Optimal. Leaf size=292 \[ -\frac {\sqrt {x} (a B-A c x)}{2 a c \left (a+c x^2\right )}-\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} c^{5/4}}+\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} c^{5/4}}-\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} a^{5/4} c^{5/4}}+\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} a^{5/4} c^{5/4}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.15, antiderivative size = 292, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {835, 841,
1182, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} c^{5/4}}+\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{5/4} c^{5/4}}-\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{8 \sqrt {2} a^{5/4} c^{5/4}}+\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{8 \sqrt {2} a^{5/4} c^{5/4}}-\frac {\sqrt {x} (a B-A c x)}{2 a c \left (a+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 210
Rule 631
Rule 642
Rule 835
Rule 841
Rule 1176
Rule 1179
Rule 1182
Rubi steps
\begin {align*} \int \frac {\sqrt {x} (A+B x)}{\left (a+c x^2\right )^2} \, dx &=-\frac {\sqrt {x} (a B-A c x)}{2 a c \left (a+c x^2\right )}+\frac {\int \frac {\frac {a B}{2}+\frac {A c x}{2}}{\sqrt {x} \left (a+c x^2\right )} \, dx}{2 a c}\\ &=-\frac {\sqrt {x} (a B-A c x)}{2 a c \left (a+c x^2\right )}+\frac {\text {Subst}\left (\int \frac {\frac {a B}{2}+\frac {1}{2} A c x^2}{a+c x^4} \, dx,x,\sqrt {x}\right )}{a c}\\ &=-\frac {\sqrt {x} (a B-A c x)}{2 a c \left (a+c x^2\right )}-\frac {\left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \text {Subst}\left (\int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx,x,\sqrt {x}\right )}{4 a c}+\frac {\left (A+\frac {\sqrt {a} B}{\sqrt {c}}\right ) \text {Subst}\left (\int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx,x,\sqrt {x}\right )}{4 a c}\\ &=-\frac {\sqrt {x} (a B-A c x)}{2 a c \left (a+c x^2\right )}+\frac {\left (A+\frac {\sqrt {a} B}{\sqrt {c}}\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {x}\right )}{8 a c}+\frac {\left (A+\frac {\sqrt {a} B}{\sqrt {c}}\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {x}\right )}{8 a c}+\frac {\left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} a^{5/4} c^{3/4}}+\frac {\left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} a^{5/4} c^{3/4}}\\ &=-\frac {\sqrt {x} (a B-A c x)}{2 a c \left (a+c x^2\right )}+\frac {\left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} a^{5/4} c^{3/4}}-\frac {\left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} a^{5/4} c^{3/4}}+\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} c^{5/4}}-\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} c^{5/4}}\\ &=-\frac {\sqrt {x} (a B-A c x)}{2 a c \left (a+c x^2\right )}-\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} c^{5/4}}+\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} c^{5/4}}+\frac {\left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} a^{5/4} c^{3/4}}-\frac {\left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} a^{5/4} c^{3/4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.73, size = 167, normalized size = 0.57 \begin {gather*} \frac {\frac {4 \sqrt [4]{a} \sqrt [4]{c} \sqrt {x} (-a B+A c x)}{a+c x^2}-\sqrt {2} \left (\sqrt {a} B+A \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}\right )+\sqrt {2} \left (\sqrt {a} B-A \sqrt {c}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}{\sqrt {a}+\sqrt {c} x}\right )}{8 a^{5/4} c^{5/4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.57, size = 247, normalized size = 0.85
method | result | size |
derivativedivides | \(\frac {\frac {A \,x^{\frac {3}{2}}}{2 a}-\frac {B \sqrt {x}}{2 c}}{c \,x^{2}+a}+\frac {\frac {B \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8}+\frac {A \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {a}{c}\right )^{\frac {1}{4}}}}{2 a c}\) | \(247\) |
default | \(\frac {\frac {A \,x^{\frac {3}{2}}}{2 a}-\frac {B \sqrt {x}}{2 c}}{c \,x^{2}+a}+\frac {\frac {B \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8}+\frac {A \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {a}{c}\right )^{\frac {1}{4}}}}{2 a c}\) | \(247\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.49, size = 272, normalized size = 0.93 \begin {gather*} \frac {A c x^{\frac {3}{2}} - B a \sqrt {x}}{2 \, {\left (a c^{2} x^{2} + a^{2} c\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (B a \sqrt {c} + A \sqrt {a} c\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} {\left (B a \sqrt {c} + A \sqrt {a} c\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {\sqrt {2} {\left (B a \sqrt {c} - A \sqrt {a} c\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (B a \sqrt {c} - A \sqrt {a} c\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}}}{16 \, a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 901 vs.
\(2 (200) = 400\).
time = 3.07, size = 901, normalized size = 3.09 \begin {gather*} -\frac {{\left (a c^{2} x^{2} + a^{2} c\right )} \sqrt {-\frac {a^{2} c^{2} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c^{5}}} + 2 \, A B}{a^{2} c^{2}}} \log \left (-{\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt {x} + {\left (A a^{4} c^{4} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c^{5}}} + B^{3} a^{3} c - A^{2} B a^{2} c^{2}\right )} \sqrt {-\frac {a^{2} c^{2} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c^{5}}} + 2 \, A B}{a^{2} c^{2}}}\right ) - {\left (a c^{2} x^{2} + a^{2} c\right )} \sqrt {-\frac {a^{2} c^{2} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c^{5}}} + 2 \, A B}{a^{2} c^{2}}} \log \left (-{\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt {x} - {\left (A a^{4} c^{4} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c^{5}}} + B^{3} a^{3} c - A^{2} B a^{2} c^{2}\right )} \sqrt {-\frac {a^{2} c^{2} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c^{5}}} + 2 \, A B}{a^{2} c^{2}}}\right ) - {\left (a c^{2} x^{2} + a^{2} c\right )} \sqrt {\frac {a^{2} c^{2} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c^{5}}} - 2 \, A B}{a^{2} c^{2}}} \log \left (-{\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt {x} + {\left (A a^{4} c^{4} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c^{5}}} - B^{3} a^{3} c + A^{2} B a^{2} c^{2}\right )} \sqrt {\frac {a^{2} c^{2} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c^{5}}} - 2 \, A B}{a^{2} c^{2}}}\right ) + {\left (a c^{2} x^{2} + a^{2} c\right )} \sqrt {\frac {a^{2} c^{2} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c^{5}}} - 2 \, A B}{a^{2} c^{2}}} \log \left (-{\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt {x} - {\left (A a^{4} c^{4} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c^{5}}} - B^{3} a^{3} c + A^{2} B a^{2} c^{2}\right )} \sqrt {\frac {a^{2} c^{2} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c^{5}}} - 2 \, A B}{a^{2} c^{2}}}\right ) - 4 \, {\left (A c x - B a\right )} \sqrt {x}}{8 \, {\left (a c^{2} x^{2} + a^{2} c\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 886 vs.
\(2 (264) = 528\).
time = 32.16, size = 886, normalized size = 3.03 \begin {gather*} \begin {cases} \tilde {\infty } \left (- \frac {2 A}{5 x^{\frac {5}{2}}} - \frac {2 B}{3 x^{\frac {3}{2}}}\right ) & \text {for}\: a = 0 \wedge c = 0 \\\frac {\frac {2 A x^{\frac {3}{2}}}{3} + \frac {2 B x^{\frac {5}{2}}}{5}}{a^{2}} & \text {for}\: c = 0 \\\frac {- \frac {2 A}{5 x^{\frac {5}{2}}} - \frac {2 B}{3 x^{\frac {3}{2}}}}{c^{2}} & \text {for}\: a = 0 \\\frac {A a \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{c}} \right )}}{8 a^{2} c \sqrt [4]{- \frac {a}{c}} + 8 a c^{2} x^{2} \sqrt [4]{- \frac {a}{c}}} - \frac {A a \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{c}} \right )}}{8 a^{2} c \sqrt [4]{- \frac {a}{c}} + 8 a c^{2} x^{2} \sqrt [4]{- \frac {a}{c}}} + \frac {2 A a \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{c}}} \right )}}{8 a^{2} c \sqrt [4]{- \frac {a}{c}} + 8 a c^{2} x^{2} \sqrt [4]{- \frac {a}{c}}} + \frac {4 A c x^{\frac {3}{2}} \sqrt [4]{- \frac {a}{c}}}{8 a^{2} c \sqrt [4]{- \frac {a}{c}} + 8 a c^{2} x^{2} \sqrt [4]{- \frac {a}{c}}} + \frac {A c x^{2} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{c}} \right )}}{8 a^{2} c \sqrt [4]{- \frac {a}{c}} + 8 a c^{2} x^{2} \sqrt [4]{- \frac {a}{c}}} - \frac {A c x^{2} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{c}} \right )}}{8 a^{2} c \sqrt [4]{- \frac {a}{c}} + 8 a c^{2} x^{2} \sqrt [4]{- \frac {a}{c}}} + \frac {2 A c x^{2} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{c}}} \right )}}{8 a^{2} c \sqrt [4]{- \frac {a}{c}} + 8 a c^{2} x^{2} \sqrt [4]{- \frac {a}{c}}} - \frac {4 B a \sqrt {x} \sqrt [4]{- \frac {a}{c}}}{8 a^{2} c \sqrt [4]{- \frac {a}{c}} + 8 a c^{2} x^{2} \sqrt [4]{- \frac {a}{c}}} - \frac {B a \sqrt {- \frac {a}{c}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{c}} \right )}}{8 a^{2} c \sqrt [4]{- \frac {a}{c}} + 8 a c^{2} x^{2} \sqrt [4]{- \frac {a}{c}}} + \frac {B a \sqrt {- \frac {a}{c}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{c}} \right )}}{8 a^{2} c \sqrt [4]{- \frac {a}{c}} + 8 a c^{2} x^{2} \sqrt [4]{- \frac {a}{c}}} + \frac {2 B a \sqrt {- \frac {a}{c}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{c}}} \right )}}{8 a^{2} c \sqrt [4]{- \frac {a}{c}} + 8 a c^{2} x^{2} \sqrt [4]{- \frac {a}{c}}} - \frac {B c x^{2} \sqrt {- \frac {a}{c}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{c}} \right )}}{8 a^{2} c \sqrt [4]{- \frac {a}{c}} + 8 a c^{2} x^{2} \sqrt [4]{- \frac {a}{c}}} + \frac {B c x^{2} \sqrt {- \frac {a}{c}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{c}} \right )}}{8 a^{2} c \sqrt [4]{- \frac {a}{c}} + 8 a c^{2} x^{2} \sqrt [4]{- \frac {a}{c}}} + \frac {2 B c x^{2} \sqrt {- \frac {a}{c}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{c}}} \right )}}{8 a^{2} c \sqrt [4]{- \frac {a}{c}} + 8 a c^{2} x^{2} \sqrt [4]{- \frac {a}{c}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.64, size = 271, normalized size = 0.93 \begin {gather*} \frac {A c x^{\frac {3}{2}} - B a \sqrt {x}}{2 \, {\left (c x^{2} + a\right )} a c} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} B a c + \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{8 \, a^{2} c^{3}} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} B a c + \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{8 \, a^{2} c^{3}} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} B a c - \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{c}}\right )}{16 \, a^{2} c^{3}} - \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} B a c - \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{c}}\right )}{16 \, a^{2} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.28, size = 652, normalized size = 2.23 \begin {gather*} 2\,\mathrm {atanh}\left (\frac {2\,A^2\,c^2\,\sqrt {x}\,\sqrt {\frac {B^2\,\sqrt {-a^5\,c^5}}{64\,a^4\,c^5}-\frac {A^2\,\sqrt {-a^5\,c^5}}{64\,a^5\,c^4}-\frac {A\,B}{32\,a^2\,c^2}}}{\frac {A\,B^2}{4}-\frac {A^3\,c}{4\,a}-\frac {B^3\,\sqrt {-a^5\,c^5}}{4\,a^2\,c^3}+\frac {A^2\,B\,\sqrt {-a^5\,c^5}}{4\,a^3\,c^2}}-\frac {2\,B^2\,c\,\sqrt {x}\,\sqrt {\frac {B^2\,\sqrt {-a^5\,c^5}}{64\,a^4\,c^5}-\frac {A^2\,\sqrt {-a^5\,c^5}}{64\,a^5\,c^4}-\frac {A\,B}{32\,a^2\,c^2}}}{\frac {A\,B^2}{4\,a}-\frac {A^3\,c}{4\,a^2}-\frac {B^3\,\sqrt {-a^5\,c^5}}{4\,a^3\,c^3}+\frac {A^2\,B\,\sqrt {-a^5\,c^5}}{4\,a^4\,c^2}}\right )\,\sqrt {-\frac {A^2\,c\,\sqrt {-a^5\,c^5}-B^2\,a\,\sqrt {-a^5\,c^5}+2\,A\,B\,a^3\,c^3}{64\,a^5\,c^5}}+2\,\mathrm {atanh}\left (\frac {2\,A^2\,c^2\,\sqrt {x}\,\sqrt {\frac {A^2\,\sqrt {-a^5\,c^5}}{64\,a^5\,c^4}-\frac {A\,B}{32\,a^2\,c^2}-\frac {B^2\,\sqrt {-a^5\,c^5}}{64\,a^4\,c^5}}}{\frac {A\,B^2}{4}-\frac {A^3\,c}{4\,a}+\frac {B^3\,\sqrt {-a^5\,c^5}}{4\,a^2\,c^3}-\frac {A^2\,B\,\sqrt {-a^5\,c^5}}{4\,a^3\,c^2}}-\frac {2\,B^2\,c\,\sqrt {x}\,\sqrt {\frac {A^2\,\sqrt {-a^5\,c^5}}{64\,a^5\,c^4}-\frac {A\,B}{32\,a^2\,c^2}-\frac {B^2\,\sqrt {-a^5\,c^5}}{64\,a^4\,c^5}}}{\frac {A\,B^2}{4\,a}-\frac {A^3\,c}{4\,a^2}+\frac {B^3\,\sqrt {-a^5\,c^5}}{4\,a^3\,c^3}-\frac {A^2\,B\,\sqrt {-a^5\,c^5}}{4\,a^4\,c^2}}\right )\,\sqrt {-\frac {B^2\,a\,\sqrt {-a^5\,c^5}-A^2\,c\,\sqrt {-a^5\,c^5}+2\,A\,B\,a^3\,c^3}{64\,a^5\,c^5}}+\frac {\frac {A\,x^{3/2}}{2\,a}-\frac {B\,\sqrt {x}}{2\,c}}{c\,x^2+a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________