Optimal. Leaf size=147 \[ \frac {b (5 b c-4 a d) \tan ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{7/2} \sqrt {b c-a d}}+\frac {\sqrt {c+d x^2} (15 b c-2 a d)}{6 a^3 c x}-\frac {5 \sqrt {c+d x^2}}{6 a^2 x^3}+\frac {\sqrt {c+d x^2}}{2 a x^3 \left (a+b x^2\right )} \]
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Rubi [A] time = 0.20, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {469, 583, 12, 377, 205} \begin {gather*} \frac {\sqrt {c+d x^2} (15 b c-2 a d)}{6 a^3 c x}+\frac {b (5 b c-4 a d) \tan ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{7/2} \sqrt {b c-a d}}-\frac {5 \sqrt {c+d x^2}}{6 a^2 x^3}+\frac {\sqrt {c+d x^2}}{2 a x^3 \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 205
Rule 377
Rule 469
Rule 583
Rubi steps
\begin {align*} \int \frac {\sqrt {c+d x^2}}{x^4 \left (a+b x^2\right )^2} \, dx &=\frac {\sqrt {c+d x^2}}{2 a x^3 \left (a+b x^2\right )}-\frac {\int \frac {-5 c-4 d x^2}{x^4 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 a}\\ &=-\frac {5 \sqrt {c+d x^2}}{6 a^2 x^3}+\frac {\sqrt {c+d x^2}}{2 a x^3 \left (a+b x^2\right )}+\frac {\int \frac {-c (15 b c-2 a d)-10 b c d x^2}{x^2 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{6 a^2 c}\\ &=-\frac {5 \sqrt {c+d x^2}}{6 a^2 x^3}+\frac {(15 b c-2 a d) \sqrt {c+d x^2}}{6 a^3 c x}+\frac {\sqrt {c+d x^2}}{2 a x^3 \left (a+b x^2\right )}-\frac {\int -\frac {3 b c^2 (5 b c-4 a d)}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{6 a^3 c^2}\\ &=-\frac {5 \sqrt {c+d x^2}}{6 a^2 x^3}+\frac {(15 b c-2 a d) \sqrt {c+d x^2}}{6 a^3 c x}+\frac {\sqrt {c+d x^2}}{2 a x^3 \left (a+b x^2\right )}+\frac {(b (5 b c-4 a d)) \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 a^3}\\ &=-\frac {5 \sqrt {c+d x^2}}{6 a^2 x^3}+\frac {(15 b c-2 a d) \sqrt {c+d x^2}}{6 a^3 c x}+\frac {\sqrt {c+d x^2}}{2 a x^3 \left (a+b x^2\right )}+\frac {(b (5 b c-4 a d)) \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}\\ &=-\frac {5 \sqrt {c+d x^2}}{6 a^2 x^3}+\frac {(15 b c-2 a d) \sqrt {c+d x^2}}{6 a^3 c x}+\frac {\sqrt {c+d x^2}}{2 a x^3 \left (a+b x^2\right )}+\frac {b (5 b c-4 a d) \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{7/2} \sqrt {b c-a d}}\\ \end {align*}
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Mathematica [A] time = 5.14, size = 120, normalized size = 0.82 \begin {gather*} \frac {b (5 b c-4 a d) \tan ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{7/2} \sqrt {b c-a d}}+\frac {\sqrt {c+d x^2} \left (3 b x^2 \left (\frac {b x^2}{a+b x^2}+4\right )-\frac {2 a \left (c+d x^2\right )}{c}\right )}{6 a^3 x^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.92, size = 170, normalized size = 1.16 \begin {gather*} \frac {\sqrt {b c-a d} \left (5 b^2 c-4 a b d\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} (a d-b c)}+\frac {\sqrt {c+d x^2} \left (-2 a^2 c-2 a^2 d x^2+10 a b c x^2-2 a b d x^4+15 b^2 c x^4\right )}{6 a^3 c x^3 \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.32, size = 602, normalized size = 4.10 \begin {gather*} \left [\frac {3 \, {\left ({\left (5 \, b^{3} c^{2} - 4 \, a b^{2} c d\right )} x^{5} + {\left (5 \, a b^{2} c^{2} - 4 \, a^{2} b c d\right )} x^{3}\right )} \sqrt {-a b c + a^{2} d} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{3} - a c x\right )} \sqrt {-a b c + a^{2} d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, {\left (2 \, a^{3} b c^{2} - 2 \, a^{4} c d - {\left (15 \, a b^{3} c^{2} - 17 \, a^{2} b^{2} c d + 2 \, a^{3} b d^{2}\right )} x^{4} - 2 \, {\left (5 \, a^{2} b^{2} c^{2} - 6 \, a^{3} b c d + a^{4} d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{24 \, {\left ({\left (a^{4} b^{2} c^{2} - a^{5} b c d\right )} x^{5} + {\left (a^{5} b c^{2} - a^{6} c d\right )} x^{3}\right )}}, \frac {3 \, {\left ({\left (5 \, b^{3} c^{2} - 4 \, a b^{2} c d\right )} x^{5} + {\left (5 \, a b^{2} c^{2} - 4 \, a^{2} b c d\right )} x^{3}\right )} \sqrt {a b c - a^{2} d} \arctan \left (\frac {\sqrt {a b c - a^{2} d} {\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left (2 \, a^{3} b c^{2} - 2 \, a^{4} c d - {\left (15 \, a b^{3} c^{2} - 17 \, a^{2} b^{2} c d + 2 \, a^{3} b d^{2}\right )} x^{4} - 2 \, {\left (5 \, a^{2} b^{2} c^{2} - 6 \, a^{3} b c d + a^{4} d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{12 \, {\left ({\left (a^{4} b^{2} c^{2} - a^{5} b c d\right )} x^{5} + {\left (a^{5} b c^{2} - a^{6} c d\right )} x^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 4.52, size = 361, normalized size = 2.46 \begin {gather*} -\frac {{\left (5 \, b^{2} c \sqrt {d} - 4 \, a b 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 \, \sqrt {a b c d - a^{2} d^{2}} a^{3}} - \frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b^{2} c \sqrt {d} - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b d^{\frac {3}{2}} - b^{2} c^{2} \sqrt {d}}{{\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 )} a^{3}} - \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} + 6 \, b c^{3} \sqrt {d} - 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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 2667, normalized size = 18.14
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 {\sqrt {d x^{2} + c}}{{\left (b x^{2} + a\right )}^{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 = -0.01 \begin {gather*} \int \frac {\sqrt {d\,x^2+c}}{x^4\,{\left (b\,x^2+a\right )}^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 {\sqrt {c + d x^{2}}}{x^{4} \left (a + b x^{2}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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