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{5 a d \sqrt {a+\frac {b}{c+d x^2}}}-\frac {\sqrt {b+a \left (c+d x^2\right )} \int \frac {x^2 \left (3 c (b+a c)+(4 b+3 a c) d x^2\right )}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{5 a d \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=-\frac {(4 b+3 a c) x \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{15 a^2 d^2 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {x^3 \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{5 a d \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\sqrt {b+a \left (c+d x^2\right )} \int \frac {c (b+a c) (4 b+3 a c) d+\left (8 b^2+13 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^2 d^3 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=-\frac {(4 b+3 a c) x \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{15 a^2 d^2 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {x^3 \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{5 a d \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left (c (b+a c) (4 b+3 a c) \sqrt {b+a \left (c+d x^2\right )}\right ) \int \frac {1}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{15 a^2 d^2 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left (\left (8 b^2+13 a b c+3 a^2 c^2\right ) \sqrt {b+a \left (c+d x^2\right )}\right ) \int \frac {x^2}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{15 a^2 d \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=-\frac {(4 b+3 a c) x \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{15 a^2 d^2 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {x^3 \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{5 a d \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left (8 b^2+13 a b c+3 a^2 c^2\right ) x \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{15 a^3 d^2 \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}+\frac {c^{3/2} (4 b+3 a c) \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{15 a^2 d^{5/2} \left (c+d x^2\right ) \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}} \sqrt {a+\frac {b}{c+d x^2}}}-\frac {\left (c \left (8 b^2+13 a b c+3 a^2 c^2\right ) \sqrt {b+a \left (c+d x^2\right )}\right ) \int \frac {\sqrt {b+a c+a d x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{15 a^3 d^2 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=-\frac {(4 b+3 a c) x \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{15 a^2 d^2 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {x^3 \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{5 a d \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left (8 b^2+13 a b c+3 a^2 c^2\right ) x \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{15 a^3 d^2 \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}-\frac {\sqrt {c} \left (8 b^2+13 a b c+3 a^2 c^2\right ) \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{15 a^3 d^{5/2} \left (c+d x^2\right ) \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}} \sqrt {a+\frac {b}{c+d x^2}}}+\frac {c^{3/2} (4 b+3 a c) \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{15 a^2 d^{5/2} \left (c+d x^2\right ) \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}} \sqrt {a+\frac {b}{c+d x^2}}}\\ \end {align*}
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Mathematica [C] time = 0.79, size = 297, normalized size = 0.67 \[ -\frac {\sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (x \left (c+d x^2\right ) \sqrt {\frac {a d}{a c+b}} \left (3 a^2 \left (c^2-d^2 x^4\right )+a b \left (7 c+d x^2\right )+4 b^2\right )+i c \left (3 a^2 c^2+13 a b c+8 b^2\right ) \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {a c+a d x^2+b}{a c+b}} E\left (i \sinh ^{-1}\left (\sqrt {\frac {a d}{b+a c}} x\right )|\frac {b}{a c}+1\right )-2 i b c (3 a c+2 b) \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {a c+a d x^2+b}{a c+b}} F\left (i \sinh ^{-1}\left (\sqrt {\frac {a d}{b+a c}} x\right )|\frac {b}{a c}+1\right )\right )}{15 a^2 d^2 \sqrt {\frac {a d}{a c+b}} \left (a \left (c+d x^2\right )+b\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.80, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (d x^{6} + c x^{4}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{a d x^{2} + a c + b}, x\right ) \]
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 \[ \int \frac {x^{4}}{\sqrt {a + \frac {b}{d x^{2} + c}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 665, normalized size = 1.50 \[ \frac {\left (3 \sqrt {-\frac {a d}{a c +b}}\, a^{2} d^{3} x^{7}+3 \sqrt {-\frac {a d}{a c +b}}\, a^{2} c \,d^{2} x^{5}-\sqrt {-\frac {a d}{a c +b}}\, a b \,d^{2} x^{5}-3 \sqrt {-\frac {a d}{a c +b}}\, a^{2} c^{2} d \,x^{3}-8 \sqrt {-\frac {a d}{a c +b}}\, a b c d \,x^{3}-3 \sqrt {-\frac {a d}{a c +b}}\, a^{2} c^{3} x +3 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, a^{2} c^{3} \EllipticE \left (\sqrt {-\frac {a d}{a c +b}}\, x , \sqrt {\frac {a c +b}{a c}}\right )-4 \sqrt {-\frac {a d}{a c +b}}\, b^{2} d \,x^{3}-7 \sqrt {-\frac {a d}{a c +b}}\, a b \,c^{2} x +13 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, a b \,c^{2} \EllipticE \left (\sqrt {-\frac {a d}{a c +b}}\, x , \sqrt {\frac {a c +b}{a c}}\right )-6 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, a b \,c^{2} \EllipticF \left (\sqrt {-\frac {a d}{a c +b}}\, x , \sqrt {\frac {a c +b}{a c}}\right )-4 \sqrt {-\frac {a d}{a c +b}}\, b^{2} c x +8 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, b^{2} c \EllipticE \left (\sqrt {-\frac {a d}{a c +b}}\, x , \sqrt {\frac {a c +b}{a c}}\right )-4 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, b^{2} c \EllipticF \left (\sqrt {-\frac {a d}{a c +b}}\, x , \sqrt {\frac {a c +b}{a c}}\right )\right ) \left (d \,x^{2}+c \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{15 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {-\frac {a d}{a c +b}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, a^{2} d^{2}} \]
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 \[ \int \frac {x^{4}}{\sqrt {a + \frac {b}{d x^{2} + c}}}\,{d x} \]
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 \[ \int \frac {x^4}{\sqrt {a+\frac {b}{d\,x^2+c}}} \,d x \]
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 \[ \int \frac {x^{4}}{\sqrt {\frac {a c + a d x^{2} + b}{c + d x^{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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