Optimal. Leaf size=311 \[ -\frac {2 \sqrt {d} \sqrt {a+b x^2} (2 b c-a d) E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 c^{3/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {2 \sqrt {a+b x^2} \sqrt {c+d x^2} (2 b c-a d)}{3 c^2 x}+\frac {2 d x \sqrt {a+b x^2} (2 b c-a d)}{3 c^2 \sqrt {c+d x^2}}+\frac {b \sqrt {a+b x^2} (3 b c-a d) F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a \sqrt {c} \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c x^3} \]
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Rubi [A] time = 0.29, antiderivative size = 311, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {474, 583, 531, 418, 492, 411} \[ -\frac {2 \sqrt {a+b x^2} \sqrt {c+d x^2} (2 b c-a d)}{3 c^2 x}+\frac {2 d x \sqrt {a+b x^2} (2 b c-a d)}{3 c^2 \sqrt {c+d x^2}}-\frac {2 \sqrt {d} \sqrt {a+b x^2} (2 b c-a d) E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 c^{3/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c x^3}+\frac {b \sqrt {a+b x^2} (3 b c-a d) F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a \sqrt {c} \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]
Antiderivative was successfully verified.
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Rule 411
Rule 418
Rule 474
Rule 492
Rule 531
Rule 583
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{3/2}}{x^4 \sqrt {c+d x^2}} \, dx &=-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c x^3}+\frac {\int \frac {2 a (2 b c-a d)+b (3 b c-a d) x^2}{x^2 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 c}\\ &=-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c x^3}-\frac {2 (2 b c-a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c^2 x}-\frac {\int \frac {-a b c (3 b c-a d)-2 a b d (2 b c-a d) x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 a c^2}\\ &=-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c x^3}-\frac {2 (2 b c-a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c^2 x}+\frac {(2 b d (2 b c-a d)) \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 c^2}+\frac {(b (3 b c-a d)) \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 c}\\ &=\frac {2 d (2 b c-a d) x \sqrt {a+b x^2}}{3 c^2 \sqrt {c+d x^2}}-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c x^3}-\frac {2 (2 b c-a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c^2 x}+\frac {b (3 b c-a d) \sqrt {a+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a \sqrt {c} \sqrt {d} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {(2 d (2 b c-a d)) \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{3 c}\\ &=\frac {2 d (2 b c-a d) x \sqrt {a+b x^2}}{3 c^2 \sqrt {c+d x^2}}-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c x^3}-\frac {2 (2 b c-a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c^2 x}-\frac {2 \sqrt {d} (2 b c-a d) \sqrt {a+b x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 c^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {b (3 b c-a d) \sqrt {a+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a \sqrt {c} \sqrt {d} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}\\ \end {align*}
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Mathematica [C] time = 0.38, size = 227, normalized size = 0.73 \[ \frac {\sqrt {\frac {b}{a}} \left (a+b x^2\right ) \left (c+d x^2\right ) \left (-a c+2 a d x^2-4 b c x^2\right )-i b c x^3 \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} (a d-b c) F\left (i \sinh ^{-1}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+2 i b c x^3 \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} (a d-2 b c) E\left (i \sinh ^{-1}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )}{3 c^2 x^3 \sqrt {\frac {b}{a}} \sqrt {a+b x^2} \sqrt {c+d x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c}}{d x^{6} + c x^{4}}, x\right ) \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{\sqrt {d x^{2} + c} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 433, normalized size = 1.39 \[ \frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (2 \sqrt {-\frac {b}{a}}\, a b \,d^{2} x^{6}-4 \sqrt {-\frac {b}{a}}\, b^{2} c d \,x^{6}+2 \sqrt {-\frac {b}{a}}\, a^{2} d^{2} x^{4}-3 \sqrt {-\frac {b}{a}}\, a b c d \,x^{4}-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, a b c d \,x^{3} \EllipticE \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )+\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, a b c d \,x^{3} \EllipticF \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )-4 \sqrt {-\frac {b}{a}}\, b^{2} c^{2} x^{4}+4 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, b^{2} c^{2} x^{3} \EllipticE \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )-\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, b^{2} c^{2} x^{3} \EllipticF \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )+\sqrt {-\frac {b}{a}}\, a^{2} c d \,x^{2}-5 \sqrt {-\frac {b}{a}}\, a b \,c^{2} x^{2}-\sqrt {-\frac {b}{a}}\, a^{2} c^{2}\right )}{3 \left (x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c \right ) \sqrt {-\frac {b}{a}}\, c^{2} x^{3}} \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{\sqrt {d x^{2} + c} x^{4}}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^2+a\right )}^{3/2}}{x^4\,\sqrt {d\,x^2+c}} \,d x \]
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 \[ \int \frac {\left (a + b x^{2}\right )^{\frac {3}{2}}}{x^{4} \sqrt {c + d x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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