Optimal. Leaf size=223 \[ -\frac {x^7}{8 c \left (a+c x^4\right )^2}-\frac {7 x^3}{32 c^2 \left (a+c x^4\right )}-\frac {21 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} \sqrt [4]{a} c^{11/4}}+\frac {21 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} \sqrt [4]{a} c^{11/4}}+\frac {21 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} \sqrt [4]{a} c^{11/4}}-\frac {21 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} \sqrt [4]{a} c^{11/4}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.11, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {294, 303,
1176, 631, 210, 1179, 642} \begin {gather*} -\frac {21 \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} \sqrt [4]{a} c^{11/4}}+\frac {21 \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt {2} \sqrt [4]{a} c^{11/4}}+\frac {21 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{128 \sqrt {2} \sqrt [4]{a} c^{11/4}}-\frac {21 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{128 \sqrt {2} \sqrt [4]{a} c^{11/4}}-\frac {7 x^3}{32 c^2 \left (a+c x^4\right )}-\frac {x^7}{8 c \left (a+c x^4\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 210
Rule 294
Rule 303
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {x^{10}}{\left (a+c x^4\right )^3} \, dx &=-\frac {x^7}{8 c \left (a+c x^4\right )^2}+\frac {7 \int \frac {x^6}{\left (a+c x^4\right )^2} \, dx}{8 c}\\ &=-\frac {x^7}{8 c \left (a+c x^4\right )^2}-\frac {7 x^3}{32 c^2 \left (a+c x^4\right )}+\frac {21 \int \frac {x^2}{a+c x^4} \, dx}{32 c^2}\\ &=-\frac {x^7}{8 c \left (a+c x^4\right )^2}-\frac {7 x^3}{32 c^2 \left (a+c x^4\right )}-\frac {21 \int \frac {\sqrt {a}-\sqrt {c} x^2}{a+c x^4} \, dx}{64 c^{5/2}}+\frac {21 \int \frac {\sqrt {a}+\sqrt {c} x^2}{a+c x^4} \, dx}{64 c^{5/2}}\\ &=-\frac {x^7}{8 c \left (a+c x^4\right )^2}-\frac {7 x^3}{32 c^2 \left (a+c x^4\right )}+\frac {21 \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{128 c^3}+\frac {21 \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{128 c^3}+\frac {21 \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}{128 \sqrt {2} \sqrt [4]{a} c^{11/4}}+\frac {21 \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}{128 \sqrt {2} \sqrt [4]{a} c^{11/4}}\\ &=-\frac {x^7}{8 c \left (a+c x^4\right )^2}-\frac {7 x^3}{32 c^2 \left (a+c x^4\right )}+\frac {21 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} \sqrt [4]{a} c^{11/4}}-\frac {21 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} \sqrt [4]{a} c^{11/4}}+\frac {21 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} \sqrt [4]{a} c^{11/4}}-\frac {21 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} \sqrt [4]{a} c^{11/4}}\\ &=-\frac {x^7}{8 c \left (a+c x^4\right )^2}-\frac {7 x^3}{32 c^2 \left (a+c x^4\right )}-\frac {21 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} \sqrt [4]{a} c^{11/4}}+\frac {21 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} \sqrt [4]{a} c^{11/4}}+\frac {21 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} \sqrt [4]{a} c^{11/4}}-\frac {21 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} \sqrt [4]{a} c^{11/4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.06, size = 205, normalized size = 0.92 \begin {gather*} \frac {\frac {32 a c^{3/4} x^3}{\left (a+c x^4\right )^2}-\frac {88 c^{3/4} x^3}{a+c x^4}-\frac {42 \sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt [4]{a}}+\frac {42 \sqrt {2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt [4]{a}}+\frac {21 \sqrt {2} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{\sqrt [4]{a}}-\frac {21 \sqrt {2} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{\sqrt [4]{a}}}{256 c^{11/4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.15, size = 131, normalized size = 0.59
method | result | size |
risch | \(\frac {-\frac {11 x^{7}}{32 c}-\frac {7 a \,x^{3}}{32 c^{2}}}{\left (x^{4} c +a \right )^{2}}+\frac {21 \left (\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{4}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}}\right )}{128 c^{3}}\) | \(56\) |
default | \(\frac {-\frac {11 x^{7}}{32 c}-\frac {7 a \,x^{3}}{32 c^{2}}}{\left (x^{4} c +a \right )^{2}}+\frac {21 \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{256 c^{3} \left (\frac {a}{c}\right )^{\frac {1}{4}}}\) | \(131\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.51, size = 215, normalized size = 0.96 \begin {gather*} -\frac {11 \, c x^{7} + 7 \, a x^{3}}{32 \, {\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )}} + \frac {21 \, {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} - \frac {\sqrt {2} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {1}{4}} c^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {1}{4}} c^{\frac {3}{4}}}\right )}}{256 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.39, size = 240, normalized size = 1.08 \begin {gather*} -\frac {44 \, c x^{7} + 28 \, a x^{3} + 84 \, {\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )} \left (-\frac {1}{a c^{11}}\right )^{\frac {1}{4}} \arctan \left (-c^{3} x \left (-\frac {1}{a c^{11}}\right )^{\frac {1}{4}} + \sqrt {-a c^{5} \sqrt {-\frac {1}{a c^{11}}} + x^{2}} c^{3} \left (-\frac {1}{a c^{11}}\right )^{\frac {1}{4}}\right ) - 21 \, {\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )} \left (-\frac {1}{a c^{11}}\right )^{\frac {1}{4}} \log \left (a c^{8} \left (-\frac {1}{a c^{11}}\right )^{\frac {3}{4}} + x\right ) + 21 \, {\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )} \left (-\frac {1}{a c^{11}}\right )^{\frac {1}{4}} \log \left (-a c^{8} \left (-\frac {1}{a c^{11}}\right )^{\frac {3}{4}} + x\right )}{128 \, {\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.25, size = 70, normalized size = 0.31 \begin {gather*} \frac {- 7 a x^{3} - 11 c x^{7}}{32 a^{2} c^{2} + 64 a c^{3} x^{4} + 32 c^{4} x^{8}} + \operatorname {RootSum} {\left (268435456 t^{4} a c^{11} + 194481, \left ( t \mapsto t \log {\left (\frac {2097152 t^{3} a c^{8}}{9261} + x \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.51, size = 206, normalized size = 0.92 \begin {gather*} -\frac {11 \, c x^{7} + 7 \, a x^{3}}{32 \, {\left (c x^{4} + a\right )}^{2} c^{2}} + \frac {21 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{128 \, a c^{5}} + \frac {21 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{128 \, a c^{5}} - \frac {21 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{256 \, a c^{5}} + \frac {21 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{256 \, a c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.07, size = 83, normalized size = 0.37 \begin {gather*} \frac {21\,\mathrm {atan}\left (\frac {c^{1/4}\,x}{{\left (-a\right )}^{1/4}}\right )}{64\,{\left (-a\right )}^{1/4}\,c^{11/4}}-\frac {\frac {11\,x^7}{32\,c}+\frac {7\,a\,x^3}{32\,c^2}}{a^2+2\,a\,c\,x^4+c^2\,x^8}-\frac {21\,\mathrm {atanh}\left (\frac {c^{1/4}\,x}{{\left (-a\right )}^{1/4}}\right )}{64\,{\left (-a\right )}^{1/4}\,c^{11/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________