Optimal. Leaf size=405 \[ -\frac {\sqrt {c} \left (a^2 c^2-14 a b c+b^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{5 a d^{5/2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}+\frac {x \left (a^2 c^2-14 a b c+b^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{5 a d^2}-\frac {c^{3/2} (7 b-a c) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{5 d^{5/2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}+\frac {x (7 b-a c) \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{5 d^2}+\frac {6 a x^3 \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{5 d}-\frac {x^3 \left (a c+a d x^2+b\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{d} \]
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Rubi [A] time = 0.88, antiderivative size = 526, normalized size of antiderivative = 1.30, number of steps used = 9, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6722, 1975, 467, 581, 582, 531, 418, 492, 411} \[ \frac {x \left (a^2 c^2-14 a b c+b^2\right ) \sqrt {a c+a d x^2+b} \sqrt {a+\frac {b}{c+d x^2}}}{5 a d^2 \sqrt {a \left (c+d x^2\right )+b}}-\frac {\sqrt {c} \left (a^2 c^2-14 a b c+b^2\right ) \sqrt {a c+a d x^2+b} \sqrt {a+\frac {b}{c+d x^2}} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{5 a d^{5/2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}} \sqrt {a \left (c+d x^2\right )+b}}-\frac {c^{3/2} (7 b-a c) \sqrt {a c+a d x^2+b} \sqrt {a+\frac {b}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{5 d^{5/2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}} \sqrt {a \left (c+d x^2\right )+b}}+\frac {x (7 b-a c) \left (c+d x^2\right ) \sqrt {a c+a d x^2+b} \sqrt {a+\frac {b}{c+d x^2}}}{5 d^2 \sqrt {a \left (c+d x^2\right )+b}}-\frac {x^3 \left (a c+a d x^2+b\right )^{3/2} \sqrt {a+\frac {b}{c+d x^2}}}{d \sqrt {a \left (c+d x^2\right )+b}}+\frac {6 a x^3 \left (c+d x^2\right ) \sqrt {a c+a d x^2+b} \sqrt {a+\frac {b}{c+d x^2}}}{5 d \sqrt {a \left (c+d x^2\right )+b}} \]
Antiderivative was successfully verified.
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Rule 411
Rule 418
Rule 467
Rule 492
Rule 531
Rule 581
Rule 582
Rule 1975
Rule 6722
Rubi steps
\begin {align*} \int x^4 \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx &=\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {x^4 \left (b+a \left (c+d x^2\right )\right )^{3/2}}{\left (c+d x^2\right )^{3/2}} \, 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 \left (b+a c+a d x^2\right )^{3/2}}{\left (c+d x^2\right )^{3/2}} \, dx}{\sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {x^3 \left (b+a c+a d x^2\right )^{3/2} \sqrt {a+\frac {b}{c+d x^2}}}{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 \sqrt {b+a c+a d x^2} \left (3 (b+a c)+6 a d x^2\right )}{\sqrt {c+d x^2}} \, dx}{d \sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {6 a 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 {x^3 \left (b+a c+a d x^2\right )^{3/2} \sqrt {a+\frac {b}{c+d x^2}}}{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 (5 b-a c) (b+a c) d+3 a (7 b-a c) d^2 x^2\right )}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{5 d^2 \sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {(7 b-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}}}{5 d^2 \sqrt {b+a \left (c+d x^2\right )}}+\frac {6 a 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 {x^3 \left (b+a c+a d x^2\right )^{3/2} \sqrt {a+\frac {b}{c+d x^2}}}{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 {3 a c (7 b-a c) (b+a c) d^2-3 a \left (b^2-14 a b c+a^2 c^2\right ) d^3 x^2}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{15 a d^4 \sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {(7 b-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}}}{5 d^2 \sqrt {b+a \left (c+d x^2\right )}}+\frac {6 a 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 {x^3 \left (b+a c+a d x^2\right )^{3/2} \sqrt {a+\frac {b}{c+d x^2}}}{d \sqrt {b+a \left (c+d x^2\right )}}-\frac {\left (c (7 b-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}{5 d^2 \sqrt {b+a \left (c+d x^2\right )}}+\frac {\left (\left (b^2-14 a b c+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}{5 d \sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {\left (b^2-14 a b c+a^2 c^2\right ) x \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 a d^2 \sqrt {b+a \left (c+d x^2\right )}}+\frac {(7 b-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}}}{5 d^2 \sqrt {b+a \left (c+d x^2\right )}}+\frac {6 a 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 {x^3 \left (b+a c+a d x^2\right )^{3/2} \sqrt {a+\frac {b}{c+d x^2}}}{d \sqrt {b+a \left (c+d x^2\right )}}-\frac {c^{3/2} (7 b-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 )}{5 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 (b^2-14 a b c+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}{5 a d^2 \sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {\left (b^2-14 a b c+a^2 c^2\right ) x \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 a d^2 \sqrt {b+a \left (c+d x^2\right )}}+\frac {(7 b-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}}}{5 d^2 \sqrt {b+a \left (c+d x^2\right )}}+\frac {6 a 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 {x^3 \left (b+a c+a d x^2\right )^{3/2} \sqrt {a+\frac {b}{c+d x^2}}}{d \sqrt {b+a \left (c+d x^2\right )}}-\frac {\sqrt {c} \left (b^2-14 a b c+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 )}{5 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 {c^{3/2} (7 b-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 )}{5 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*}
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Mathematica [C] time = 0.85, size = 308, normalized size = 0.76 \[ \frac {\sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (x \sqrt {\frac {a d}{a c+b}} \left (-a^2 \left (c-d x^2\right ) \left (c+d x^2\right )^2+3 a b \left (2 c^2+3 c d x^2+d^2 x^4\right )+b^2 \left (7 c+2 d x^2\right )\right )-i c \left (a^2 c^2-14 a b c+b^2\right ) \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {a c+a d x^2+b}{a c+b}} E\left (i \sinh ^{-1}\left (\sqrt {\frac {a d}{b+a c}} x\right )|\frac {b}{a c}+1\right )+8 i b c (b-a c) \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {a c+a d x^2+b}{a c+b}} F\left (i \sinh ^{-1}\left (\sqrt {\frac {a d}{b+a c}} x\right )|\frac {b}{a c}+1\right )\right )}{5 d^2 \sqrt {\frac {a d}{a c+b}} \left (a \left (c+d x^2\right )+b\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a d x^{6} + {\left (a c + b\right )} x^{4}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{d x^{2} + c}, x\right ) \]
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 \[ \int {\left (a + \frac {b}{d x^{2} + c}\right )}^{\frac {3}{2}} x^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 1098, normalized size = 2.71 \[ \frac {\left (\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^{3} x^{7}+\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} c \,d^{2} x^{5}+3 \sqrt {-\frac {a d}{a c +b}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, a b \,d^{2} x^{5}-\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} c^{2} d \,x^{3}+5 \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}}\, a b c d \,x^{3}+4 \sqrt {-\frac {a d}{a c +b}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, a b c d \,x^{3}-\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} c^{3} x +\sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, a^{2} c^{3} \EllipticE \left (\sqrt {-\frac {a d}{a c +b}}\, x , \sqrt {\frac {a c +b}{a c}}\right )+2 \sqrt {-\frac {a d}{a c +b}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, b^{2} d \,x^{3}+5 \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}}\, a b \,c^{2} x +\sqrt {-\frac {a d}{a c +b}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, a b \,c^{2} x -14 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, a b \,c^{2} \EllipticE \left (\sqrt {-\frac {a d}{a c +b}}\, x , \sqrt {\frac {a c +b}{a c}}\right )+8 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, a b \,c^{2} \EllipticF \left (\sqrt {-\frac {a d}{a c +b}}\, x , \sqrt {\frac {a c +b}{a c}}\right )+5 \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}}\, b^{2} c x +2 \sqrt {-\frac {a d}{a c +b}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, b^{2} c x +\sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, b^{2} c \EllipticE \left (\sqrt {-\frac {a d}{a c +b}}\, x , \sqrt {\frac {a c +b}{a c}}\right )-8 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, b^{2} c \EllipticF \left (\sqrt {-\frac {a d}{a c +b}}\, x , \sqrt {\frac {a c +b}{a c}}\right )\right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{5 \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}}\, \left (a d \,x^{2}+a c +b \right ) 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 \[ \int {\left (a + \frac {b}{d x^{2} + c}\right )}^{\frac {3}{2}} x^{4}\,{d x} \]
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 \[ \int x^4\,{\left (a+\frac {b}{d\,x^2+c}\right )}^{3/2} \,d x \]
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 \[ \int x^{4} \left (\frac {a c + a d x^{2} + b}{c + d x^{2}}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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