Optimal. Leaf size=343 \[ \frac {c \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{2 b^{3/2}}+\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (9 \sqrt {b} d-5 \sqrt {a} f\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{12 b^{9/4} \sqrt {a+b x^4}}+\frac {3 d x \sqrt {a+b x^4}}{2 b^{3/2} \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {3 \sqrt [4]{a} d \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 b^{7/4} \sqrt {a+b x^4}}+\frac {x \left (a f-b c x-b d x^2-b e x^3\right )}{2 b^2 \sqrt {a+b x^4}}+\frac {e \sqrt {a+b x^4}}{b^2}+\frac {f x \sqrt {a+b x^4}}{3 b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.37, antiderivative size = 343, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1828, 1885, 1248, 641, 217, 206, 1888, 1198, 220, 1196} \[ \frac {x \left (a f-b c x-b d x^2-b e x^3\right )}{2 b^2 \sqrt {a+b x^4}}+\frac {c \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{2 b^{3/2}}+\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (9 \sqrt {b} d-5 \sqrt {a} f\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{12 b^{9/4} \sqrt {a+b x^4}}+\frac {3 d x \sqrt {a+b x^4}}{2 b^{3/2} \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {3 \sqrt [4]{a} d \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 b^{7/4} \sqrt {a+b x^4}}+\frac {e \sqrt {a+b x^4}}{b^2}+\frac {f x \sqrt {a+b x^4}}{3 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 217
Rule 220
Rule 641
Rule 1196
Rule 1198
Rule 1248
Rule 1828
Rule 1885
Rule 1888
Rubi steps
\begin {align*} \int \frac {x^5 \left (c+d x+e x^2+f x^3\right )}{\left (a+b x^4\right )^{3/2}} \, dx &=\frac {x \left (a f-b c x-b d x^2-b e x^3\right )}{2 b^2 \sqrt {a+b x^4}}-\frac {\int \frac {a^2 f-2 a b c x-3 a b d x^2-4 a b e x^3-2 a b f x^4}{\sqrt {a+b x^4}} \, dx}{2 a b^2}\\ &=\frac {x \left (a f-b c x-b d x^2-b e x^3\right )}{2 b^2 \sqrt {a+b x^4}}-\frac {\int \left (\frac {x \left (-2 a b c-4 a b e x^2\right )}{\sqrt {a+b x^4}}+\frac {a^2 f-3 a b d x^2-2 a b f x^4}{\sqrt {a+b x^4}}\right ) \, dx}{2 a b^2}\\ &=\frac {x \left (a f-b c x-b d x^2-b e x^3\right )}{2 b^2 \sqrt {a+b x^4}}-\frac {\int \frac {x \left (-2 a b c-4 a b e x^2\right )}{\sqrt {a+b x^4}} \, dx}{2 a b^2}-\frac {\int \frac {a^2 f-3 a b d x^2-2 a b f x^4}{\sqrt {a+b x^4}} \, dx}{2 a b^2}\\ &=\frac {x \left (a f-b c x-b d x^2-b e x^3\right )}{2 b^2 \sqrt {a+b x^4}}+\frac {f x \sqrt {a+b x^4}}{3 b^2}-\frac {\int \frac {5 a^2 b f-9 a b^2 d x^2}{\sqrt {a+b x^4}} \, dx}{6 a b^3}-\frac {\operatorname {Subst}\left (\int \frac {-2 a b c-4 a b e x}{\sqrt {a+b x^2}} \, dx,x,x^2\right )}{4 a b^2}\\ &=\frac {x \left (a f-b c x-b d x^2-b e x^3\right )}{2 b^2 \sqrt {a+b x^4}}+\frac {e \sqrt {a+b x^4}}{b^2}+\frac {f x \sqrt {a+b x^4}}{3 b^2}+\frac {c \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,x^2\right )}{2 b}-\frac {\left (3 \sqrt {a} d\right ) \int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx}{2 b^{3/2}}+\frac {\left (\sqrt {a} \left (9 \sqrt {b} d-5 \sqrt {a} f\right )\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx}{6 b^2}\\ &=\frac {x \left (a f-b c x-b d x^2-b e x^3\right )}{2 b^2 \sqrt {a+b x^4}}+\frac {e \sqrt {a+b x^4}}{b^2}+\frac {f x \sqrt {a+b x^4}}{3 b^2}+\frac {3 d x \sqrt {a+b x^4}}{2 b^{3/2} \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {3 \sqrt [4]{a} d \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 b^{7/4} \sqrt {a+b x^4}}+\frac {\sqrt [4]{a} \left (9 \sqrt {b} d-5 \sqrt {a} f\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{12 b^{9/4} \sqrt {a+b x^4}}+\frac {c \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x^2}{\sqrt {a+b x^4}}\right )}{2 b}\\ &=\frac {x \left (a f-b c x-b d x^2-b e x^3\right )}{2 b^2 \sqrt {a+b x^4}}+\frac {e \sqrt {a+b x^4}}{b^2}+\frac {f x \sqrt {a+b x^4}}{3 b^2}+\frac {3 d x \sqrt {a+b x^4}}{2 b^{3/2} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {c \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{2 b^{3/2}}-\frac {3 \sqrt [4]{a} d \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 b^{7/4} \sqrt {a+b x^4}}+\frac {\sqrt [4]{a} \left (9 \sqrt {b} d-5 \sqrt {a} f\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{12 b^{9/4} \sqrt {a+b x^4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.17, size = 176, normalized size = 0.51 \[ \frac {3 \sqrt {a} \sqrt {b} c \sqrt {\frac {b x^4}{a}+1} \sinh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )-6 b d x^3 \sqrt {\frac {b x^4}{a}+1} \, _2F_1\left (\frac {3}{4},\frac {3}{2};\frac {7}{4};-\frac {b x^4}{a}\right )-5 a f x \sqrt {\frac {b x^4}{a}+1} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\frac {b x^4}{a}\right )+6 a e+5 a f x-3 b c x^2+6 b d x^3+3 b e x^4+2 b f x^5}{6 b^2 \sqrt {a+b x^4}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (f x^{8} + e x^{7} + d x^{6} + c x^{5}\right )} \sqrt {b x^{4} + a}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (f x^{3} + e x^{2} + d x + c\right )} x^{5}}{{\left (b x^{4} + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.19, size = 358, normalized size = 1.04 \[ -\frac {d \,x^{3}}{2 \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}\, b}-\frac {c \,x^{2}}{2 \sqrt {b \,x^{4}+a}\, b}+\frac {a f x}{2 \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}\, b^{2}}-\frac {5 \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, a f \EllipticF \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{6 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, b^{2}}-\frac {3 i \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {a}\, d \EllipticE \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{2 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, b^{\frac {3}{2}}}+\frac {3 i \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {a}\, d \EllipticF \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{2 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, b^{\frac {3}{2}}}+\frac {c \ln \left (\sqrt {b}\, x^{2}+\sqrt {b \,x^{4}+a}\right )}{2 b^{\frac {3}{2}}}+\frac {\sqrt {b \,x^{4}+a}\, f x}{3 b^{2}}+\frac {\left (b \,x^{4}+2 a \right ) e}{2 \sqrt {b \,x^{4}+a}\, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{4} \, c {\left (\frac {2 \, x^{2}}{\sqrt {b x^{4} + a} b} + \frac {\log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x^{4} + a}}{x^{2}}}{\sqrt {b} + \frac {\sqrt {b x^{4} + a}}{x^{2}}}\right )}{b^{\frac {3}{2}}}\right )} + \int \frac {f x^{8} + e x^{7} + d x^{6}}{{\left (b x^{4} + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^5\,\left (f\,x^3+e\,x^2+d\,x+c\right )}{{\left (b\,x^4+a\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 43.26, size = 172, normalized size = 0.50 \[ c \left (\frac {\operatorname {asinh}{\left (\frac {\sqrt {b} x^{2}}{\sqrt {a}} \right )}}{2 b^{\frac {3}{2}}} - \frac {x^{2}}{2 \sqrt {a} b \sqrt {1 + \frac {b x^{4}}{a}}}\right ) + e \left (\begin {cases} \frac {a}{b^{2} \sqrt {a + b x^{4}}} + \frac {x^{4}}{2 b \sqrt {a + b x^{4}}} & \text {for}\: b \neq 0 \\\frac {x^{8}}{8 a^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) + \frac {d x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {11}{4}\right )} + \frac {f x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {13}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________