Optimal. Leaf size=394 \[ -\frac {a^{7/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (7 \sqrt {a} f+5 \sqrt {b} d\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{105 b^{7/4} \sqrt {a+b x^4}}+\frac {2 a^{9/4} f \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 b^{7/4} \sqrt {a+b x^4}}-\frac {a^2 e \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{16 b^{3/2}}-\frac {2 a^2 f x \sqrt {a+b x^4}}{15 b^{3/2} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {\left (a+b x^4\right )^{3/2} \left (4 c+3 e x^2\right )}{24 b}+\frac {1}{63} x^5 \sqrt {a+b x^4} \left (9 d+7 f x^2\right )+\frac {2 a d x \sqrt {a+b x^4}}{21 b}-\frac {a e x^2 \sqrt {a+b x^4}}{16 b}+\frac {2 a f x^3 \sqrt {a+b x^4}}{45 b} \]
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Rubi [A] time = 0.33, antiderivative size = 394, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {1833, 1252, 780, 195, 217, 206, 1274, 1280, 1198, 220, 1196} \[ -\frac {a^{7/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (7 \sqrt {a} f+5 \sqrt {b} d\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{105 b^{7/4} \sqrt {a+b x^4}}-\frac {a^2 e \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{16 b^{3/2}}-\frac {2 a^2 f x \sqrt {a+b x^4}}{15 b^{3/2} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {2 a^{9/4} f \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 b^{7/4} \sqrt {a+b x^4}}+\frac {\left (a+b x^4\right )^{3/2} \left (4 c+3 e x^2\right )}{24 b}+\frac {1}{63} x^5 \sqrt {a+b x^4} \left (9 d+7 f x^2\right )+\frac {2 a d x \sqrt {a+b x^4}}{21 b}-\frac {a e x^2 \sqrt {a+b x^4}}{16 b}+\frac {2 a f x^3 \sqrt {a+b x^4}}{45 b} \]
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 220
Rule 780
Rule 1196
Rule 1198
Rule 1252
Rule 1274
Rule 1280
Rule 1833
Rubi steps
\begin {align*} \int x^3 \left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4} \, dx &=\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} \operatorname {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) \operatorname {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 ) \operatorname {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 ) \operatorname {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*}
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Mathematica [C] time = 0.66, size = 215, normalized size = 0.55 \[ \frac {\sqrt {a+b x^4} \left (63 e \left (\sqrt {b} x^2 \left (a+2 b x^4\right )-\frac {a^{3/2} \sinh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{\sqrt {\frac {b x^4}{a}+1}}\right )+168 \sqrt {b} c \left (a+b x^4\right )-\frac {144 a \sqrt {b} d x \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {5}{4};-\frac {b x^4}{a}\right )}{\sqrt {\frac {b x^4}{a}+1}}+144 \sqrt {b} d x \left (a+b x^4\right )-\frac {112 a \sqrt {b} f x^3 \, _2F_1\left (-\frac {1}{2},\frac {3}{4};\frac {7}{4};-\frac {b x^4}{a}\right )}{\sqrt {\frac {b x^4}{a}+1}}+112 \sqrt {b} f x^3 \left (a+b x^4\right )\right )}{1008 b^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (f x^{6} + e x^{5} + d x^{4} + c x^{3}\right )} \sqrt {b x^{4} + a}, x\right ) \]
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 \[ \int \sqrt {b x^{4} + a} {\left (f x^{3} + e x^{2} + d x + c\right )} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.19, size = 380, normalized size = 0.96 \[ \frac {\sqrt {b \,x^{4}+a}\, f \,x^{7}}{9}+\frac {\sqrt {b \,x^{4}+a}\, d \,x^{5}}{7}+\frac {2 \sqrt {b \,x^{4}+a}\, a f \,x^{3}}{45 b}+\frac {2 i \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, a^{\frac {5}{2}} f \EllipticE \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{15 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, b^{\frac {3}{2}}}-\frac {2 i \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, a^{\frac {5}{2}} f \EllipticF \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{15 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, b^{\frac {3}{2}}}-\frac {2 \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, a^{2} d \EllipticF \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{21 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, b}-\frac {\sqrt {b \,x^{4}+a}\, a e \,x^{2}}{16 b}-\frac {a^{2} e \ln \left (\sqrt {b}\, x^{2}+\sqrt {b \,x^{4}+a}\right )}{16 b^{\frac {3}{2}}}+\frac {2 \sqrt {b \,x^{4}+a}\, a d x}{21 b}+\frac {\left (b \,x^{4}+a \right )^{\frac {3}{2}} e \,x^{2}}{8 b}+\frac {\left (b \,x^{4}+a \right )^{\frac {3}{2}} c}{6 b} \]
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 \[ \frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}} c}{6 \, b} + \int {\left (f x^{6} + e x^{5} + d x^{4}\right )} \sqrt {b x^{4} + a}\,{d x} \]
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 \[ \int x^3\,\sqrt {b\,x^4+a}\,\left (f\,x^3+e\,x^2+d\,x+c\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.51, size = 212, normalized size = 0.54 \[ \frac {a^{\frac {3}{2}} e x^{2}}{16 b \sqrt {1 + \frac {b x^{4}}{a}}} + \frac {\sqrt {a} d x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} + \frac {3 \sqrt {a} e x^{6}}{16 \sqrt {1 + \frac {b x^{4}}{a}}} + \frac {\sqrt {a} f x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {11}{4}\right )} - \frac {a^{2} e \operatorname {asinh}{\left (\frac {\sqrt {b} x^{2}}{\sqrt {a}} \right )}}{16 b^{\frac {3}{2}}} + c \left (\begin {cases} \frac {\sqrt {a} x^{4}}{4} & \text {for}\: b = 0 \\\frac {\left (a + b x^{4}\right )^{\frac {3}{2}}}{6 b} & \text {otherwise} \end {cases}\right ) + \frac {b e x^{10}}{8 \sqrt {a} \sqrt {1 + \frac {b x^{4}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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