Optimal. Leaf size=494 \[ -\frac {d^{5/2} \left (a^2 c^2+16 a b c+16 b^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{5 c^{7/2} (a c+b) \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}+\frac {d^3 x \left (a^2 c^2+16 a b c+16 b^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{5 c^4 (a c+b)}-\frac {d^2 \left (a^2 c^2+16 a b c+16 b^2\right ) \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{5 c^4 x (a c+b)}+\frac {a d^{5/2} (a c+8 b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{5 c^{5/2} (a c+b) \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}+\frac {d (a c+8 b) \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{5 c^3 x^3}-\frac {(a c+6 b) \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{5 c^2 x^5}+\frac {b \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{c x^5} \]
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Rubi [A] time = 1.08, antiderivative size = 648, normalized size of antiderivative = 1.31, number of steps used = 10, 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} \[ \frac {d^3 x \left (a^2 c^2+16 a b c+16 b^2\right ) \sqrt {a c+a d x^2+b} \sqrt {a+\frac {b}{c+d x^2}}}{5 c^4 (a c+b) \sqrt {a \left (c+d x^2\right )+b}}-\frac {d^2 \left (a^2 c^2+16 a b c+16 b^2\right ) \left (c+d x^2\right ) \sqrt {a c+a d x^2+b} \sqrt {a+\frac {b}{c+d x^2}}}{5 c^4 x (a c+b) \sqrt {a \left (c+d x^2\right )+b}}-\frac {d^{5/2} \left (a^2 c^2+16 a b c+16 b^2\right ) \sqrt {a c+a d x^2+b} \sqrt {a+\frac {b}{c+d x^2}} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{5 c^{7/2} (a c+b) \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}} \sqrt {a \left (c+d x^2\right )+b}}+\frac {a d^{5/2} (a c+8 b) \sqrt {a c+a d x^2+b} \sqrt {a+\frac {b}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{5 c^{5/2} (a c+b) \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}} \sqrt {a \left (c+d x^2\right )+b}}+\frac {d (a c+8 b) \left (c+d x^2\right ) \sqrt {a c+a d x^2+b} \sqrt {a+\frac {b}{c+d x^2}}}{5 c^3 x^3 \sqrt {a \left (c+d x^2\right )+b}}-\frac {(a c+6 b) \left (c+d x^2\right ) \sqrt {a c+a d x^2+b} \sqrt {a+\frac {b}{c+d x^2}}}{5 c^2 x^5 \sqrt {a \left (c+d x^2\right )+b}}+\frac {b \sqrt {a c+a d x^2+b} \sqrt {a+\frac {b}{c+d x^2}}}{c x^5 \sqrt {a \left (c+d x^2\right )+b}} \]
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 {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x^6} \, dx &=\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {\left (b+a \left (c+d x^2\right )\right )^{3/2}}{x^6 \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 {\left (b+a c+a d x^2\right )^{3/2}}{x^6 \left (c+d x^2\right )^{3/2}} \, dx}{\sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {b \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{c x^5 \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 {-(b+a c) (6 b+a c) d-a (5 b+a c) d^2 x^2}{x^6 \sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{c d \sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {b \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{c x^5 \sqrt {b+a \left (c+d x^2\right )}}-\frac {(6 b+a c) \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 c^2 x^5 \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 (b+a c)^2 (8 b+a c) d^2-3 a (b+a c) (6 b+a c) d^3 x^2}{x^4 \sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{5 c^2 (b+a c) d \sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {b \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{c x^5 \sqrt {b+a \left (c+d x^2\right )}}-\frac {(6 b+a c) \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 c^2 x^5 \sqrt {b+a \left (c+d x^2\right )}}+\frac {(8 b+a c) d \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 c^3 x^3 \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 (b+a c)^2 \left (16 b^2+16 a b c+a^2 c^2\right ) d^3-3 a (b+a c)^2 (8 b+a c) d^4 x^2}{x^2 \sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{15 c^3 (b+a c)^2 d \sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {b \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{c x^5 \sqrt {b+a \left (c+d x^2\right )}}-\frac {(6 b+a c) \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 c^2 x^5 \sqrt {b+a \left (c+d x^2\right )}}+\frac {(8 b+a c) d \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 c^3 x^3 \sqrt {b+a \left (c+d x^2\right )}}-\frac {\left (16 b^2+16 a b c+a^2 c^2\right ) d^2 \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 c^4 (b+a c) x \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 (b+a c)^3 (8 b+a c) d^4+3 a (b+a c)^2 \left (16 b^2+16 a b c+a^2 c^2\right ) d^5 x^2}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{15 c^4 (b+a c)^3 d \sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {b \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{c x^5 \sqrt {b+a \left (c+d x^2\right )}}-\frac {(6 b+a c) \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 c^2 x^5 \sqrt {b+a \left (c+d x^2\right )}}+\frac {(8 b+a c) d \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 c^3 x^3 \sqrt {b+a \left (c+d x^2\right )}}-\frac {\left (16 b^2+16 a b c+a^2 c^2\right ) d^2 \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 c^4 (b+a c) x \sqrt {b+a \left (c+d x^2\right )}}+\frac {\left (a (8 b+a c) d^3 \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 c^3 \sqrt {b+a \left (c+d x^2\right )}}+\frac {\left (a \left (16 b^2+16 a b c+a^2 c^2\right ) d^4 \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 c^4 (b+a c) \sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {b \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{c x^5 \sqrt {b+a \left (c+d x^2\right )}}+\frac {\left (16 b^2+16 a b c+a^2 c^2\right ) d^3 x \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 c^4 (b+a c) \sqrt {b+a \left (c+d x^2\right )}}-\frac {(6 b+a c) \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 c^2 x^5 \sqrt {b+a \left (c+d x^2\right )}}+\frac {(8 b+a c) d \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 c^3 x^3 \sqrt {b+a \left (c+d x^2\right )}}-\frac {\left (16 b^2+16 a b c+a^2 c^2\right ) d^2 \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 c^4 (b+a c) x \sqrt {b+a \left (c+d x^2\right )}}+\frac {a (8 b+a c) d^{5/2} \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 c^{5/2} (b+a c) \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 (\left (16 b^2+16 a b c+a^2 c^2\right ) d^3 \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 c^3 (b+a c) \sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {b \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{c x^5 \sqrt {b+a \left (c+d x^2\right )}}+\frac {\left (16 b^2+16 a b c+a^2 c^2\right ) d^3 x \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 c^4 (b+a c) \sqrt {b+a \left (c+d x^2\right )}}-\frac {(6 b+a c) \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 c^2 x^5 \sqrt {b+a \left (c+d x^2\right )}}+\frac {(8 b+a c) d \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 c^3 x^3 \sqrt {b+a \left (c+d x^2\right )}}-\frac {\left (16 b^2+16 a b c+a^2 c^2\right ) d^2 \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 c^4 (b+a c) x \sqrt {b+a \left (c+d x^2\right )}}-\frac {\left (16 b^2+16 a b c+a^2 c^2\right ) d^{5/2} \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 c^{7/2} (b+a c) \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 {a (8 b+a c) d^{5/2} \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 c^{5/2} (b+a c) \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 = 1.11, size = 430, normalized size = 0.87 \[ -\frac {\sqrt {\frac {a d}{a c+b}} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (i a c d^3 x^5 \left (a^2 c^2+16 a b c+16 b^2\right ) \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {a c+a d x^2+b}{a c+b}} E\left (i \sinh ^{-1}\left (\sqrt {\frac {a d}{b+a c}} x\right )|\frac {b}{a c}+1\right )+\sqrt {\frac {a d}{a c+b}} \left (a^3 c^2 \left (c^4+c^3 d x^2+c d^3 x^6+d^4 x^8\right )+a^2 b c \left (3 c^4+5 c^2 d^2 x^4+24 c d^3 x^6+16 d^4 x^8\right )+a b^2 \left (3 c^4-3 c^3 d x^2+13 c^2 d^2 x^4+40 c d^3 x^6+16 d^4 x^8\right )+b^3 \left (c^3-2 c^2 d x^2+8 c d^2 x^4+16 d^3 x^6\right )\right )-i a b c d^3 x^5 (7 a c+8 b) \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {a c+a d x^2+b}{a c+b}} F\left (i \sinh ^{-1}\left (\sqrt {\frac {a d}{b+a c}} x\right )|\frac {b}{a c}+1\right )\right )}{5 a c^4 d x^5 \left (a \left (c+d x^2\right )+b\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.79, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a d x^{2} + a c + b\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{d x^{8} + c x^{6}}, x\right ) \]
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 \[ \int \frac {{\left (a + \frac {b}{d x^{2} + c}\right )}^{\frac {3}{2}}}{x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 1666, normalized size = 3.37 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (a + \frac {b}{d x^{2} + c}\right )}^{\frac {3}{2}}}{x^{6}}\,{d x} \]
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 \[ \int \frac {{\left (a+\frac {b}{d\,x^2+c}\right )}^{3/2}}{x^6} \,d x \]
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 \[ \int \frac {\left (\frac {a c + a d x^{2} + b}{c + d x^{2}}\right )^{\frac {3}{2}}}{x^{6}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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