Optimal. Leaf size=96 \[ \frac {d \sqrt {c+d x^2}}{b}-\frac {c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a}+\frac {(b c-a d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{a b^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.08, antiderivative size = 96, 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, 86, 162,
65, 214} \begin {gather*} \frac {(b c-a d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{a b^{3/2}}-\frac {c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a}+\frac {d \sqrt {c+d x^2}}{b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 65
Rule 86
Rule 162
Rule 214
Rule 457
Rubi steps
\begin {align*} \int \frac {\left (c+d x^2\right )^{3/2}}{x \left (a+b x^2\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(c+d x)^{3/2}}{x (a+b x)} \, dx,x,x^2\right )\\ &=\frac {d \sqrt {c+d x^2}}{b}+\frac {\text {Subst}\left (\int \frac {b c^2+d (2 b c-a d) x}{x (a+b x) \sqrt {c+d x}} \, dx,x,x^2\right )}{2 b}\\ &=\frac {d \sqrt {c+d x^2}}{b}+\frac {c^2 \text {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^2\right )}{2 a}-\frac {(b c-a d)^2 \text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^2\right )}{2 a b}\\ &=\frac {d \sqrt {c+d x^2}}{b}+\frac {c^2 \text {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{a d}-\frac {(b c-a d)^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 b d}\\ &=\frac {d \sqrt {c+d x^2}}{b}-\frac {c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a}+\frac {(b c-a d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{a b^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.17, size = 103, normalized size = 1.07 \begin {gather*} \frac {a \sqrt {b} d \sqrt {c+d x^2}-(-b c+a d)^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )-b^{3/2} c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1292\) vs.
\(2(78)=156\).
time = 0.10, size = 1293, normalized size = 13.47
method | result | size |
default | \(-\frac {\frac {\left (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}\right )^{\frac {3}{2}}}{3}+\frac {d \sqrt {-a b}\, \left (\frac {\left (2 d \left (x -\frac {\sqrt {-a b}}{b}\right )+\frac {2 d \sqrt {-a b}}{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}}}{4 d}+\frac {\left (-\frac {4 d \left (a d -b c \right )}{b}+\frac {4 d^{2} a}{b}\right ) \ln \left (\frac {\frac {d \sqrt {-a b}}{b}+d \left (x -\frac {\sqrt {-a b}}{b}\right )}{\sqrt {d}}+\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}}\right )}{8 d^{\frac {3}{2}}}\right )}{b}-\frac {\left (a d -b c \right ) \left (\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 {\sqrt {d}\, \sqrt {-a b}\, \ln \left (\frac {\frac {d \sqrt {-a b}}{b}+d \left (x -\frac {\sqrt {-a b}}{b}\right )}{\sqrt {d}}+\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}}\right )}{b}+\frac {\left (a d -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 )}{b \sqrt {-\frac {a d -b c}{b}}}\right )}{b}}{2 a}-\frac {\frac {\left (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}\right )^{\frac {3}{2}}}{3}-\frac {d \sqrt {-a b}\, \left (\frac {\left (2 d \left (x +\frac {\sqrt {-a b}}{b}\right )-\frac {2 d \sqrt {-a b}}{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}}}{4 d}+\frac {\left (-\frac {4 d \left (a d -b c \right )}{b}+\frac {4 d^{2} a}{b}\right ) \ln \left (\frac {-\frac {d \sqrt {-a b}}{b}+d \left (x +\frac {\sqrt {-a b}}{b}\right )}{\sqrt {d}}+\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}}\right )}{8 d^{\frac {3}{2}}}\right )}{b}-\frac {\left (a d -b c \right ) \left (\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 {\sqrt {d}\, \sqrt {-a b}\, \ln \left (\frac {-\frac {d \sqrt {-a b}}{b}+d \left (x +\frac {\sqrt {-a b}}{b}\right )}{\sqrt {d}}+\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}}\right )}{b}+\frac {\left (a d -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 )}{b \sqrt {-\frac {a d -b c}{b}}}\right )}{b}}{2 a}+\frac {\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}}}{3}+c \left (\sqrt {d \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )\right )}{a}\) | \(1293\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 2.34, size = 682, normalized size = 7.10 \begin {gather*} \left [\frac {2 \, b c^{\frac {3}{2}} \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) + 4 \, \sqrt {d x^{2} + c} a d - {\left (b c - a d\right )} \sqrt {\frac {b c - a d}{b}} \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 (b^{2} d x^{2} + 2 \, b^{2} c - a b d\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b c - a d}{b}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{4 \, a b}, \frac {4 \, b \sqrt {-c} c \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) + 4 \, \sqrt {d x^{2} + c} a d - {\left (b c - a d\right )} \sqrt {\frac {b c - a d}{b}} \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 (b^{2} d x^{2} + 2 \, b^{2} c - a b d\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b c - a d}{b}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{4 \, a b}, \frac {b c^{\frac {3}{2}} \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) + 2 \, \sqrt {d x^{2} + c} a d + {\left (b c - a d\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b c - a d}{b}}}{2 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x^{2}\right )}}\right )}{2 \, a b}, \frac {2 \, b \sqrt {-c} c \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) + 2 \, \sqrt {d x^{2} + c} a d + {\left (b c - a d\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b c - a d}{b}}}{2 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x^{2}\right )}}\right )}{2 \, a b}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 11.13, size = 92, normalized size = 0.96 \begin {gather*} \frac {d \sqrt {c + d x^{2}}}{b} + \frac {c^{2} \operatorname {atan}{\left (\frac {\sqrt {c + d x^{2}}}{\sqrt {- c}} \right )}}{a \sqrt {- c}} - \frac {\left (a d - b c\right )^{2} \operatorname {atan}{\left (\frac {\sqrt {c + d x^{2}}}{\sqrt {\frac {a d - b c}{b}}} \right )}}{a b^{2} \sqrt {\frac {a d - b c}{b}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.58, size = 110, normalized size = 1.15 \begin {gather*} \frac {c^{2} \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{a \sqrt {-c}} + \frac {\sqrt {d x^{2} + c} d}{b} - \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.42, size = 711, normalized size = 7.41 \begin {gather*} \frac {d\,\sqrt {d\,x^2+c}}{b}-\frac {\mathrm {atanh}\left (\frac {2\,a^3\,d^6\,\sqrt {d\,x^2+c}\,\sqrt {c^3}}{2\,a^3\,c^2\,d^6-8\,a^2\,b\,c^3\,d^5+12\,a\,b^2\,c^4\,d^4-6\,b^3\,c^5\,d^3}+\frac {8\,a^2\,c\,d^5\,\sqrt {d\,x^2+c}\,\sqrt {c^3}}{8\,a^2\,c^3\,d^5+6\,b^2\,c^5\,d^3-\frac {2\,a^3\,c^2\,d^6}{b}-12\,a\,b\,c^4\,d^4}+\frac {6\,b^2\,c^3\,d^3\,\sqrt {d\,x^2+c}\,\sqrt {c^3}}{8\,a^2\,c^3\,d^5+6\,b^2\,c^5\,d^3-\frac {2\,a^3\,c^2\,d^6}{b}-12\,a\,b\,c^4\,d^4}-\frac {12\,a\,b\,c^2\,d^4\,\sqrt {d\,x^2+c}\,\sqrt {c^3}}{8\,a^2\,c^3\,d^5+6\,b^2\,c^5\,d^3-\frac {2\,a^3\,c^2\,d^6}{b}-12\,a\,b\,c^4\,d^4}\right )\,\sqrt {c^3}}{a}+\frac {\mathrm {atanh}\left (\frac {6\,c^3\,d^3\,\sqrt {d\,x^2+c}\,\sqrt {-a^3\,b^3\,d^3+3\,a^2\,b^4\,c\,d^2-3\,a\,b^5\,c^2\,d+b^6\,c^3}}{6\,b^3\,c^5\,d^3-10\,a^3\,c^2\,d^6-18\,a\,b^2\,c^4\,d^4+20\,a^2\,b\,c^3\,d^5+\frac {2\,a^4\,c\,d^7}{b}}-\frac {6\,a\,c^2\,d^4\,\sqrt {d\,x^2+c}\,\sqrt {-a^3\,b^3\,d^3+3\,a^2\,b^4\,c\,d^2-3\,a\,b^5\,c^2\,d+b^6\,c^3}}{2\,a^4\,c\,d^7-10\,a^3\,b\,c^2\,d^6+20\,a^2\,b^2\,c^3\,d^5-18\,a\,b^3\,c^4\,d^4+6\,b^4\,c^5\,d^3}+\frac {2\,a^2\,c\,d^5\,\sqrt {d\,x^2+c}\,\sqrt {-a^3\,b^3\,d^3+3\,a^2\,b^4\,c\,d^2-3\,a\,b^5\,c^2\,d+b^6\,c^3}}{2\,a^4\,b\,c\,d^7-10\,a^3\,b^2\,c^2\,d^6+20\,a^2\,b^3\,c^3\,d^5-18\,a\,b^4\,c^4\,d^4+6\,b^5\,c^5\,d^3}\right )\,\sqrt {-b^3\,{\left (a\,d-b\,c\right )}^3}}{a\,b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________