Optimal. Leaf size=490 \[ \frac {\sqrt {c} d^{3/2} (3 b-a c) \left (a c+a d x^2+b\right ) F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 (a c+b)^3 \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}-\frac {\sqrt {c} d^{3/2} (7 b-a c) \left (a c+a d x^2+b\right ) E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 (a c+b)^3 \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}+\frac {d^2 x (7 b-a c) \left (a c+a d x^2+b\right )}{3 (a c+b)^3 \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}-\frac {d (7 b-a c) \left (a c+a d x^2+b\right )}{3 x (a c+b)^3 \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}+\frac {(3 b-a c) \left (a c+a d x^2+b\right )}{3 a x^3 (a c+b)^2 \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}-\frac {b}{a x^3 (a c+b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}} \]
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Rubi [A] time = 0.87, antiderivative size = 567, normalized size of antiderivative = 1.16, number of steps used = 9, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {6722, 1975, 468, 583, 531, 418, 492, 411} \[ \frac {d^2 x (7 b-a c) \sqrt {a c+a d x^2+b} \sqrt {a \left (c+d x^2\right )+b}}{3 (a c+b)^3 \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\sqrt {c} d^{3/2} (3 b-a c) \sqrt {a c+a d x^2+b} \sqrt {a \left (c+d x^2\right )+b} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 (a c+b)^3 \left (c+d x^2\right ) \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}} \sqrt {a+\frac {b}{c+d x^2}}}-\frac {\sqrt {c} d^{3/2} (7 b-a c) \sqrt {a c+a d x^2+b} \sqrt {a \left (c+d x^2\right )+b} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 (a c+b)^3 \left (c+d x^2\right ) \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}} \sqrt {a+\frac {b}{c+d x^2}}}-\frac {d (7 b-a c) \sqrt {a c+a d x^2+b} \sqrt {a \left (c+d x^2\right )+b}}{3 x (a c+b)^3 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {(3 b-a c) \sqrt {a c+a d x^2+b} \sqrt {a \left (c+d x^2\right )+b}}{3 a x^3 (a c+b)^2 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {b \sqrt {a \left (c+d x^2\right )+b}}{a x^3 (a c+b) \sqrt {a c+a d x^2+b} \sqrt {a+\frac {b}{c+d x^2}}} \]
Antiderivative was successfully verified.
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Rule 411
Rule 418
Rule 468
Rule 492
Rule 531
Rule 583
Rule 1975
Rule 6722
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx &=\frac {\sqrt {b+a \left (c+d x^2\right )} \int \frac {\left (c+d x^2\right )^{3/2}}{x^4 \left (b+a \left (c+d x^2\right )\right )^{3/2}} \, 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 {\left (c+d x^2\right )^{3/2}}{x^4 \left (b+a c+a d x^2\right )^{3/2}} \, dx}{\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=-\frac {b \sqrt {b+a \left (c+d x^2\right )}}{a (b+a c) x^3 \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}-\frac {\sqrt {b+a \left (c+d x^2\right )} \int \frac {c (3 b-a c) d+(2 b-a c) d^2 x^2}{x^4 \sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{a (b+a c) d \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=-\frac {b \sqrt {b+a \left (c+d x^2\right )}}{a (b+a c) x^3 \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}+\frac {(3 b-a c) \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{3 a (b+a c)^2 x^3 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\sqrt {b+a \left (c+d x^2\right )} \int \frac {a c^2 (7 b-a c) d^2+a c (3 b-a c) d^3 x^2}{x^2 \sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{3 a c (b+a c)^2 d \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=-\frac {b \sqrt {b+a \left (c+d x^2\right )}}{a (b+a c) x^3 \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}+\frac {(3 b-a c) \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{3 a (b+a c)^2 x^3 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {(7 b-a c) d \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{3 (b+a c)^3 x \sqrt {a+\frac {b}{c+d x^2}}}-\frac {\sqrt {b+a \left (c+d x^2\right )} \int \frac {-a c^2 (3 b-a c) (b+a c) d^3-a^2 c^2 (7 b-a c) d^4 x^2}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{3 a c^2 (b+a c)^3 d \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=-\frac {b \sqrt {b+a \left (c+d x^2\right )}}{a (b+a c) x^3 \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}+\frac {(3 b-a c) \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{3 a (b+a c)^2 x^3 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {(7 b-a c) d \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{3 (b+a c)^3 x \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left ((3 b-a c) d^2 \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}{3 (b+a c)^2 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left (a (7 b-a c) d^3 \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}{3 (b+a c)^3 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=-\frac {b \sqrt {b+a \left (c+d x^2\right )}}{a (b+a c) x^3 \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}+\frac {(3 b-a c) \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{3 a (b+a c)^2 x^3 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {(7 b-a c) d \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{3 (b+a c)^3 x \sqrt {a+\frac {b}{c+d x^2}}}+\frac {(7 b-a c) d^2 x \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{3 (b+a c)^3 \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\sqrt {c} (3 b-a c) d^{3/2} \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 )}{3 (b+a c)^3 \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 (7 b-a c) d^2 \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}{3 (b+a c)^3 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=-\frac {b \sqrt {b+a \left (c+d x^2\right )}}{a (b+a c) x^3 \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}+\frac {(3 b-a c) \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{3 a (b+a c)^2 x^3 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {(7 b-a c) d \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{3 (b+a c)^3 x \sqrt {a+\frac {b}{c+d x^2}}}+\frac {(7 b-a c) d^2 x \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{3 (b+a c)^3 \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}-\frac {\sqrt {c} (7 b-a c) d^{3/2} \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 )}{3 (b+a c)^3 \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 {\sqrt {c} (3 b-a c) d^{3/2} \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 )}{3 (b+a c)^3 \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 = 0.90, size = 319, normalized size = 0.65 \[ -\frac {\sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\left (c+d x^2\right ) \sqrt {\frac {a d}{a c+b}} \left (a^2 c \left (c^2-d^2 x^4\right )+a b \left (2 c^2+4 c d x^2+7 d^2 x^4\right )+b^2 \left (c+4 d x^2\right )\right )+i b d^2 x^3 (3 b-5 a c) \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {a c+a d x^2+b}{a c+b}} F\left (i \sinh ^{-1}\left (\sqrt {\frac {a d}{b+a c}} x\right )|\frac {b}{a c}+1\right )-i a c d^2 x^3 (a c-7 b) \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {a c+a d x^2+b}{a c+b}} E\left (i \sinh ^{-1}\left (\sqrt {\frac {a d}{b+a c}} x\right )|\frac {b}{a c}+1\right )\right )}{3 x^3 (a c+b)^3 \sqrt {\frac {a d}{a c+b}} \left (a \left (c+d x^2\right )+b\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (d^{2} x^{4} + 2 \, c d x^{2} + c^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{a^{2} d^{2} x^{8} + 2 \, {\left (a^{2} c + a b\right )} d x^{6} + {\left (a^{2} c^{2} + 2 \, a b c + b^{2}\right )} x^{4}}, x\right ) \]
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 \[ \int \frac {1}{{\left (a + \frac {b}{d x^{2} + c}\right )}^{\frac {3}{2}} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 1082, normalized size = 2.21 \[ \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} c \,d^{3} x^{6}-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 \,d^{3} x^{6}-3 \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 \,d^{3} x^{6}+\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^{2} x^{4}-\sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, a^{2} c^{2} d^{2} x^{3} \EllipticE \left (\sqrt {-\frac {a d}{a c +b}}\, x , \sqrt {\frac {a c +b}{a c}}\right )-8 \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^{2} x^{4}-3 \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^{2} x^{4}+7 \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, a b c \,d^{2} x^{3} \EllipticE \left (\sqrt {-\frac {a d}{a c +b}}\, x , \sqrt {\frac {a c +b}{a c}}\right )-5 \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, a b c \,d^{2} x^{3} \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}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, a^{2} c^{3} d \,x^{2}-4 \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^{2} x^{4}+3 \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, b^{2} d^{2} x^{3} \EllipticF \left (\sqrt {-\frac {a d}{a c +b}}\, x , \sqrt {\frac {a c +b}{a c}}\right )-6 \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} d \,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 )}\, a^{2} c^{4}-5 \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 d \,x^{2}-2 \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^{3}-\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^{2}\right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{3 \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 c +b \right )^{3} \left (a d \,x^{2}+a c +b \right ) x^{3}} \]
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 \[ \int \frac {1}{{\left (a + \frac {b}{d x^{2} + c}\right )}^{\frac {3}{2}} x^{4}}\,{d x} \]
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 \[ \int \frac {1}{x^4\,{\left (a+\frac {b}{d\,x^2+c}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{x^{4} \left (\frac {a c + a d x^{2} + b}{c + d x^{2}}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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