Integrand size = 30, antiderivative size = 394 \[ \int x^3 \left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4} \, dx=\frac {2 a d x \sqrt {a+b x^4}}{21 b}-\frac {a e x^2 \sqrt {a+b x^4}}{16 b}+\frac {2 a f x^3 \sqrt {a+b x^4}}{45 b}-\frac {2 a^2 f x \sqrt {a+b x^4}}{15 b^{3/2} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {1}{63} x^5 \left (9 d+7 f x^2\right ) \sqrt {a+b x^4}+\frac {\left (4 c+3 e x^2\right ) \left (a+b x^4\right )^{3/2}}{24 b}-\frac {a^2 e \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{16 b^{3/2}}+\frac {2 a^{9/4} f \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 b^{7/4} \sqrt {a+b x^4}}-\frac {a^{7/4} \left (5 \sqrt {b} d+7 \sqrt {a} f\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{105 b^{7/4} \sqrt {a+b x^4}} \]
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Time = 0.22 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {1847, 1266, 794, 201, 223, 212, 1288, 1294, 1212, 226, 1210} \[ \int x^3 \left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4} \, dx=-\frac {a^{7/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (7 \sqrt {a} f+5 \sqrt {b} d\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{105 b^{7/4} \sqrt {a+b x^4}}+\frac {2 a^{9/4} f \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 b^{7/4} \sqrt {a+b x^4}}-\frac {a^2 e \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{16 b^{3/2}}-\frac {2 a^2 f x \sqrt {a+b x^4}}{15 b^{3/2} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {\left (a+b x^4\right )^{3/2} \left (4 c+3 e x^2\right )}{24 b}+\frac {1}{63} x^5 \sqrt {a+b x^4} \left (9 d+7 f x^2\right )+\frac {2 a d x \sqrt {a+b x^4}}{21 b}-\frac {a e x^2 \sqrt {a+b x^4}}{16 b}+\frac {2 a f x^3 \sqrt {a+b x^4}}{45 b} \]
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Rule 201
Rule 212
Rule 223
Rule 226
Rule 794
Rule 1210
Rule 1212
Rule 1266
Rule 1288
Rule 1294
Rule 1847
Rubi steps \begin{align*} \text {integral}& = \int \left (x^3 \left (c+e x^2\right ) \sqrt {a+b x^4}+x^4 \left (d+f x^2\right ) \sqrt {a+b x^4}\right ) \, dx \\ & = \int x^3 \left (c+e x^2\right ) \sqrt {a+b x^4} \, dx+\int x^4 \left (d+f x^2\right ) \sqrt {a+b x^4} \, dx \\ & = \frac {1}{63} x^5 \left (9 d+7 f x^2\right ) \sqrt {a+b x^4}+\frac {1}{2} \text {Subst}\left (\int x (c+e x) \sqrt {a+b x^2} \, dx,x,x^2\right )+\frac {1}{63} (2 a) \int \frac {x^4 \left (9 d+7 f x^2\right )}{\sqrt {a+b x^4}} \, dx \\ & = \frac {2 a f x^3 \sqrt {a+b x^4}}{45 b}+\frac {1}{63} x^5 \left (9 d+7 f x^2\right ) \sqrt {a+b x^4}+\frac {\left (4 c+3 e x^2\right ) \left (a+b x^4\right )^{3/2}}{24 b}-\frac {(2 a) \int \frac {x^2 \left (21 a f-45 b d x^2\right )}{\sqrt {a+b x^4}} \, dx}{315 b}-\frac {(a e) \text {Subst}\left (\int \sqrt {a+b x^2} \, dx,x,x^2\right )}{8 b} \\ & = \frac {2 a d x \sqrt {a+b x^4}}{21 b}-\frac {a e x^2 \sqrt {a+b x^4}}{16 b}+\frac {2 a f x^3 \sqrt {a+b x^4}}{45 b}+\frac {1}{63} x^5 \left (9 d+7 f x^2\right ) \sqrt {a+b x^4}+\frac {\left (4 c+3 e x^2\right ) \left (a+b x^4\right )^{3/2}}{24 b}+\frac {(2 a) \int \frac {-45 a b d-63 a b f x^2}{\sqrt {a+b x^4}} \, dx}{945 b^2}-\frac {\left (a^2 e\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,x^2\right )}{16 b} \\ & = \frac {2 a d x \sqrt {a+b x^4}}{21 b}-\frac {a e x^2 \sqrt {a+b x^4}}{16 b}+\frac {2 a f x^3 \sqrt {a+b x^4}}{45 b}+\frac {1}{63} x^5 \left (9 d+7 f x^2\right ) \sqrt {a+b x^4}+\frac {\left (4 c+3 e x^2\right ) \left (a+b x^4\right )^{3/2}}{24 b}-\frac {\left (a^2 e\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x^2}{\sqrt {a+b x^4}}\right )}{16 b}+\frac {\left (2 a^{5/2} f\right ) \int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx}{15 b^{3/2}}-\frac {\left (2 a^2 \left (5 \sqrt {b} d+7 \sqrt {a} f\right )\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx}{105 b^{3/2}} \\ & = \frac {2 a d x \sqrt {a+b x^4}}{21 b}-\frac {a e x^2 \sqrt {a+b x^4}}{16 b}+\frac {2 a f x^3 \sqrt {a+b x^4}}{45 b}-\frac {2 a^2 f x \sqrt {a+b x^4}}{15 b^{3/2} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {1}{63} x^5 \left (9 d+7 f x^2\right ) \sqrt {a+b x^4}+\frac {\left (4 c+3 e x^2\right ) \left (a+b x^4\right )^{3/2}}{24 b}-\frac {a^2 e \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{16 b^{3/2}}+\frac {2 a^{9/4} f \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 )}{15 b^{7/4} \sqrt {a+b x^4}}-\frac {a^{7/4} \left (5 \sqrt {b} d+7 \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 )}{105 b^{7/4} \sqrt {a+b x^4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.59 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.55 \[ \int x^3 \left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4} \, dx=\frac {\sqrt {a+b x^4} \left (168 \sqrt {b} c \left (a+b x^4\right )+144 \sqrt {b} d x \left (a+b x^4\right )+112 \sqrt {b} f x^3 \left (a+b x^4\right )+63 e \left (\sqrt {b} x^2 \left (a+2 b x^4\right )-\frac {a^{3/2} \text {arcsinh}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{\sqrt {1+\frac {b x^4}{a}}}\right )-\frac {144 a \sqrt {b} d x \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},-\frac {b x^4}{a}\right )}{\sqrt {1+\frac {b x^4}{a}}}-\frac {112 a \sqrt {b} f x^3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {b x^4}{a}\right )}{\sqrt {1+\frac {b x^4}{a}}}\right )}{1008 b^{3/2}} \]
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Result contains complex when optimal does not.
Time = 2.03 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.68
method | result | size |
risch | \(\frac {\left (560 b f \,x^{7}+630 b e \,x^{6}+720 b d \,x^{5}+840 b c \,x^{4}+224 a f \,x^{3}+315 a e \,x^{2}+480 a d x +840 a c \right ) \sqrt {b \,x^{4}+a}}{5040 b}-\frac {a^{2} \left (\frac {80 d \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {112 i f \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}+\frac {105 e \ln \left (x^{2} \sqrt {b}+\sqrt {b \,x^{4}+a}\right )}{2 \sqrt {b}}\right )}{840 b}\) | \(269\) |
default | \(f \left (\frac {x^{7} \sqrt {b \,x^{4}+a}}{9}+\frac {2 a \,x^{3} \sqrt {b \,x^{4}+a}}{45 b}-\frac {2 i a^{\frac {5}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{15 b^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+e \left (\frac {x^{2} \left (b \,x^{4}+a \right )^{\frac {3}{2}}}{8 b}-\frac {a \,x^{2} \sqrt {b \,x^{4}+a}}{16 b}-\frac {a^{2} \ln \left (x^{2} \sqrt {b}+\sqrt {b \,x^{4}+a}\right )}{16 b^{\frac {3}{2}}}\right )+d \left (\frac {x^{5} \sqrt {b \,x^{4}+a}}{7}+\frac {2 a x \sqrt {b \,x^{4}+a}}{21 b}-\frac {2 a^{2} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{21 b \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+\frac {c \left (b \,x^{4}+a \right )^{\frac {3}{2}}}{6 b}\) | \(321\) |
elliptic | \(\frac {f \,x^{7} \sqrt {b \,x^{4}+a}}{9}+\frac {e \,x^{6} \sqrt {b \,x^{4}+a}}{8}+\frac {d \,x^{5} \sqrt {b \,x^{4}+a}}{7}+\frac {c \,x^{4} \sqrt {b \,x^{4}+a}}{6}+\frac {2 a f \,x^{3} \sqrt {b \,x^{4}+a}}{45 b}+\frac {a e \,x^{2} \sqrt {b \,x^{4}+a}}{16 b}+\frac {2 a d x \sqrt {b \,x^{4}+a}}{21 b}+\frac {a c \sqrt {b \,x^{4}+a}}{6 b}-\frac {2 a^{2} d \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{21 b \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {a^{2} e \ln \left (2 x^{2} \sqrt {b}+2 \sqrt {b \,x^{4}+a}\right )}{16 b^{\frac {3}{2}}}-\frac {2 i a^{\frac {5}{2}} f \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{15 b^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\) | \(337\) |
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Time = 0.28 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.52 \[ \int x^3 \left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4} \, dx=-\frac {1344 \, a^{2} \sqrt {b} f x \left (-\frac {a}{b}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (-\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - 315 \, a^{2} \sqrt {b} e x \log \left (-2 \, b x^{4} + 2 \, \sqrt {b x^{4} + a} \sqrt {b} x^{2} - a\right ) + 192 \, {\left (5 \, a b d - 7 \, a^{2} f\right )} \sqrt {b} x \left (-\frac {a}{b}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - 2 \, {\left (560 \, b^{2} f x^{8} + 630 \, b^{2} e x^{7} + 720 \, b^{2} d x^{6} + 840 \, b^{2} c x^{5} + 224 \, a b f x^{4} + 315 \, a b e x^{3} + 480 \, a b d x^{2} + 840 \, a b c x - 672 \, a^{2} f\right )} \sqrt {b x^{4} + a}}{10080 \, b^{2} x} \]
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Time = 3.35 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.54 \[ \int x^3 \left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4} \, dx=\frac {a^{\frac {3}{2}} e x^{2}}{16 b \sqrt {1 + \frac {b x^{4}}{a}}} + \frac {\sqrt {a} d x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} + \frac {3 \sqrt {a} e x^{6}}{16 \sqrt {1 + \frac {b x^{4}}{a}}} + \frac {\sqrt {a} f x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {11}{4}\right )} - \frac {a^{2} e \operatorname {asinh}{\left (\frac {\sqrt {b} x^{2}}{\sqrt {a}} \right )}}{16 b^{\frac {3}{2}}} + c \left (\begin {cases} \frac {\sqrt {a} x^{4}}{4} & \text {for}\: b = 0 \\\frac {\left (a + b x^{4}\right )^{\frac {3}{2}}}{6 b} & \text {otherwise} \end {cases}\right ) + \frac {b e x^{10}}{8 \sqrt {a} \sqrt {1 + \frac {b x^{4}}{a}}} \]
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\[ \int x^3 \left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4} \, dx=\int { \sqrt {b x^{4} + a} {\left (f x^{3} + e x^{2} + d x + c\right )} x^{3} \,d x } \]
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\[ \int x^3 \left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4} \, dx=\int { \sqrt {b x^{4} + a} {\left (f x^{3} + e x^{2} + d x + c\right )} x^{3} \,d x } \]
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Timed out. \[ \int x^3 \left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4} \, dx=\int x^3\,\sqrt {b\,x^4+a}\,\left (f\,x^3+e\,x^2+d\,x+c\right ) \,d x \]
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