Optimal. Leaf size=146 \[ -\frac {c \sqrt {a+b x^2}}{6 a x^6}+\frac {(5 b c-6 a d) \sqrt {a+b x^2}}{24 a^2 x^4}-\frac {\left (5 b^2 c-6 a b d+8 a^2 e\right ) \sqrt {a+b x^2}}{16 a^3 x^2}+\frac {\left (5 b^3 c-6 a b^2 d+8 a^2 b e-16 a^3 f\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 a^{7/2}} \]
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Rubi [A]
time = 0.20, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1813, 1635,
911, 1171, 393, 214} \begin {gather*} \frac {\sqrt {a+b x^2} (5 b c-6 a d)}{24 a^2 x^4}-\frac {\sqrt {a+b x^2} \left (8 a^2 e-6 a b d+5 b^2 c\right )}{16 a^3 x^2}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) \left (-16 a^3 f+8 a^2 b e-6 a b^2 d+5 b^3 c\right )}{16 a^{7/2}}-\frac {c \sqrt {a+b x^2}}{6 a x^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 393
Rule 911
Rule 1171
Rule 1635
Rule 1813
Rubi steps
\begin {align*} \int \frac {c+d x^2+e x^4+f x^6}{x^7 \sqrt {a+b x^2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {c+d x+e x^2+f x^3}{x^4 \sqrt {a+b x}} \, dx,x,x^2\right )\\ &=-\frac {c \sqrt {a+b x^2}}{6 a x^6}-\frac {\text {Subst}\left (\int \frac {\frac {1}{2} (5 b c-6 a d)-3 a e x-3 a f x^2}{x^3 \sqrt {a+b x}} \, dx,x,x^2\right )}{6 a}\\ &=-\frac {c \sqrt {a+b x^2}}{6 a x^6}-\frac {\text {Subst}\left (\int \frac {\frac {\frac {1}{2} b^2 (5 b c-6 a d)+3 a^2 b e-3 a^3 f}{b^2}-\frac {\left (3 a b e-6 a^2 f\right ) x^2}{b^2}-\frac {3 a f x^4}{b^2}}{\left (-\frac {a}{b}+\frac {x^2}{b}\right )^3} \, dx,x,\sqrt {a+b x^2}\right )}{3 a b}\\ &=-\frac {c \sqrt {a+b x^2}}{6 a x^6}+\frac {(5 b c-6 a d) \sqrt {a+b x^2}}{24 a^2 x^4}-\frac {\text {Subst}\left (\int \frac {-\frac {3}{2} \left (5 b c-6 a d+\frac {8 a^2 e}{b}-\frac {8 a^3 f}{b^2}\right )-\frac {12 a^2 f x^2}{b^2}}{\left (-\frac {a}{b}+\frac {x^2}{b}\right )^2} \, dx,x,\sqrt {a+b x^2}\right )}{12 a^2}\\ &=-\frac {c \sqrt {a+b x^2}}{6 a x^6}+\frac {(5 b c-6 a d) \sqrt {a+b x^2}}{24 a^2 x^4}-\frac {\left (5 b^2 c-6 a b d+8 a^2 e\right ) \sqrt {a+b x^2}}{16 a^3 x^2}+\frac {\left (b^2 \left (\frac {12 a^3 f}{b^3}-\frac {3 \left (5 b c-6 a d+\frac {8 a^2 e}{b}-\frac {8 a^3 f}{b^2}\right )}{2 b}\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{24 a^3}\\ &=-\frac {c \sqrt {a+b x^2}}{6 a x^6}+\frac {(5 b c-6 a d) \sqrt {a+b x^2}}{24 a^2 x^4}-\frac {\left (5 b^2 c-6 a b d+8 a^2 e\right ) \sqrt {a+b x^2}}{16 a^3 x^2}+\frac {\left (5 b^3 c-6 a b^2 d+8 a^2 b e-16 a^3 f\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 a^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 126, normalized size = 0.86 \begin {gather*} \frac {\sqrt {a+b x^2} \left (-8 a^2 c+10 a b c x^2-12 a^2 d x^2-15 b^2 c x^4+18 a b d x^4-24 a^2 e x^4\right )}{48 a^3 x^6}+\frac {\left (5 b^3 c-6 a b^2 d+8 a^2 b e-16 a^3 f\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 a^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 250, normalized size = 1.71
method | result | size |
risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (24 a^{2} e \,x^{4}-18 a b d \,x^{4}+15 b^{2} c \,x^{4}+12 a^{2} d \,x^{2}-10 a b c \,x^{2}+8 a^{2} c \right )}{48 a^{3} x^{6}}-\frac {f \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{\sqrt {a}}+\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right ) b e}{2 a^{\frac {3}{2}}}-\frac {3 \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right ) b^{2} d}{8 a^{\frac {5}{2}}}+\frac {5 \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right ) b^{3} c}{16 a^{\frac {7}{2}}}\) | \(192\) |
default | \(d \left (-\frac {\sqrt {b \,x^{2}+a}}{4 a \,x^{4}}-\frac {3 b \left (-\frac {\sqrt {b \,x^{2}+a}}{2 a \,x^{2}}+\frac {b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{4 a}\right )+c \left (-\frac {\sqrt {b \,x^{2}+a}}{6 a \,x^{6}}-\frac {5 b \left (-\frac {\sqrt {b \,x^{2}+a}}{4 a \,x^{4}}-\frac {3 b \left (-\frac {\sqrt {b \,x^{2}+a}}{2 a \,x^{2}}+\frac {b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{4 a}\right )}{6 a}\right )+e \left (-\frac {\sqrt {b \,x^{2}+a}}{2 a \,x^{2}}+\frac {b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )-\frac {f \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{\sqrt {a}}\) | \(250\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 195, normalized size = 1.34 \begin {gather*} \frac {5 \, b^{3} c \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{16 \, a^{\frac {7}{2}}} - \frac {3 \, b^{2} d \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{8 \, a^{\frac {5}{2}}} - \frac {f \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{\sqrt {a}} + \frac {b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right ) e}{2 \, a^{\frac {3}{2}}} - \frac {5 \, \sqrt {b x^{2} + a} b^{2} c}{16 \, a^{3} x^{2}} + \frac {3 \, \sqrt {b x^{2} + a} b d}{8 \, a^{2} x^{2}} - \frac {\sqrt {b x^{2} + a} e}{2 \, a x^{2}} + \frac {5 \, \sqrt {b x^{2} + a} b c}{24 \, a^{2} x^{4}} - \frac {\sqrt {b x^{2} + a} d}{4 \, a x^{4}} - \frac {\sqrt {b x^{2} + a} c}{6 \, a x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.15, size = 281, normalized size = 1.92 \begin {gather*} \left [\frac {3 \, {\left (8 \, a^{2} b x^{6} e + {\left (5 \, b^{3} c - 6 \, a b^{2} d - 16 \, a^{3} f\right )} x^{6}\right )} \sqrt {a} \log \left (-\frac {b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (24 \, a^{3} x^{4} e + 3 \, {\left (5 \, a b^{2} c - 6 \, a^{2} b d\right )} x^{4} + 8 \, a^{3} c - 2 \, {\left (5 \, a^{2} b c - 6 \, a^{3} d\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{96 \, a^{4} x^{6}}, -\frac {3 \, {\left (8 \, a^{2} b x^{6} e + {\left (5 \, b^{3} c - 6 \, a b^{2} d - 16 \, a^{3} f\right )} x^{6}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (24 \, a^{3} x^{4} e + 3 \, {\left (5 \, a b^{2} c - 6 \, a^{2} b d\right )} x^{4} + 8 \, a^{3} c - 2 \, {\left (5 \, a^{2} b c - 6 \, a^{3} d\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{48 \, a^{4} x^{6}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 303 vs.
\(2 (141) = 282\).
time = 85.98, size = 303, normalized size = 2.08 \begin {gather*} - \frac {c}{6 \sqrt {b} x^{7} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {d}{4 \sqrt {b} x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {\sqrt {b} c}{24 a x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {\sqrt {b} d}{8 a x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {\sqrt {b} e \sqrt {\frac {a}{b x^{2}} + 1}}{2 a x} - \frac {5 b^{\frac {3}{2}} c}{48 a^{2} x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {3 b^{\frac {3}{2}} d}{8 a^{2} x \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {5 b^{\frac {5}{2}} c}{16 a^{3} x \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {f \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{\sqrt {a}} + \frac {b e \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{2 a^{\frac {3}{2}}} - \frac {3 b^{2} d \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{8 a^{\frac {5}{2}}} + \frac {5 b^{3} c \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{16 a^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.08, size = 232, normalized size = 1.59 \begin {gather*} -\frac {\frac {3 \, {\left (5 \, b^{4} c - 6 \, a b^{3} d - 16 \, a^{3} b f + 8 \, a^{2} b^{2} e\right )} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{3}} + \frac {15 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{4} c - 40 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a b^{4} c + 33 \, \sqrt {b x^{2} + a} a^{2} b^{4} c - 18 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a b^{3} d + 48 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} b^{3} d - 30 \, \sqrt {b x^{2} + a} a^{3} b^{3} d + 24 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2} b^{2} e - 48 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3} b^{2} e + 24 \, \sqrt {b x^{2} + a} a^{4} b^{2} e}{a^{3} b^{3} x^{6}}}{48 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.54, size = 199, normalized size = 1.36 \begin {gather*} \frac {5\,c\,{\left (b\,x^2+a\right )}^{3/2}}{6\,a^2\,x^6}-\frac {11\,c\,\sqrt {b\,x^2+a}}{16\,a\,x^6}-\frac {f\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {5\,c\,{\left (b\,x^2+a\right )}^{5/2}}{16\,a^3\,x^6}-\frac {5\,d\,\sqrt {b\,x^2+a}}{8\,a\,x^4}+\frac {3\,d\,{\left (b\,x^2+a\right )}^{3/2}}{8\,a^2\,x^4}-\frac {e\,\sqrt {b\,x^2+a}}{2\,a\,x^2}+\frac {b\,e\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{2\,a^{3/2}}-\frac {3\,b^2\,d\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{8\,a^{5/2}}-\frac {b^3\,c\,\mathrm {atan}\left (\frac {\sqrt {b\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,5{}\mathrm {i}}{16\,a^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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