Optimal. Leaf size=277 \[ \frac {b^3 (5 b c-8 a d) \tan ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{7/2} (b c-a d)^{5/2}}-\frac {\sqrt {c+d x^2} \left (8 a^2 d^2-4 a b c d+5 b^2 c^2\right )}{6 a^2 c^2 x^3 (b c-a d)^2}+\frac {\sqrt {c+d x^2} \left (16 a^3 d^3-8 a^2 b c d^2-14 a b^2 c^2 d+15 b^3 c^3\right )}{6 a^3 c^3 x (b c-a d)^2}+\frac {b}{2 a x^3 \left (a+b x^2\right ) \sqrt {c+d x^2} (b c-a d)}+\frac {d (2 a d+b c)}{2 a c x^3 \sqrt {c+d x^2} (b c-a d)^2} \]
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Rubi [A] time = 0.40, antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {472, 579, 583, 12, 377, 205} \begin {gather*} \frac {\sqrt {c+d x^2} \left (-8 a^2 b c d^2+16 a^3 d^3-14 a b^2 c^2 d+15 b^3 c^3\right )}{6 a^3 c^3 x (b c-a d)^2}-\frac {\sqrt {c+d x^2} \left (8 a^2 d^2-4 a b c d+5 b^2 c^2\right )}{6 a^2 c^2 x^3 (b c-a d)^2}+\frac {b^3 (5 b c-8 a d) \tan ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{7/2} (b c-a d)^{5/2}}+\frac {b}{2 a x^3 \left (a+b x^2\right ) \sqrt {c+d x^2} (b c-a d)}+\frac {d (2 a d+b c)}{2 a c x^3 \sqrt {c+d x^2} (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 205
Rule 377
Rule 472
Rule 579
Rule 583
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx &=\frac {b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \sqrt {c+d x^2}}-\frac {\int \frac {-5 b c+2 a d-6 b d x^2}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx}{2 a (b c-a d)}\\ &=\frac {d (b c+2 a d)}{2 a c (b c-a d)^2 x^3 \sqrt {c+d x^2}}+\frac {b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \sqrt {c+d x^2}}-\frac {\int \frac {-5 b^2 c^2+4 a b c d-8 a^2 d^2-4 b d (b c+2 a d) x^2}{x^4 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 a c (b c-a d)^2}\\ &=\frac {d (b c+2 a d)}{2 a c (b c-a d)^2 x^3 \sqrt {c+d x^2}}+\frac {b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \sqrt {c+d x^2}}-\frac {\left (5 b^2 c^2-4 a b c d+8 a^2 d^2\right ) \sqrt {c+d x^2}}{6 a^2 c^2 (b c-a d)^2 x^3}+\frac {\int \frac {-15 b^3 c^3+14 a b^2 c^2 d+8 a^2 b c d^2-16 a^3 d^3-2 b d \left (5 b^2 c^2-4 a b c d+8 a^2 d^2\right ) x^2}{x^2 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{6 a^2 c^2 (b c-a d)^2}\\ &=\frac {d (b c+2 a d)}{2 a c (b c-a d)^2 x^3 \sqrt {c+d x^2}}+\frac {b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \sqrt {c+d x^2}}-\frac {\left (5 b^2 c^2-4 a b c d+8 a^2 d^2\right ) \sqrt {c+d x^2}}{6 a^2 c^2 (b c-a d)^2 x^3}+\frac {\left (15 b^3 c^3-14 a b^2 c^2 d-8 a^2 b c d^2+16 a^3 d^3\right ) \sqrt {c+d x^2}}{6 a^3 c^3 (b c-a d)^2 x}-\frac {\int -\frac {3 b^3 c^3 (5 b c-8 a d)}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{6 a^3 c^3 (b c-a d)^2}\\ &=\frac {d (b c+2 a d)}{2 a c (b c-a d)^2 x^3 \sqrt {c+d x^2}}+\frac {b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \sqrt {c+d x^2}}-\frac {\left (5 b^2 c^2-4 a b c d+8 a^2 d^2\right ) \sqrt {c+d x^2}}{6 a^2 c^2 (b c-a d)^2 x^3}+\frac {\left (15 b^3 c^3-14 a b^2 c^2 d-8 a^2 b c d^2+16 a^3 d^3\right ) \sqrt {c+d x^2}}{6 a^3 c^3 (b c-a d)^2 x}+\frac {\left (b^3 (5 b c-8 a d)\right ) \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 a^3 (b c-a d)^2}\\ &=\frac {d (b c+2 a d)}{2 a c (b c-a d)^2 x^3 \sqrt {c+d x^2}}+\frac {b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \sqrt {c+d x^2}}-\frac {\left (5 b^2 c^2-4 a b c d+8 a^2 d^2\right ) \sqrt {c+d x^2}}{6 a^2 c^2 (b c-a d)^2 x^3}+\frac {\left (15 b^3 c^3-14 a b^2 c^2 d-8 a^2 b c d^2+16 a^3 d^3\right ) \sqrt {c+d x^2}}{6 a^3 c^3 (b c-a d)^2 x}+\frac {\left (b^3 (5 b c-8 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{2 a^3 (b c-a d)^2}\\ &=\frac {d (b c+2 a d)}{2 a c (b c-a d)^2 x^3 \sqrt {c+d x^2}}+\frac {b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \sqrt {c+d x^2}}-\frac {\left (5 b^2 c^2-4 a b c d+8 a^2 d^2\right ) \sqrt {c+d x^2}}{6 a^2 c^2 (b c-a d)^2 x^3}+\frac {\left (15 b^3 c^3-14 a b^2 c^2 d-8 a^2 b c d^2+16 a^3 d^3\right ) \sqrt {c+d x^2}}{6 a^3 c^3 (b c-a d)^2 x}+\frac {b^3 (5 b c-8 a d) \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{7/2} (b c-a d)^{5/2}}\\ \end {align*}
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Mathematica [A] time = 5.49, size = 167, normalized size = 0.60 \begin {gather*} \frac {b^3 (5 b c-8 a d) \tan ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{7/2} (b c-a d)^{5/2}}+\sqrt {c+d x^2} \left (\frac {\frac {b^4 x}{2 \left (a+b x^2\right ) (b c-a d)^2}+\frac {2 b}{c^2 x}}{a^3}-\frac {c-5 d x^2}{3 a^2 c^3 x^3}+\frac {d^4 x}{c^3 \left (c+d x^2\right ) (b c-a d)^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.93, size = 324, normalized size = 1.17 \begin {gather*} \frac {\left (8 a b^3 d-5 b^4 c\right ) \tan ^{-1}\left (\frac {a \sqrt {d}-b x \sqrt {c+d x^2}+b \sqrt {d} x^2}{\sqrt {a} \sqrt {b c-a d}}\right )}{2 a^{7/2} (b c-a d)^{5/2}}+\frac {-2 a^4 c^2 d^2+8 a^4 c d^3 x^2+16 a^4 d^4 x^4+4 a^3 b c^3 d-6 a^3 b c^2 d^2 x^2+16 a^3 b d^4 x^6-2 a^2 b^2 c^4-12 a^2 b^2 c^3 d x^2-18 a^2 b^2 c^2 d^2 x^4-8 a^2 b^2 c d^3 x^6+10 a b^3 c^4 x^2-4 a b^3 c^3 d x^4-14 a b^3 c^2 d^2 x^6+15 b^4 c^4 x^4+15 b^4 c^3 d x^6}{6 a^3 c^3 x^3 \left (a+b x^2\right ) \sqrt {c+d x^2} (a d-b c)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 3.02, size = 1252, normalized size = 4.52
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 5.81, size = 486, normalized size = 1.75 \begin {gather*} \frac {d^{4} x}{{\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2}\right )} \sqrt {d x^{2} + c}} - \frac {{\left (5 \, b^{4} c \sqrt {d} - 8 \, a b^{3} d^{\frac {3}{2}}\right )} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{2 \, {\left (a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2}\right )} \sqrt {a b c d - a^{2} d^{2}}} - \frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b^{4} c \sqrt {d} - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b^{3} d^{\frac {3}{2}} - b^{4} c^{2} \sqrt {d}}{{\left (a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2}\right )} {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d + b c^{2}\right )}} - \frac {2 \, {\left (6 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b c \sqrt {d} + 3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a d^{\frac {3}{2}} - 12 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c^{2} \sqrt {d} - 12 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a c d^{\frac {3}{2}} + 6 \, b c^{3} \sqrt {d} + 5 \, a c^{2} d^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c\right )}^{3} a^{3} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 1608, normalized size = 5.81
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 \begin {gather*} \int \frac {1}{{\left (b x^{2} + a\right )}^{2} {\left (d x^{2} + c\right )}^{\frac {3}{2}} x^{4}}\,{d x} \end {gather*}
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 \begin {gather*} \int \frac {1}{x^4\,{\left (b\,x^2+a\right )}^2\,{\left (d\,x^2+c\right )}^{3/2}} \,d x \end {gather*}
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 \begin {gather*} \int \frac {1}{x^{4} \left (a + b x^{2}\right )^{2} \left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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