Optimal. Leaf size=418 \[ -\frac {a^{7/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (7 \sqrt {a} e+5 \sqrt {b} c\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{105 b^{7/4} \sqrt {a+b x^4}}+\frac {2 a^{9/4} e \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 b^{7/4} \sqrt {a+b x^4}}-\frac {a^2 d \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{16 b^{3/2}}-\frac {2 a^2 e x \sqrt {a+b x^4}}{15 b^{3/2} \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {\left (a+b x^4\right )^{3/2} \left (8 a f-15 b d x^2\right )}{120 b^2}+\frac {1}{63} x^5 \sqrt {a+b x^4} \left (9 c+7 e x^2\right )+\frac {2 a c x \sqrt {a+b x^4}}{21 b}-\frac {a d x^2 \sqrt {a+b x^4}}{16 b}+\frac {2 a e x^3 \sqrt {a+b x^4}}{45 b}+\frac {f x^4 \left (a+b x^4\right )^{3/2}}{10 b} \]
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Rubi [A] time = 0.38, antiderivative size = 418, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 12, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {1833, 1274, 1280, 1198, 220, 1196, 1252, 833, 780, 195, 217, 206} \[ -\frac {a^{7/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (7 \sqrt {a} e+5 \sqrt {b} c\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{105 b^{7/4} \sqrt {a+b x^4}}-\frac {a^2 d \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{16 b^{3/2}}-\frac {2 a^2 e x \sqrt {a+b x^4}}{15 b^{3/2} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {2 a^{9/4} e \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 b^{7/4} \sqrt {a+b x^4}}-\frac {\left (a+b x^4\right )^{3/2} \left (8 a f-15 b d x^2\right )}{120 b^2}+\frac {1}{63} x^5 \sqrt {a+b x^4} \left (9 c+7 e x^2\right )+\frac {2 a c x \sqrt {a+b x^4}}{21 b}-\frac {a d x^2 \sqrt {a+b x^4}}{16 b}+\frac {2 a e x^3 \sqrt {a+b x^4}}{45 b}+\frac {f x^4 \left (a+b x^4\right )^{3/2}}{10 b} \]
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 220
Rule 780
Rule 833
Rule 1196
Rule 1198
Rule 1252
Rule 1274
Rule 1280
Rule 1833
Rubi steps
\begin {align*} \int x^4 \left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4} \, dx &=\int \left (x^4 \left (c+e x^2\right ) \sqrt {a+b x^4}+x^5 \left (d+f x^2\right ) \sqrt {a+b x^4}\right ) \, dx\\ &=\int x^4 \left (c+e x^2\right ) \sqrt {a+b x^4} \, dx+\int x^5 \left (d+f x^2\right ) \sqrt {a+b x^4} \, dx\\ &=\frac {1}{63} x^5 \left (9 c+7 e x^2\right ) \sqrt {a+b x^4}+\frac {1}{2} \operatorname {Subst}\left (\int x^2 (d+f x) \sqrt {a+b x^2} \, dx,x,x^2\right )+\frac {1}{63} (2 a) \int \frac {x^4 \left (9 c+7 e x^2\right )}{\sqrt {a+b x^4}} \, dx\\ &=\frac {2 a e x^3 \sqrt {a+b x^4}}{45 b}+\frac {1}{63} x^5 \left (9 c+7 e x^2\right ) \sqrt {a+b x^4}+\frac {f x^4 \left (a+b x^4\right )^{3/2}}{10 b}+\frac {\operatorname {Subst}\left (\int x (-2 a f+5 b d x) \sqrt {a+b x^2} \, dx,x,x^2\right )}{10 b}-\frac {(2 a) \int \frac {x^2 \left (21 a e-45 b c x^2\right )}{\sqrt {a+b x^4}} \, dx}{315 b}\\ &=\frac {2 a c x \sqrt {a+b x^4}}{21 b}+\frac {2 a e x^3 \sqrt {a+b x^4}}{45 b}+\frac {1}{63} x^5 \left (9 c+7 e x^2\right ) \sqrt {a+b x^4}+\frac {f x^4 \left (a+b x^4\right )^{3/2}}{10 b}-\frac {\left (8 a f-15 b d x^2\right ) \left (a+b x^4\right )^{3/2}}{120 b^2}+\frac {(2 a) \int \frac {-45 a b c-63 a b e x^2}{\sqrt {a+b x^4}} \, dx}{945 b^2}-\frac {(a d) \operatorname {Subst}\left (\int \sqrt {a+b x^2} \, dx,x,x^2\right )}{8 b}\\ &=\frac {2 a c x \sqrt {a+b x^4}}{21 b}-\frac {a d x^2 \sqrt {a+b x^4}}{16 b}+\frac {2 a e x^3 \sqrt {a+b x^4}}{45 b}+\frac {1}{63} x^5 \left (9 c+7 e x^2\right ) \sqrt {a+b x^4}+\frac {f x^4 \left (a+b x^4\right )^{3/2}}{10 b}-\frac {\left (8 a f-15 b d x^2\right ) \left (a+b x^4\right )^{3/2}}{120 b^2}-\frac {\left (a^2 d\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,x^2\right )}{16 b}+\frac {\left (2 a^{5/2} e\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} c+7 \sqrt {a} e\right )\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx}{105 b^{3/2}}\\ &=\frac {2 a c x \sqrt {a+b x^4}}{21 b}-\frac {a d x^2 \sqrt {a+b x^4}}{16 b}+\frac {2 a e x^3 \sqrt {a+b x^4}}{45 b}-\frac {2 a^2 e 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 c+7 e x^2\right ) \sqrt {a+b x^4}+\frac {f x^4 \left (a+b x^4\right )^{3/2}}{10 b}-\frac {\left (8 a f-15 b d x^2\right ) \left (a+b x^4\right )^{3/2}}{120 b^2}+\frac {2 a^{9/4} e \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} c+7 \sqrt {a} e\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}}-\frac {\left (a^2 d\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x^2}{\sqrt {a+b x^4}}\right )}{16 b}\\ &=\frac {2 a c x \sqrt {a+b x^4}}{21 b}-\frac {a d x^2 \sqrt {a+b x^4}}{16 b}+\frac {2 a e x^3 \sqrt {a+b x^4}}{45 b}-\frac {2 a^2 e 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 c+7 e x^2\right ) \sqrt {a+b x^4}+\frac {f x^4 \left (a+b x^4\right )^{3/2}}{10 b}-\frac {\left (8 a f-15 b d x^2\right ) \left (a+b x^4\right )^{3/2}}{120 b^2}-\frac {a^2 d \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{16 b^{3/2}}+\frac {2 a^{9/4} e \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} c+7 \sqrt {a} e\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*}
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Mathematica [C] time = 0.73, size = 202, normalized size = 0.48 \[ \frac {\sqrt {a+b x^4} \left (-\frac {315 a^{3/2} \sqrt {b} d \sinh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{\sqrt {\frac {b x^4}{a}+1}}-\frac {720 a b c x \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {5}{4};-\frac {b x^4}{a}\right )}{\sqrt {\frac {b x^4}{a}+1}}+720 b c x \left (a+b x^4\right )+315 b d x^2 \left (a+2 b x^4\right )-\frac {560 a b e x^3 \, _2F_1\left (-\frac {1}{2},\frac {3}{4};\frac {7}{4};-\frac {b x^4}{a}\right )}{\sqrt {\frac {b x^4}{a}+1}}+560 b e x^3 \left (a+b x^4\right )+168 f \left (a+b x^4\right ) \left (3 b x^4-2 a\right )\right )}{5040 b^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (f x^{7} + e x^{6} + d x^{5} + c x^{4}\right )} \sqrt {b x^{4} + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b x^{4} + a} {\left (f x^{3} + e x^{2} + d x + c\right )} x^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.21, size = 390, normalized size = 0.93 \[ \frac {\sqrt {b \,x^{4}+a}\, e \,x^{7}}{9}+\frac {\sqrt {b \,x^{4}+a}\, c \,x^{5}}{7}+\frac {2 \sqrt {b \,x^{4}+a}\, a e \,x^{3}}{45 b}+\frac {2 i \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, a^{\frac {5}{2}} e \EllipticE \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{15 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, b^{\frac {3}{2}}}-\frac {2 i \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, a^{\frac {5}{2}} e \EllipticF \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{15 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, b^{\frac {3}{2}}}-\frac {2 \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, a^{2} c \EllipticF \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{21 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, b}-\frac {\sqrt {b \,x^{4}+a}\, a d \,x^{2}}{16 b}-\frac {a^{2} d \ln \left (\sqrt {b}\, x^{2}+\sqrt {b \,x^{4}+a}\right )}{16 b^{\frac {3}{2}}}+\frac {2 \sqrt {b \,x^{4}+a}\, a c x}{21 b}+\frac {\left (b \,x^{4}+a \right )^{\frac {3}{2}} d \,x^{2}}{8 b}-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{2}} \left (-3 b \,x^{4}+2 a \right ) f}{30 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b x^{4} + a} {\left (f x^{3} + e x^{2} + d x + c\right )} x^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^4\,\sqrt {b\,x^4+a}\,\left (f\,x^3+e\,x^2+d\,x+c\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.54, size = 252, normalized size = 0.60 \[ \frac {a^{\frac {3}{2}} d x^{2}}{16 b \sqrt {1 + \frac {b x^{4}}{a}}} + \frac {\sqrt {a} c x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} + \frac {3 \sqrt {a} d x^{6}}{16 \sqrt {1 + \frac {b x^{4}}{a}}} + \frac {\sqrt {a} e x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {11}{4}\right )} - \frac {a^{2} d \operatorname {asinh}{\left (\frac {\sqrt {b} x^{2}}{\sqrt {a}} \right )}}{16 b^{\frac {3}{2}}} + f \left (\begin {cases} - \frac {a^{2} \sqrt {a + b x^{4}}}{15 b^{2}} + \frac {a x^{4} \sqrt {a + b x^{4}}}{30 b} + \frac {x^{8} \sqrt {a + b x^{4}}}{10} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{8}}{8} & \text {otherwise} \end {cases}\right ) + \frac {b d x^{10}}{8 \sqrt {a} \sqrt {1 + \frac {b x^{4}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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