Integrand size = 24, antiderivative size = 159 \[ \int \frac {\sqrt {c+d x^2}}{x^3 \left (a+b x^2\right )^2} \, dx=-\frac {b \sqrt {c+d x^2}}{a^2 \left (a+b x^2\right )}-\frac {\sqrt {c+d x^2}}{2 a x^2 \left (a+b x^2\right )}+\frac {(4 b c-a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^3 \sqrt {c}}-\frac {\sqrt {b} (4 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 a^3 \sqrt {b c-a d}} \]
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Time = 0.14 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {457, 101, 156, 162, 65, 214} \[ \int \frac {\sqrt {c+d x^2}}{x^3 \left (a+b x^2\right )^2} \, dx=-\frac {\sqrt {b} (4 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 a^3 \sqrt {b c-a d}}+\frac {(4 b c-a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^3 \sqrt {c}}-\frac {b \sqrt {c+d x^2}}{a^2 \left (a+b x^2\right )}-\frac {\sqrt {c+d x^2}}{2 a x^2 \left (a+b x^2\right )} \]
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Rule 65
Rule 101
Rule 156
Rule 162
Rule 214
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {c+d x}}{x^2 (a+b x)^2} \, dx,x,x^2\right ) \\ & = -\frac {\sqrt {c+d x^2}}{2 a 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} \\ & = -\frac {b \sqrt {c+d x^2}}{a^2 \left (a+b x^2\right )}-\frac {\sqrt {c+d x^2}}{2 a 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)-b d (b c-a d) x}{x (a+b x) \sqrt {c+d x}} \, dx,x,x^2\right )}{2 a^2 (b c-a d)} \\ & = -\frac {b \sqrt {c+d x^2}}{a^2 \left (a+b x^2\right )}-\frac {\sqrt {c+d x^2}}{2 a x^2 \left (a+b x^2\right )}+\frac {(b (4 b c-3 a d)) \text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^2\right )}{4 a^3}-\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} \\ & = -\frac {b \sqrt {c+d x^2}}{a^2 \left (a+b x^2\right )}-\frac {\sqrt {c+d x^2}}{2 a x^2 \left (a+b x^2\right )}+\frac {(b (4 b c-3 a d)) \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}-\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 d} \\ & = -\frac {b \sqrt {c+d x^2}}{a^2 \left (a+b x^2\right )}-\frac {\sqrt {c+d x^2}}{2 a 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 \sqrt {c}}-\frac {\sqrt {b} (4 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 a^3 \sqrt {b c-a d}} \\ \end{align*}
Time = 0.66 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt {c+d x^2}}{x^3 \left (a+b x^2\right )^2} \, dx=\frac {-\frac {a \left (a+2 b x^2\right ) \sqrt {c+d x^2}}{x^2 \left (a+b x^2\right )}+\frac {\sqrt {b} (4 b c-3 a d) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{\sqrt {-b c+a d}}+\frac {(4 b c-a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{\sqrt {c}}}{2 a^3} \]
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Time = 3.10 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.96
method | result | size |
pseudoelliptic | \(-\frac {-4 \left (b \,x^{2}+a \right ) x^{2} b \left (c^{\frac {3}{2}} b -\frac {3 a d \sqrt {c}}{4}\right ) \arctan \left (\frac {b \sqrt {d \,x^{2}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )+\sqrt {\left (a d -b c \right ) b}\, \left (x^{2} \left (b \,x^{2}+a \right ) \left (a d -4 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{\sqrt {c}}\right )+a \sqrt {c}\, \sqrt {d \,x^{2}+c}\, \left (2 b \,x^{2}+a \right )\right )}{2 \sqrt {\left (a d -b c \right ) b}\, \sqrt {c}\, a^{3} \left (b \,x^{2}+a \right ) x^{2}}\) | \(153\) |
risch | \(-\frac {\sqrt {d \,x^{2}+c}}{2 a^{2} x^{2}}-\frac {-\frac {\left (-a d +4 b c \right ) \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )}{a \sqrt {c}}-\frac {\left (a d -2 b c \right ) \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 \sqrt {-\frac {a d -b c}{b}}}-\frac {\left (a d -2 b c \right ) \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 \sqrt {-\frac {a d -b c}{b}}}-\frac {\left (a d -b c \right ) \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 )}{2 \sqrt {-a b}}+\frac {\left (a d -b c \right ) \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 )}{2 \sqrt {-a b}}}{2 a^{2}}\) | \(905\) |
default | \(\text {Expression too large to display}\) | \(2071\) |
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Time = 0.40 (sec) , antiderivative size = 1043, normalized size of antiderivative = 6.56 \[ \int \frac {\sqrt {c+d x^2}}{x^3 \left (a+b x^2\right )^2} \, dx=\left [-\frac {{\left ({\left (4 \, b^{2} c^{2} - 3 \, a b c d\right )} x^{4} + {\left (4 \, a b c^{2} - 3 \, a^{2} c 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^{2} c - a b d\right )} x^{4} + {\left (4 \, a b c - a^{2} d\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 (2 \, a b c x^{2} + a^{2} c\right )} \sqrt {d x^{2} + c}}{8 \, {\left (a^{3} b c x^{4} + a^{4} c x^{2}\right )}}, -\frac {4 \, {\left ({\left (4 \, b^{2} c - a b d\right )} x^{4} + {\left (4 \, a b c - a^{2} d\right )} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) + {\left ({\left (4 \, b^{2} c^{2} - 3 \, a b c d\right )} x^{4} + {\left (4 \, a b c^{2} - 3 \, a^{2} c 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 (2 \, a b c x^{2} + a^{2} c\right )} \sqrt {d x^{2} + c}}{8 \, {\left (a^{3} b c x^{4} + a^{4} c x^{2}\right )}}, \frac {{\left ({\left (4 \, b^{2} c^{2} - 3 \, a b c d\right )} x^{4} + {\left (4 \, a b c^{2} - 3 \, a^{2} c 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^{2} c - a b d\right )} x^{4} + {\left (4 \, a b c - a^{2} d\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 (2 \, a b c x^{2} + a^{2} c\right )} \sqrt {d x^{2} + c}}{4 \, {\left (a^{3} b c x^{4} + a^{4} c x^{2}\right )}}, \frac {{\left ({\left (4 \, b^{2} c^{2} - 3 \, a b c d\right )} x^{4} + {\left (4 \, a b c^{2} - 3 \, a^{2} c 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^{2} c - a b d\right )} x^{4} + {\left (4 \, a b c - a^{2} d\right )} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) - 2 \, {\left (2 \, a b c x^{2} + a^{2} c\right )} \sqrt {d x^{2} + c}}{4 \, {\left (a^{3} b c x^{4} + a^{4} c x^{2}\right )}}\right ] \]
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\[ \int \frac {\sqrt {c+d x^2}}{x^3 \left (a+b x^2\right )^2} \, dx=\int \frac {\sqrt {c + d x^{2}}}{x^{3} \left (a + b x^{2}\right )^{2}}\, dx \]
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\[ \int \frac {\sqrt {c+d x^2}}{x^3 \left (a+b x^2\right )^2} \, dx=\int { \frac {\sqrt {d x^{2} + c}}{{\left (b x^{2} + a\right )}^{2} x^{3}} \,d x } \]
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Time = 0.32 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.15 \[ \int \frac {\sqrt {c+d x^2}}{x^3 \left (a+b x^2\right )^2} \, dx=\frac {{\left (4 \, b^{2} c - 3 \, a b d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{2 \, \sqrt {-b^{2} c + a b d} a^{3}} - \frac {{\left (4 \, b c - a d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{2 \, a^{3} \sqrt {-c}} - \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b d - 2 \, \sqrt {d x^{2} + c} b c d + \sqrt {d x^{2} + c} a d^{2}}{2 \, {\left ({\left (d x^{2} + c\right )}^{2} b - 2 \, {\left (d x^{2} + c\right )} b c + b c^{2} + {\left (d x^{2} + c\right )} a d - a c d\right )} a^{2}} \]
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Time = 6.20 (sec) , antiderivative size = 1193, normalized size of antiderivative = 7.50 \[ \int \frac {\sqrt {c+d x^2}}{x^3 \left (a+b x^2\right )^2} \, dx=\frac {\mathrm {atanh}\left (\frac {b^2\,d^6\,\sqrt {d\,x^2+c}}{4\,c^{3/2}\,\left (\frac {b^3\,d^5}{a}-\frac {b^2\,d^6}{4\,c}\right )}-\frac {b^3\,d^5\,\sqrt {d\,x^2+c}}{\sqrt {c}\,\left (b^3\,d^5-\frac {a\,b^2\,d^6}{4\,c}\right )}\right )\,\left (a\,d-4\,b\,c\right )}{2\,a^3\,\sqrt {c}}-\frac {\frac {b\,d\,{\left (d\,x^2+c\right )}^{3/2}}{a^2}+\frac {d\,\sqrt {d\,x^2+c}\,\left (a\,d-2\,b\,c\right )}{2\,a^2}}{\left (d\,x^2+c\right )\,\left (a\,d-2\,b\,c\right )+b\,{\left (d\,x^2+c\right )}^2+b\,c^2-a\,c\,d}+\frac {\mathrm {atan}\left (\frac {\frac {\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (\frac {\sqrt {d\,x^2+c}\,\left (5\,a^2\,b^3\,d^4-16\,a\,b^4\,c\,d^3+16\,b^5\,c^2\,d^2\right )}{a^4}-\frac {\left (\frac {2\,a^7\,b^2\,d^4-4\,a^6\,b^3\,c\,d^3}{a^6}-\frac {\left (8\,a^7\,b^2\,d^3-16\,a^6\,b^3\,c\,d^2\right )\,\sqrt {d\,x^2+c}\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (3\,a\,d-4\,b\,c\right )}{4\,a^4\,\left (a^4\,d-a^3\,b\,c\right )}\right )\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (3\,a\,d-4\,b\,c\right )}{4\,\left (a^4\,d-a^3\,b\,c\right )}\right )\,\left (3\,a\,d-4\,b\,c\right )\,1{}\mathrm {i}}{4\,\left (a^4\,d-a^3\,b\,c\right )}+\frac {\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (\frac {\sqrt {d\,x^2+c}\,\left (5\,a^2\,b^3\,d^4-16\,a\,b^4\,c\,d^3+16\,b^5\,c^2\,d^2\right )}{a^4}+\frac {\left (\frac {2\,a^7\,b^2\,d^4-4\,a^6\,b^3\,c\,d^3}{a^6}+\frac {\left (8\,a^7\,b^2\,d^3-16\,a^6\,b^3\,c\,d^2\right )\,\sqrt {d\,x^2+c}\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (3\,a\,d-4\,b\,c\right )}{4\,a^4\,\left (a^4\,d-a^3\,b\,c\right )}\right )\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (3\,a\,d-4\,b\,c\right )}{4\,\left (a^4\,d-a^3\,b\,c\right )}\right )\,\left (3\,a\,d-4\,b\,c\right )\,1{}\mathrm {i}}{4\,\left (a^4\,d-a^3\,b\,c\right )}}{\frac {\frac {3\,a^2\,b^3\,d^5}{2}-8\,a\,b^4\,c\,d^4+8\,b^5\,c^2\,d^3}{a^6}-\frac {\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (\frac {\sqrt {d\,x^2+c}\,\left (5\,a^2\,b^3\,d^4-16\,a\,b^4\,c\,d^3+16\,b^5\,c^2\,d^2\right )}{a^4}-\frac {\left (\frac {2\,a^7\,b^2\,d^4-4\,a^6\,b^3\,c\,d^3}{a^6}-\frac {\left (8\,a^7\,b^2\,d^3-16\,a^6\,b^3\,c\,d^2\right )\,\sqrt {d\,x^2+c}\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (3\,a\,d-4\,b\,c\right )}{4\,a^4\,\left (a^4\,d-a^3\,b\,c\right )}\right )\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (3\,a\,d-4\,b\,c\right )}{4\,\left (a^4\,d-a^3\,b\,c\right )}\right )\,\left (3\,a\,d-4\,b\,c\right )}{4\,\left (a^4\,d-a^3\,b\,c\right )}+\frac {\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (\frac {\sqrt {d\,x^2+c}\,\left (5\,a^2\,b^3\,d^4-16\,a\,b^4\,c\,d^3+16\,b^5\,c^2\,d^2\right )}{a^4}+\frac {\left (\frac {2\,a^7\,b^2\,d^4-4\,a^6\,b^3\,c\,d^3}{a^6}+\frac {\left (8\,a^7\,b^2\,d^3-16\,a^6\,b^3\,c\,d^2\right )\,\sqrt {d\,x^2+c}\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (3\,a\,d-4\,b\,c\right )}{4\,a^4\,\left (a^4\,d-a^3\,b\,c\right )}\right )\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (3\,a\,d-4\,b\,c\right )}{4\,\left (a^4\,d-a^3\,b\,c\right )}\right )\,\left (3\,a\,d-4\,b\,c\right )}{4\,\left (a^4\,d-a^3\,b\,c\right )}}\right )\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (3\,a\,d-4\,b\,c\right )\,1{}\mathrm {i}}{2\,\left (a^4\,d-a^3\,b\,c\right )} \]
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