Optimal. Leaf size=242 \[ -\frac {3 \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} b^{7/4} c^{5/4}}+\frac {3 \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} b^{7/4} c^{5/4}}-\frac {3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{7/4} c^{5/4}}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt {2} b^{7/4} c^{5/4}}+\frac {\sqrt {x}}{16 b c \left (b+c x^2\right )}-\frac {\sqrt {x}}{4 c \left (b+c x^2\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.18, antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {1584, 288, 290, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac {3 \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} b^{7/4} c^{5/4}}+\frac {3 \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} b^{7/4} c^{5/4}}-\frac {3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{7/4} c^{5/4}}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt {2} b^{7/4} c^{5/4}}+\frac {\sqrt {x}}{16 b c \left (b+c x^2\right )}-\frac {\sqrt {x}}{4 c \left (b+c x^2\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 211
Rule 288
Rule 290
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1584
Rubi steps
\begin {align*} \int \frac {x^{15/2}}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac {x^{3/2}}{\left (b+c x^2\right )^3} \, dx\\ &=-\frac {\sqrt {x}}{4 c \left (b+c x^2\right )^2}+\frac {\int \frac {1}{\sqrt {x} \left (b+c x^2\right )^2} \, dx}{8 c}\\ &=-\frac {\sqrt {x}}{4 c \left (b+c x^2\right )^2}+\frac {\sqrt {x}}{16 b c \left (b+c x^2\right )}+\frac {3 \int \frac {1}{\sqrt {x} \left (b+c x^2\right )} \, dx}{32 b c}\\ &=-\frac {\sqrt {x}}{4 c \left (b+c x^2\right )^2}+\frac {\sqrt {x}}{16 b c \left (b+c x^2\right )}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{b+c x^4} \, dx,x,\sqrt {x}\right )}{16 b c}\\ &=-\frac {\sqrt {x}}{4 c \left (b+c x^2\right )^2}+\frac {\sqrt {x}}{16 b c \left (b+c x^2\right )}+\frac {3 \operatorname {Subst}\left (\int \frac {\sqrt {b}-\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{32 b^{3/2} c}+\frac {3 \operatorname {Subst}\left (\int \frac {\sqrt {b}+\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{32 b^{3/2} c}\\ &=-\frac {\sqrt {x}}{4 c \left (b+c x^2\right )^2}+\frac {\sqrt {x}}{16 b c \left (b+c x^2\right )}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {b}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {x}\right )}{64 b^{3/2} c^{3/2}}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {b}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {x}\right )}{64 b^{3/2} c^{3/2}}-\frac {3 \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{b}}{\sqrt [4]{c}}+2 x}{-\frac {\sqrt {b}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} b^{7/4} c^{5/4}}-\frac {3 \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{b}}{\sqrt [4]{c}}-2 x}{-\frac {\sqrt {b}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} b^{7/4} c^{5/4}}\\ &=-\frac {\sqrt {x}}{4 c \left (b+c x^2\right )^2}+\frac {\sqrt {x}}{16 b c \left (b+c x^2\right )}-\frac {3 \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{7/4} c^{5/4}}+\frac {3 \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{7/4} c^{5/4}}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{7/4} c^{5/4}}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{7/4} c^{5/4}}\\ &=-\frac {\sqrt {x}}{4 c \left (b+c x^2\right )^2}+\frac {\sqrt {x}}{16 b c \left (b+c x^2\right )}-\frac {3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{7/4} c^{5/4}}+\frac {3 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{7/4} c^{5/4}}-\frac {3 \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{7/4} c^{5/4}}+\frac {3 \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{7/4} c^{5/4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.11, size = 223, normalized size = 0.92 \[ \frac {-\frac {3 \sqrt {2} \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{b^{7/4}}+\frac {3 \sqrt {2} \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{b^{7/4}}-\frac {6 \sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{b^{7/4}}+\frac {6 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{b^{7/4}}+\frac {8 \sqrt [4]{c} \sqrt {x}}{b^2+b c x^2}-\frac {32 \sqrt [4]{c} \sqrt {x}}{\left (b+c x^2\right )^2}}{128 c^{5/4}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.83, size = 257, normalized size = 1.06 \[ \frac {12 \, {\left (b c^{3} x^{4} + 2 \, b^{2} c^{2} x^{2} + b^{3} c\right )} \left (-\frac {1}{b^{7} c^{5}}\right )^{\frac {1}{4}} \arctan \left (\sqrt {b^{4} c^{2} \sqrt {-\frac {1}{b^{7} c^{5}}} + x} b^{5} c^{4} \left (-\frac {1}{b^{7} c^{5}}\right )^{\frac {3}{4}} - b^{5} c^{4} \sqrt {x} \left (-\frac {1}{b^{7} c^{5}}\right )^{\frac {3}{4}}\right ) + 3 \, {\left (b c^{3} x^{4} + 2 \, b^{2} c^{2} x^{2} + b^{3} c\right )} \left (-\frac {1}{b^{7} c^{5}}\right )^{\frac {1}{4}} \log \left (b^{2} c \left (-\frac {1}{b^{7} c^{5}}\right )^{\frac {1}{4}} + \sqrt {x}\right ) - 3 \, {\left (b c^{3} x^{4} + 2 \, b^{2} c^{2} x^{2} + b^{3} c\right )} \left (-\frac {1}{b^{7} c^{5}}\right )^{\frac {1}{4}} \log \left (-b^{2} c \left (-\frac {1}{b^{7} c^{5}}\right )^{\frac {1}{4}} + \sqrt {x}\right ) + 4 \, {\left (c x^{2} - 3 \, b\right )} \sqrt {x}}{64 \, {\left (b c^{3} x^{4} + 2 \, b^{2} c^{2} x^{2} + b^{3} c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 211, normalized size = 0.87 \[ \frac {3 \, \sqrt {2} \left (b c^{3}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{64 \, b^{2} c^{2}} + \frac {3 \, \sqrt {2} \left (b c^{3}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{64 \, b^{2} c^{2}} + \frac {3 \, \sqrt {2} \left (b c^{3}\right )^{\frac {1}{4}} \log \left (\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{128 \, b^{2} c^{2}} - \frac {3 \, \sqrt {2} \left (b c^{3}\right )^{\frac {1}{4}} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{128 \, b^{2} c^{2}} + \frac {c x^{\frac {5}{2}} - 3 \, b \sqrt {x}}{16 \, {\left (c x^{2} + b\right )}^{2} b c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 169, normalized size = 0.70 \[ \frac {3 \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )}{64 b^{2} c}+\frac {3 \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )}{64 b^{2} c}+\frac {3 \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {b}{c}}}\right )}{128 b^{2} c}+\frac {\frac {x^{\frac {5}{2}}}{16 b}-\frac {3 \sqrt {x}}{16 c}}{\left (c \,x^{2}+b \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 2.96, size = 221, normalized size = 0.91 \[ \frac {c x^{\frac {5}{2}} - 3 \, b \sqrt {x}}{16 \, {\left (b c^{3} x^{4} + 2 \, b^{2} c^{2} x^{2} + b^{3} c\right )}} + \frac {3 \, {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {b} \sqrt {\sqrt {b} \sqrt {c}}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {b} \sqrt {\sqrt {b} \sqrt {c}}} + \frac {\sqrt {2} \log \left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {3}{4}} c^{\frac {1}{4}}} - \frac {\sqrt {2} \log \left (-\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {3}{4}} c^{\frac {1}{4}}}\right )}}{128 \, b c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.10, size = 85, normalized size = 0.35 \[ \frac {\frac {x^{5/2}}{16\,b}-\frac {3\,\sqrt {x}}{16\,c}}{b^2+2\,b\,c\,x^2+c^2\,x^4}+\frac {3\,\mathrm {atan}\left (\frac {c^{1/4}\,\sqrt {x}}{{\left (-b\right )}^{1/4}}\right )}{32\,{\left (-b\right )}^{7/4}\,c^{5/4}}+\frac {3\,\mathrm {atanh}\left (\frac {c^{1/4}\,\sqrt {x}}{{\left (-b\right )}^{1/4}}\right )}{32\,{\left (-b\right )}^{7/4}\,c^{5/4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________