Optimal. Leaf size=340 \[ -\frac {3 \left (\sqrt {b} d-\sqrt {a} f\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{7/4} b^{7/4}}+\frac {3 \left (\sqrt {b} d-\sqrt {a} f\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{7/4} b^{7/4}}-\frac {3 \left (\sqrt {a} f+\sqrt {b} d\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{7/4} b^{7/4}}+\frac {3 \left (\sqrt {a} f+\sqrt {b} d\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt {2} a^{7/4} b^{7/4}}+\frac {e \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{3/2} b^{3/2}}-\frac {c+d x+e x^2+f x^3}{8 b \left (a+b x^4\right )^2}+\frac {x \left (d+2 e x+3 f x^2\right )}{32 a b \left (a+b x^4\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.33, antiderivative size = 340, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {1823, 1855, 1876, 275, 205, 1168, 1162, 617, 204, 1165, 628} \[ -\frac {3 \left (\sqrt {b} d-\sqrt {a} f\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{7/4} b^{7/4}}+\frac {3 \left (\sqrt {b} d-\sqrt {a} f\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{7/4} b^{7/4}}-\frac {3 \left (\sqrt {a} f+\sqrt {b} d\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{7/4} b^{7/4}}+\frac {3 \left (\sqrt {a} f+\sqrt {b} d\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt {2} a^{7/4} b^{7/4}}+\frac {e \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{3/2} b^{3/2}}-\frac {c+d x+e x^2+f x^3}{8 b \left (a+b x^4\right )^2}+\frac {x \left (d+2 e x+3 f x^2\right )}{32 a b \left (a+b x^4\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 205
Rule 275
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1168
Rule 1823
Rule 1855
Rule 1876
Rubi steps
\begin {align*} \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{\left (a+b x^4\right )^3} \, dx &=-\frac {c+d x+e x^2+f x^3}{8 b \left (a+b x^4\right )^2}+\frac {\int \frac {d+2 e x+3 f x^2}{\left (a+b x^4\right )^2} \, dx}{8 b}\\ &=-\frac {c+d x+e x^2+f x^3}{8 b \left (a+b x^4\right )^2}+\frac {x \left (d+2 e x+3 f x^2\right )}{32 a b \left (a+b x^4\right )}-\frac {\int \frac {-3 d-4 e x-3 f x^2}{a+b x^4} \, dx}{32 a b}\\ &=-\frac {c+d x+e x^2+f x^3}{8 b \left (a+b x^4\right )^2}+\frac {x \left (d+2 e x+3 f x^2\right )}{32 a b \left (a+b x^4\right )}-\frac {\int \left (-\frac {4 e x}{a+b x^4}+\frac {-3 d-3 f x^2}{a+b x^4}\right ) \, dx}{32 a b}\\ &=-\frac {c+d x+e x^2+f x^3}{8 b \left (a+b x^4\right )^2}+\frac {x \left (d+2 e x+3 f x^2\right )}{32 a b \left (a+b x^4\right )}-\frac {\int \frac {-3 d-3 f x^2}{a+b x^4} \, dx}{32 a b}+\frac {e \int \frac {x}{a+b x^4} \, dx}{8 a b}\\ &=-\frac {c+d x+e x^2+f x^3}{8 b \left (a+b x^4\right )^2}+\frac {x \left (d+2 e x+3 f x^2\right )}{32 a b \left (a+b x^4\right )}+\frac {e \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,x^2\right )}{16 a b}+\frac {\left (3 \left (\frac {\sqrt {b} d}{\sqrt {a}}-f\right )\right ) \int \frac {\sqrt {a} \sqrt {b}-b x^2}{a+b x^4} \, dx}{64 a b^2}+\frac {\left (3 \left (\frac {\sqrt {b} d}{\sqrt {a}}+f\right )\right ) \int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx}{64 a b^2}\\ &=-\frac {c+d x+e x^2+f x^3}{8 b \left (a+b x^4\right )^2}+\frac {x \left (d+2 e x+3 f x^2\right )}{32 a b \left (a+b x^4\right )}+\frac {e \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{3/2} b^{3/2}}+\frac {\left (3 \left (\frac {\sqrt {b} d}{\sqrt {a}}+f\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{128 a b^2}+\frac {\left (3 \left (\frac {\sqrt {b} d}{\sqrt {a}}+f\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{128 a b^2}-\frac {\left (3 \left (\sqrt {b} d-\sqrt {a} f\right )\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx}{128 \sqrt {2} a^{7/4} b^{7/4}}-\frac {\left (3 \left (\sqrt {b} d-\sqrt {a} f\right )\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx}{128 \sqrt {2} a^{7/4} b^{7/4}}\\ &=-\frac {c+d x+e x^2+f x^3}{8 b \left (a+b x^4\right )^2}+\frac {x \left (d+2 e x+3 f x^2\right )}{32 a b \left (a+b x^4\right )}+\frac {e \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{3/2} b^{3/2}}-\frac {3 \left (\sqrt {b} d-\sqrt {a} f\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{7/4} b^{7/4}}+\frac {3 \left (\sqrt {b} d-\sqrt {a} f\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{7/4} b^{7/4}}+\frac {\left (3 \left (\sqrt {b} d+\sqrt {a} f\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{7/4} b^{7/4}}-\frac {\left (3 \left (\sqrt {b} d+\sqrt {a} f\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{7/4} b^{7/4}}\\ &=-\frac {c+d x+e x^2+f x^3}{8 b \left (a+b x^4\right )^2}+\frac {x \left (d+2 e x+3 f x^2\right )}{32 a b \left (a+b x^4\right )}+\frac {e \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{3/2} b^{3/2}}-\frac {3 \left (\sqrt {b} d+\sqrt {a} f\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{7/4} b^{7/4}}+\frac {3 \left (\sqrt {b} d+\sqrt {a} f\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{7/4} b^{7/4}}-\frac {3 \left (\sqrt {b} d-\sqrt {a} f\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{7/4} b^{7/4}}+\frac {3 \left (\sqrt {b} d-\sqrt {a} f\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{7/4} b^{7/4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.39, size = 329, normalized size = 0.97 \[ \frac {-\frac {2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (8 \sqrt [4]{a} \sqrt [4]{b} e+3 \sqrt {2} \sqrt {a} f+3 \sqrt {2} \sqrt {b} d\right )}{a^{7/4}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (-8 \sqrt [4]{a} \sqrt [4]{b} e+3 \sqrt {2} \sqrt {a} f+3 \sqrt {2} \sqrt {b} d\right )}{a^{7/4}}+\frac {3 \sqrt {2} \left (\sqrt {a} f-\sqrt {b} d\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{a^{7/4}}+\frac {3 \sqrt {2} \left (\sqrt {b} d-\sqrt {a} f\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{a^{7/4}}-\frac {32 b^{3/4} (c+x (d+x (e+f x)))}{\left (a+b x^4\right )^2}+\frac {8 b^{3/4} x (d+x (2 e+3 f x))}{a \left (a+b x^4\right )}}{256 b^{7/4}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [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]
________________________________________________________________________________________
giac [A] time = 0.29, size = 338, normalized size = 0.99 \[ \frac {3 \, b f x^{7} + 2 \, b x^{6} e + b d x^{5} - a f x^{3} - 2 \, a x^{2} e - 3 \, a d x - 4 \, a c}{32 \, {\left (b x^{4} + a\right )}^{2} a b} + \frac {\sqrt {2} {\left (4 \, \sqrt {2} \sqrt {a b} b^{2} e + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} d + 3 \, \left (a b^{3}\right )^{\frac {3}{4}} f\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \, a^{2} b^{4}} + \frac {\sqrt {2} {\left (4 \, \sqrt {2} \sqrt {a b} b^{2} e + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} d + 3 \, \left (a b^{3}\right )^{\frac {3}{4}} f\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \, a^{2} b^{4}} + \frac {3 \, \sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{2} d - \left (a b^{3}\right )^{\frac {3}{4}} f\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{256 \, a^{2} b^{4}} - \frac {3 \, \sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{2} d - \left (a b^{3}\right )^{\frac {3}{4}} f\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{256 \, a^{2} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 373, normalized size = 1.10 \[ \frac {e \arctan \left (\sqrt {\frac {b}{a}}\, x^{2}\right )}{16 \sqrt {a b}\, a b}+\frac {3 \sqrt {2}\, f \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{128 \left (\frac {a}{b}\right )^{\frac {1}{4}} a \,b^{2}}+\frac {3 \sqrt {2}\, f \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{128 \left (\frac {a}{b}\right )^{\frac {1}{4}} a \,b^{2}}+\frac {3 \sqrt {2}\, f \ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}\right )}{256 \left (\frac {a}{b}\right )^{\frac {1}{4}} a \,b^{2}}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, d \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{128 a^{2} b}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, d \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{128 a^{2} b}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, d \ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}\right )}{256 a^{2} b}+\frac {\frac {3 f \,x^{7}}{32 a}+\frac {e \,x^{6}}{16 a}+\frac {d \,x^{5}}{32 a}-\frac {f \,x^{3}}{32 b}-\frac {e \,x^{2}}{16 b}-\frac {3 d x}{32 b}-\frac {c}{8 b}}{\left (b \,x^{4}+a \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 3.08, size = 343, normalized size = 1.01 \[ \frac {3 \, b f x^{7} + 2 \, b e x^{6} + b d x^{5} - a f x^{3} - 2 \, a e x^{2} - 3 \, a d x - 4 \, a c}{32 \, {\left (a b^{3} x^{8} + 2 \, a^{2} b^{2} x^{4} + a^{3} b\right )}} + \frac {\frac {3 \, \sqrt {2} {\left (\sqrt {b} d - \sqrt {a} f\right )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}} - \frac {3 \, \sqrt {2} {\left (\sqrt {b} d - \sqrt {a} f\right )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}} + \frac {2 \, {\left (3 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {3}{4}} d + 3 \, \sqrt {2} a^{\frac {3}{4}} b^{\frac {1}{4}} f - 8 \, \sqrt {a} \sqrt {b} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {3}{4}}} + \frac {2 \, {\left (3 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {3}{4}} d + 3 \, \sqrt {2} a^{\frac {3}{4}} b^{\frac {1}{4}} f + 8 \, \sqrt {a} \sqrt {b} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {3}{4}}}}{256 \, a b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.40, size = 521, normalized size = 1.53 \[ \left (\sum _{k=1}^4\ln \left (-\mathrm {root}\left (268435456\,a^7\,b^7\,z^4+589824\,a^4\,b^4\,d\,f\,z^2+524288\,a^4\,b^4\,e^2\,z^2+18432\,a^3\,b^2\,e\,f^2\,z-18432\,a^2\,b^3\,d^2\,e\,z-576\,a\,b\,d\,e^2\,f+162\,a\,b\,d^2\,f^2+256\,a\,b\,e^4+81\,a^2\,f^4+81\,b^2\,d^4,z,k\right )\,\left (\mathrm {root}\left (268435456\,a^7\,b^7\,z^4+589824\,a^4\,b^4\,d\,f\,z^2+524288\,a^4\,b^4\,e^2\,z^2+18432\,a^3\,b^2\,e\,f^2\,z-18432\,a^2\,b^3\,d^2\,e\,z-576\,a\,b\,d\,e^2\,f+162\,a\,b\,d^2\,f^2+256\,a\,b\,e^4+81\,a^2\,f^4+81\,b^2\,d^4,z,k\right )\,\left (\frac {3\,b^2\,d}{2}-2\,b^2\,e\,x\right )+\frac {3\,e\,f}{32\,a}+\frac {x\,\left (144\,a\,b^2\,d^2-144\,a^2\,b\,f^2\right )}{4096\,a^3\,b}\right )-\frac {3\,\left (9\,b\,d^2\,f-16\,b\,d\,e^2+9\,a\,f^3\right )}{32768\,a^3\,b^2}+\frac {x\,\left (8\,e^3-9\,d\,e\,f\right )}{4096\,a^3\,b}\right )\,\mathrm {root}\left (268435456\,a^7\,b^7\,z^4+589824\,a^4\,b^4\,d\,f\,z^2+524288\,a^4\,b^4\,e^2\,z^2+18432\,a^3\,b^2\,e\,f^2\,z-18432\,a^2\,b^3\,d^2\,e\,z-576\,a\,b\,d\,e^2\,f+162\,a\,b\,d^2\,f^2+256\,a\,b\,e^4+81\,a^2\,f^4+81\,b^2\,d^4,z,k\right )\right )-\frac {\frac {c}{8\,b}-\frac {d\,x^5}{32\,a}-\frac {e\,x^6}{16\,a}+\frac {e\,x^2}{16\,b}-\frac {3\,f\,x^7}{32\,a}+\frac {f\,x^3}{32\,b}+\frac {3\,d\,x}{32\,b}}{a^2+2\,a\,b\,x^4+b^2\,x^8} \]
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]
________________________________________________________________________________________