Optimal. Leaf size=185 \[ -\frac {b (2 b c-a d) \sqrt {c+d x^2}}{2 a^2 c (b c-a d) \left (a+b x^2\right )}-\frac {\sqrt {c+d x^2}}{2 a c x^2 \left (a+b x^2\right )}+\frac {(4 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^3 c^{3/2}}-\frac {b^{3/2} (4 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 a^3 (b c-a d)^{3/2}} \]
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Rubi [A]
time = 0.16, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {457, 105, 156,
162, 65, 214} \begin {gather*} -\frac {b^{3/2} (4 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 a^3 (b c-a d)^{3/2}}+\frac {(a d+4 b c) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^3 c^{3/2}}-\frac {b \sqrt {c+d x^2} (2 b c-a d)}{2 a^2 c \left (a+b x^2\right ) (b c-a d)}-\frac {\sqrt {c+d x^2}}{2 a c x^2 \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 105
Rule 156
Rule 162
Rule 214
Rule 457
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 (a+b x)^2 \sqrt {c+d x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {c+d x^2}}{2 a c x^2 \left (a+b x^2\right )}-\frac {\text {Subst}\left (\int \frac {\frac {1}{2} (4 b c+a d)+\frac {3 b d x}{2}}{x (a+b x)^2 \sqrt {c+d x}} \, dx,x,x^2\right )}{2 a c}\\ &=-\frac {b (2 b c-a d) \sqrt {c+d x^2}}{2 a^2 c (b c-a d) \left (a+b x^2\right )}-\frac {\sqrt {c+d x^2}}{2 a c x^2 \left (a+b x^2\right )}-\frac {\text {Subst}\left (\int \frac {\frac {1}{2} (b c-a d) (4 b c+a d)+\frac {1}{2} b d (2 b c-a d) x}{x (a+b x) \sqrt {c+d x}} \, dx,x,x^2\right )}{2 a^2 c (b c-a d)}\\ &=-\frac {b (2 b c-a d) \sqrt {c+d x^2}}{2 a^2 c (b c-a d) \left (a+b x^2\right )}-\frac {\sqrt {c+d x^2}}{2 a c x^2 \left (a+b x^2\right )}+\frac {\left (b^2 (4 b c-5 a d)\right ) \text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^2\right )}{4 a^3 (b c-a d)}-\frac {(4 b c+a d) \text {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^2\right )}{4 a^3 c}\\ &=-\frac {b (2 b c-a d) \sqrt {c+d x^2}}{2 a^2 c (b c-a d) \left (a+b x^2\right )}-\frac {\sqrt {c+d x^2}}{2 a c x^2 \left (a+b x^2\right )}+\frac {\left (b^2 (4 b c-5 a d)\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{2 a^3 d (b c-a d)}-\frac {(4 b c+a d) \text {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{2 a^3 c d}\\ &=-\frac {b (2 b c-a d) \sqrt {c+d x^2}}{2 a^2 c (b c-a d) \left (a+b x^2\right )}-\frac {\sqrt {c+d x^2}}{2 a c x^2 \left (a+b x^2\right )}+\frac {(4 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^3 c^{3/2}}-\frac {b^{3/2} (4 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 a^3 (b c-a d)^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.75, size = 163, normalized size = 0.88 \begin {gather*} \frac {\frac {a \sqrt {c+d x^2} \left (-a^2 d+2 b^2 c x^2+a b \left (c-d x^2\right )\right )}{c (-b c+a d) x^2 \left (a+b x^2\right )}-\frac {b^{3/2} (4 b c-5 a d) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{(-b c+a d)^{3/2}}+\frac {(4 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{c^{3/2}}}{2 a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(902\) vs.
\(2(157)=314\).
time = 0.14, size = 903, normalized size = 4.88
method | result | size |
risch | \(-\frac {\sqrt {d \,x^{2}+c}}{2 c \,a^{2} x^{2}}-\frac {b^{2} \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{4 a^{2} \sqrt {-a b}\, \left (a d -b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}-\frac {b d \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{4 a^{2} \left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}+\frac {\ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right ) d}{2 a^{2} c^{\frac {3}{2}}}+\frac {2 b \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )}{a^{3} \sqrt {c}}-\frac {b \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{a^{3} \sqrt {-\frac {a d -b c}{b}}}-\frac {b \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{a^{3} \sqrt {-\frac {a d -b c}{b}}}+\frac {b^{2} \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{4 a^{2} \sqrt {-a b}\, \left (a d -b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}-\frac {b d \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{4 a^{2} \left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}\) | \(899\) |
default | \(-\frac {b \left (\frac {b \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{\left (a d -b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}+\frac {d \sqrt {-a b}\, \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}\right )}{4 a^{2} \sqrt {-a b}}-\frac {b \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{a^{3} \sqrt {-\frac {a d -b c}{b}}}-\frac {b \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{a^{3} \sqrt {-\frac {a d -b c}{b}}}+\frac {-\frac {\sqrt {d \,x^{2}+c}}{2 c \,x^{2}}+\frac {d \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )}{2 c^{\frac {3}{2}}}}{a^{2}}+\frac {2 b \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )}{a^{3} \sqrt {c}}+\frac {b \left (\frac {b \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{\left (a d -b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}-\frac {d \sqrt {-a b}\, \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}\right )}{4 a^{2} \sqrt {-a b}}\) | \(903\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.23, size = 1407, normalized size = 7.61 \begin {gather*} \left [\frac {{\left ({\left (4 \, b^{3} c^{3} - 5 \, a b^{2} c^{2} d\right )} x^{4} + {\left (4 \, a b^{2} c^{3} - 5 \, a^{2} b c^{2} d\right )} x^{2}\right )} \sqrt {\frac {b}{b c - a d}} \log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} - 4 \, {\left (2 \, b^{2} c^{2} - 3 \, a b c d + a^{2} d^{2} + {\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 2 \, {\left ({\left (4 \, b^{3} c^{2} - 3 \, a b^{2} c d - a^{2} b d^{2}\right )} x^{4} + {\left (4 \, a b^{2} c^{2} - 3 \, a^{2} b c d - a^{3} d^{2}\right )} x^{2}\right )} \sqrt {c} \log \left (-\frac {d x^{2} + 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) - 4 \, {\left (a^{2} b c^{2} - a^{3} c d + {\left (2 \, a b^{2} c^{2} - a^{2} b c d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{8 \, {\left ({\left (a^{3} b^{2} c^{3} - a^{4} b c^{2} d\right )} x^{4} + {\left (a^{4} b c^{3} - a^{5} c^{2} d\right )} x^{2}\right )}}, -\frac {4 \, {\left ({\left (4 \, b^{3} c^{2} - 3 \, a b^{2} c d - a^{2} b d^{2}\right )} x^{4} + {\left (4 \, a b^{2} c^{2} - 3 \, a^{2} b c d - a^{3} d^{2}\right )} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) - {\left ({\left (4 \, b^{3} c^{3} - 5 \, a b^{2} c^{2} d\right )} x^{4} + {\left (4 \, a b^{2} c^{3} - 5 \, a^{2} b c^{2} d\right )} x^{2}\right )} \sqrt {\frac {b}{b c - a d}} \log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} - 4 \, {\left (2 \, b^{2} c^{2} - 3 \, a b c d + a^{2} d^{2} + {\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \, {\left (a^{2} b c^{2} - a^{3} c d + {\left (2 \, a b^{2} c^{2} - a^{2} b c d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{8 \, {\left ({\left (a^{3} b^{2} c^{3} - a^{4} b c^{2} d\right )} x^{4} + {\left (a^{4} b c^{3} - a^{5} c^{2} d\right )} x^{2}\right )}}, \frac {{\left ({\left (4 \, b^{3} c^{3} - 5 \, a b^{2} c^{2} d\right )} x^{4} + {\left (4 \, a b^{2} c^{3} - 5 \, a^{2} b c^{2} d\right )} x^{2}\right )} \sqrt {-\frac {b}{b c - a d}} \arctan \left (\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b}{b c - a d}}}{2 \, {\left (b d x^{2} + b c\right )}}\right ) + {\left ({\left (4 \, b^{3} c^{2} - 3 \, a b^{2} c d - a^{2} b d^{2}\right )} x^{4} + {\left (4 \, a b^{2} c^{2} - 3 \, a^{2} b c d - a^{3} d^{2}\right )} x^{2}\right )} \sqrt {c} \log \left (-\frac {d x^{2} + 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) - 2 \, {\left (a^{2} b c^{2} - a^{3} c d + {\left (2 \, a b^{2} c^{2} - a^{2} b c d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{4 \, {\left ({\left (a^{3} b^{2} c^{3} - a^{4} b c^{2} d\right )} x^{4} + {\left (a^{4} b c^{3} - a^{5} c^{2} d\right )} x^{2}\right )}}, \frac {{\left ({\left (4 \, b^{3} c^{3} - 5 \, a b^{2} c^{2} d\right )} x^{4} + {\left (4 \, a b^{2} c^{3} - 5 \, a^{2} b c^{2} d\right )} x^{2}\right )} \sqrt {-\frac {b}{b c - a d}} \arctan \left (\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b}{b c - a d}}}{2 \, {\left (b d x^{2} + b c\right )}}\right ) - 2 \, {\left ({\left (4 \, b^{3} c^{2} - 3 \, a b^{2} c d - a^{2} b d^{2}\right )} x^{4} + {\left (4 \, a b^{2} c^{2} - 3 \, a^{2} b c d - a^{3} d^{2}\right )} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) - 2 \, {\left (a^{2} b c^{2} - a^{3} c d + {\left (2 \, a b^{2} c^{2} - a^{2} b c d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{4 \, {\left ({\left (a^{3} b^{2} c^{3} - a^{4} b c^{2} d\right )} x^{4} + {\left (a^{4} b c^{3} - a^{5} c^{2} d\right )} x^{2}\right )}}\right ] \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^{3} \left (a + b x^{2}\right )^{2} \sqrt {c + d x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.55, size = 257, normalized size = 1.39 \begin {gather*} \frac {{\left (4 \, b^{3} c - 5 \, a b^{2} d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{2 \, {\left (a^{3} b c - a^{4} d\right )} \sqrt {-b^{2} c + a b d}} - \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} c d - 2 \, \sqrt {d x^{2} + c} b^{2} c^{2} d - {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b d^{2} + 2 \, \sqrt {d x^{2} + c} a b c d^{2} - \sqrt {d x^{2} + c} a^{2} d^{3}}{2 \, {\left (a^{2} b c^{2} - a^{3} c d\right )} {\left ({\left (d x^{2} + c\right )}^{2} b - 2 \, {\left (d x^{2} + c\right )} b c + b c^{2} + {\left (d x^{2} + c\right )} a d - a c d\right )}} - \frac {{\left (4 \, b c + a d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{2 \, a^{3} \sqrt {-c} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.58, size = 2500, normalized size = 13.51 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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