Optimal. Leaf size=107 \[ -\frac {d}{c (b c-a d) \sqrt {c+d x^2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a c^{3/2}}+\frac {b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{a (b c-a d)^{3/2}} \]
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Rubi [A]
time = 0.08, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {457, 87, 162,
65, 214} \begin {gather*} \frac {b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{a (b c-a d)^{3/2}}-\frac {d}{c \sqrt {c+d x^2} (b c-a d)}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a c^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 87
Rule 162
Rule 214
Rule 457
Rubi steps
\begin {align*} \int \frac {1}{x \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x (a+b x) (c+d x)^{3/2}} \, dx,x,x^2\right )\\ &=-\frac {d}{c (b c-a d) \sqrt {c+d x^2}}+\frac {\text {Subst}\left (\int \frac {b c-a d-b d x}{x (a+b x) \sqrt {c+d x}} \, dx,x,x^2\right )}{2 c (b c-a d)}\\ &=-\frac {d}{c (b c-a d) \sqrt {c+d x^2}}+\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^2\right )}{2 a c}-\frac {b^2 \text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^2\right )}{2 a (b c-a d)}\\ &=-\frac {d}{c (b c-a d) \sqrt {c+d x^2}}+\frac {\text {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{a c d}-\frac {b^2 \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{a d (b c-a d)}\\ &=-\frac {d}{c (b c-a d) \sqrt {c+d x^2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a c^{3/2}}+\frac {b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{a (b c-a d)^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.25, size = 106, normalized size = 0.99 \begin {gather*} \frac {d}{c (-b c+a d) \sqrt {c+d x^2}}+\frac {b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{a (-b c+a d)^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a c^{3/2}} \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. \(772\) vs.
\(2(89)=178\).
time = 0.10, size = 773, normalized size = 7.22
method | result | size |
default | \(-\frac {-\frac {b}{\left (a d -b c \right ) \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}}}+\frac {2 d \sqrt {-a b}\, \left (2 d \left (x -\frac {\sqrt {-a b}}{b}\right )+\frac {2 d \sqrt {-a b}}{b}\right )}{\left (a d -b c \right ) \left (-\frac {4 d \left (a d -b c \right )}{b}+\frac {4 d^{2} a}{b}\right ) \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}}}+\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 )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}}{2 a}-\frac {-\frac {b}{\left (a d -b c \right ) \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}}}-\frac {2 d \sqrt {-a b}\, \left (2 d \left (x +\frac {\sqrt {-a b}}{b}\right )-\frac {2 d \sqrt {-a b}}{b}\right )}{\left (a d -b c \right ) \left (-\frac {4 d \left (a d -b c \right )}{b}+\frac {4 d^{2} a}{b}\right ) \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}}}+\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 )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}}{2 a}+\frac {\frac {1}{c \sqrt {d \,x^{2}+c}}-\frac {\ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )}{c^{\frac {3}{2}}}}{a}\) | \(773\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 192 vs.
\(2 (89) = 178\).
time = 1.64, size = 959, normalized size = 8.96 \begin {gather*} \left [-\frac {4 \, \sqrt {d x^{2} + c} a c d + {\left (b c^{2} d x^{2} + b c^{3}\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 (b c^{2} - a c d + {\left (b c d - a 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 b c^{4} - a^{2} c^{3} d + {\left (a b c^{3} d - a^{2} c^{2} d^{2}\right )} x^{2}\right )}}, -\frac {4 \, \sqrt {d x^{2} + c} a c d - 4 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) + {\left (b c^{2} d x^{2} + b c^{3}\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 b c^{4} - a^{2} c^{3} d + {\left (a b c^{3} d - a^{2} c^{2} d^{2}\right )} x^{2}\right )}}, -\frac {2 \, \sqrt {d x^{2} + c} a c d + {\left (b c^{2} d x^{2} + b c^{3}\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 (b c^{2} - a c d + {\left (b c d - a 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 b c^{4} - a^{2} c^{3} d + {\left (a b c^{3} d - a^{2} c^{2} d^{2}\right )} x^{2}\right )}}, -\frac {2 \, \sqrt {d x^{2} + c} a c d + {\left (b c^{2} d x^{2} + b c^{3}\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 (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right )}{2 \, {\left (a b c^{4} - a^{2} c^{3} d + {\left (a b c^{3} d - a^{2} c^{2} d^{2}\right )} x^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 6.95, size = 94, normalized size = 0.88 \begin {gather*} \frac {d}{c \sqrt {c + d x^{2}} \left (a d - b c\right )} + \frac {b \operatorname {atan}{\left (\frac {\sqrt {c + d x^{2}}}{\sqrt {\frac {a d - b c}{b}}} \right )}}{a \sqrt {\frac {a d - b c}{b}} \left (a d - b c\right )} + \frac {\operatorname {atan}{\left (\frac {\sqrt {c + d x^{2}}}{\sqrt {- c}} \right )}}{a c \sqrt {- c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.77, size = 110, normalized size = 1.03 \begin {gather*} -\frac {b^{2} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{{\left (a b c - a^{2} d\right )} \sqrt {-b^{2} c + a b d}} - \frac {d}{{\left (b c^{2} - a c d\right )} \sqrt {d x^{2} + c}} + \frac {\arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{a \sqrt {-c} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.82, size = 2296, normalized size = 21.46 \begin {gather*} -\frac {d}{\sqrt {d\,x^2+c}\,\left (b\,c^2-a\,c\,d\right )}-\frac {\mathrm {atanh}\left (\frac {6\,b^7\,c^7\,d^3\,\sqrt {d\,x^2+c}}{\sqrt {c^3}\,\left (-2\,a^5\,b^2\,c\,d^8+12\,a^4\,b^3\,c^2\,d^7-30\,a^3\,b^4\,c^3\,d^6+38\,a^2\,b^5\,c^4\,d^5-24\,a\,b^6\,c^5\,d^4+6\,b^7\,c^6\,d^3\right )}-\frac {24\,a\,b^6\,c^6\,d^4\,\sqrt {d\,x^2+c}}{\sqrt {c^3}\,\left (-2\,a^5\,b^2\,c\,d^8+12\,a^4\,b^3\,c^2\,d^7-30\,a^3\,b^4\,c^3\,d^6+38\,a^2\,b^5\,c^4\,d^5-24\,a\,b^6\,c^5\,d^4+6\,b^7\,c^6\,d^3\right )}+\frac {38\,a^2\,b^5\,c^5\,d^5\,\sqrt {d\,x^2+c}}{\sqrt {c^3}\,\left (-2\,a^5\,b^2\,c\,d^8+12\,a^4\,b^3\,c^2\,d^7-30\,a^3\,b^4\,c^3\,d^6+38\,a^2\,b^5\,c^4\,d^5-24\,a\,b^6\,c^5\,d^4+6\,b^7\,c^6\,d^3\right )}-\frac {30\,a^3\,b^4\,c^4\,d^6\,\sqrt {d\,x^2+c}}{\sqrt {c^3}\,\left (-2\,a^5\,b^2\,c\,d^8+12\,a^4\,b^3\,c^2\,d^7-30\,a^3\,b^4\,c^3\,d^6+38\,a^2\,b^5\,c^4\,d^5-24\,a\,b^6\,c^5\,d^4+6\,b^7\,c^6\,d^3\right )}+\frac {12\,a^4\,b^3\,c^3\,d^7\,\sqrt {d\,x^2+c}}{\sqrt {c^3}\,\left (-2\,a^5\,b^2\,c\,d^8+12\,a^4\,b^3\,c^2\,d^7-30\,a^3\,b^4\,c^3\,d^6+38\,a^2\,b^5\,c^4\,d^5-24\,a\,b^6\,c^5\,d^4+6\,b^7\,c^6\,d^3\right )}-\frac {2\,a^5\,b^2\,c^2\,d^8\,\sqrt {d\,x^2+c}}{\sqrt {c^3}\,\left (-2\,a^5\,b^2\,c\,d^8+12\,a^4\,b^3\,c^2\,d^7-30\,a^3\,b^4\,c^3\,d^6+38\,a^2\,b^5\,c^4\,d^5-24\,a\,b^6\,c^5\,d^4+6\,b^7\,c^6\,d^3\right )}\right )}{a\,\sqrt {c^3}}+\frac {\mathrm {atan}\left (\frac {\frac {\sqrt {-b^3\,{\left (a\,d-b\,c\right )}^3}\,\left (\frac {\sqrt {d\,x^2+c}\,\left (-2\,a^5\,b^3\,c^3\,d^7+10\,a^4\,b^4\,c^4\,d^6-22\,a^3\,b^5\,c^5\,d^5+26\,a^2\,b^6\,c^6\,d^4-16\,a\,b^7\,c^7\,d^3+4\,b^8\,c^8\,d^2\right )}{2}-\frac {\sqrt {-b^3\,{\left (a\,d-b\,c\right )}^3}\,\left (18\,a^3\,b^6\,c^8\,d^4-4\,a^2\,b^7\,c^9\,d^3-32\,a^4\,b^5\,c^7\,d^5+28\,a^5\,b^4\,c^6\,d^6-12\,a^6\,b^3\,c^5\,d^7+2\,a^7\,b^2\,c^4\,d^8+\frac {\sqrt {-b^3\,{\left (a\,d-b\,c\right )}^3}\,\sqrt {d\,x^2+c}\,\left (8\,a^8\,b^2\,c^5\,d^8-56\,a^7\,b^3\,c^6\,d^7+160\,a^6\,b^4\,c^7\,d^6-240\,a^5\,b^5\,c^8\,d^5+200\,a^4\,b^6\,c^9\,d^4-88\,a^3\,b^7\,c^{10}\,d^3+16\,a^2\,b^8\,c^{11}\,d^2\right )}{4\,a\,{\left (a\,d-b\,c\right )}^3}\right )}{2\,a\,{\left (a\,d-b\,c\right )}^3}\right )\,1{}\mathrm {i}}{a\,{\left (a\,d-b\,c\right )}^3}+\frac {\sqrt {-b^3\,{\left (a\,d-b\,c\right )}^3}\,\left (\frac {\sqrt {d\,x^2+c}\,\left (-2\,a^5\,b^3\,c^3\,d^7+10\,a^4\,b^4\,c^4\,d^6-22\,a^3\,b^5\,c^5\,d^5+26\,a^2\,b^6\,c^6\,d^4-16\,a\,b^7\,c^7\,d^3+4\,b^8\,c^8\,d^2\right )}{2}-\frac {\sqrt {-b^3\,{\left (a\,d-b\,c\right )}^3}\,\left (4\,a^2\,b^7\,c^9\,d^3-18\,a^3\,b^6\,c^8\,d^4+32\,a^4\,b^5\,c^7\,d^5-28\,a^5\,b^4\,c^6\,d^6+12\,a^6\,b^3\,c^5\,d^7-2\,a^7\,b^2\,c^4\,d^8+\frac {\sqrt {-b^3\,{\left (a\,d-b\,c\right )}^3}\,\sqrt {d\,x^2+c}\,\left (8\,a^8\,b^2\,c^5\,d^8-56\,a^7\,b^3\,c^6\,d^7+160\,a^6\,b^4\,c^7\,d^6-240\,a^5\,b^5\,c^8\,d^5+200\,a^4\,b^6\,c^9\,d^4-88\,a^3\,b^7\,c^{10}\,d^3+16\,a^2\,b^8\,c^{11}\,d^2\right )}{4\,a\,{\left (a\,d-b\,c\right )}^3}\right )}{2\,a\,{\left (a\,d-b\,c\right )}^3}\right )\,1{}\mathrm {i}}{a\,{\left (a\,d-b\,c\right )}^3}}{2\,b^7\,c^6\,d^3-6\,a\,b^6\,c^5\,d^4+6\,a^2\,b^5\,c^4\,d^5-2\,a^3\,b^4\,c^3\,d^6+\frac {\sqrt {-b^3\,{\left (a\,d-b\,c\right )}^3}\,\left (\frac {\sqrt {d\,x^2+c}\,\left (-2\,a^5\,b^3\,c^3\,d^7+10\,a^4\,b^4\,c^4\,d^6-22\,a^3\,b^5\,c^5\,d^5+26\,a^2\,b^6\,c^6\,d^4-16\,a\,b^7\,c^7\,d^3+4\,b^8\,c^8\,d^2\right )}{2}-\frac {\sqrt {-b^3\,{\left (a\,d-b\,c\right )}^3}\,\left (18\,a^3\,b^6\,c^8\,d^4-4\,a^2\,b^7\,c^9\,d^3-32\,a^4\,b^5\,c^7\,d^5+28\,a^5\,b^4\,c^6\,d^6-12\,a^6\,b^3\,c^5\,d^7+2\,a^7\,b^2\,c^4\,d^8+\frac {\sqrt {-b^3\,{\left (a\,d-b\,c\right )}^3}\,\sqrt {d\,x^2+c}\,\left (8\,a^8\,b^2\,c^5\,d^8-56\,a^7\,b^3\,c^6\,d^7+160\,a^6\,b^4\,c^7\,d^6-240\,a^5\,b^5\,c^8\,d^5+200\,a^4\,b^6\,c^9\,d^4-88\,a^3\,b^7\,c^{10}\,d^3+16\,a^2\,b^8\,c^{11}\,d^2\right )}{4\,a\,{\left (a\,d-b\,c\right )}^3}\right )}{2\,a\,{\left (a\,d-b\,c\right )}^3}\right )}{a\,{\left (a\,d-b\,c\right )}^3}-\frac {\sqrt {-b^3\,{\left (a\,d-b\,c\right )}^3}\,\left (\frac {\sqrt {d\,x^2+c}\,\left (-2\,a^5\,b^3\,c^3\,d^7+10\,a^4\,b^4\,c^4\,d^6-22\,a^3\,b^5\,c^5\,d^5+26\,a^2\,b^6\,c^6\,d^4-16\,a\,b^7\,c^7\,d^3+4\,b^8\,c^8\,d^2\right )}{2}-\frac {\sqrt {-b^3\,{\left (a\,d-b\,c\right )}^3}\,\left (4\,a^2\,b^7\,c^9\,d^3-18\,a^3\,b^6\,c^8\,d^4+32\,a^4\,b^5\,c^7\,d^5-28\,a^5\,b^4\,c^6\,d^6+12\,a^6\,b^3\,c^5\,d^7-2\,a^7\,b^2\,c^4\,d^8+\frac {\sqrt {-b^3\,{\left (a\,d-b\,c\right )}^3}\,\sqrt {d\,x^2+c}\,\left (8\,a^8\,b^2\,c^5\,d^8-56\,a^7\,b^3\,c^6\,d^7+160\,a^6\,b^4\,c^7\,d^6-240\,a^5\,b^5\,c^8\,d^5+200\,a^4\,b^6\,c^9\,d^4-88\,a^3\,b^7\,c^{10}\,d^3+16\,a^2\,b^8\,c^{11}\,d^2\right )}{4\,a\,{\left (a\,d-b\,c\right )}^3}\right )}{2\,a\,{\left (a\,d-b\,c\right )}^3}\right )}{a\,{\left (a\,d-b\,c\right )}^3}}\right )\,\sqrt {-b^3\,{\left (a\,d-b\,c\right )}^3}\,1{}\mathrm {i}}{a\,{\left (a\,d-b\,c\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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