3.497 \(\int \frac{\left (c+d x+e x^2+f x^3\right ) \sqrt{a+b x^4}}{x^{10}} \, dx\)

Optimal. Leaf size=425 \[ -\frac{b^{7/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (5 \sqrt{a} e+7 \sqrt{b} c\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{105 a^{7/4} \sqrt{a+b x^4}}+\frac{2 b^{9/4} c \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 a^{7/4} \sqrt{a+b x^4}}+\frac{b^2 d \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{16 a^{3/2}}-\frac{2 b^{5/2} c x \sqrt{a+b x^4}}{15 a^2 \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{2 b^2 c \sqrt{a+b x^4}}{15 a^2 x}-\frac{1}{504} \sqrt{a+b x^4} \left (\frac{56 c}{x^9}+\frac{63 d}{x^8}+\frac{72 e}{x^7}+\frac{84 f}{x^6}\right )-\frac{2 b c \sqrt{a+b x^4}}{45 a x^5}-\frac{b d \sqrt{a+b x^4}}{16 a x^4}-\frac{2 b e \sqrt{a+b x^4}}{21 a x^3}-\frac{b f \sqrt{a+b x^4}}{6 a x^2} \]

[Out]

-(((56*c)/x^9 + (63*d)/x^8 + (72*e)/x^7 + (84*f)/x^6)*Sqrt[a + b*x^4])/504 - (2*
b*c*Sqrt[a + b*x^4])/(45*a*x^5) - (b*d*Sqrt[a + b*x^4])/(16*a*x^4) - (2*b*e*Sqrt
[a + b*x^4])/(21*a*x^3) - (b*f*Sqrt[a + b*x^4])/(6*a*x^2) + (2*b^2*c*Sqrt[a + b*
x^4])/(15*a^2*x) - (2*b^(5/2)*c*x*Sqrt[a + b*x^4])/(15*a^2*(Sqrt[a] + Sqrt[b]*x^
2)) + (b^2*d*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]])/(16*a^(3/2)) + (2*b^(9/4)*c*(Sqrt
[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticE[2*ArcTa
n[(b^(1/4)*x)/a^(1/4)], 1/2])/(15*a^(7/4)*Sqrt[a + b*x^4]) - (b^(7/4)*(7*Sqrt[b]
*c + 5*Sqrt[a]*e)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^
2)^2]*EllipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(105*a^(7/4)*Sqrt[a + b*x^4
])

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Rubi [A]  time = 1.18464, antiderivative size = 425, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 13, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.433 \[ -\frac{b^{7/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (5 \sqrt{a} e+7 \sqrt{b} c\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{105 a^{7/4} \sqrt{a+b x^4}}+\frac{2 b^{9/4} c \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 a^{7/4} \sqrt{a+b x^4}}+\frac{b^2 d \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{16 a^{3/2}}-\frac{2 b^{5/2} c x \sqrt{a+b x^4}}{15 a^2 \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{2 b^2 c \sqrt{a+b x^4}}{15 a^2 x}-\frac{1}{504} \sqrt{a+b x^4} \left (\frac{56 c}{x^9}+\frac{63 d}{x^8}+\frac{72 e}{x^7}+\frac{84 f}{x^6}\right )-\frac{2 b c \sqrt{a+b x^4}}{45 a x^5}-\frac{b d \sqrt{a+b x^4}}{16 a x^4}-\frac{2 b e \sqrt{a+b x^4}}{21 a x^3}-\frac{b f \sqrt{a+b x^4}}{6 a x^2} \]

Antiderivative was successfully verified.

[In]  Int[((c + d*x + e*x^2 + f*x^3)*Sqrt[a + b*x^4])/x^10,x]

[Out]

-(((56*c)/x^9 + (63*d)/x^8 + (72*e)/x^7 + (84*f)/x^6)*Sqrt[a + b*x^4])/504 - (2*
b*c*Sqrt[a + b*x^4])/(45*a*x^5) - (b*d*Sqrt[a + b*x^4])/(16*a*x^4) - (2*b*e*Sqrt
[a + b*x^4])/(21*a*x^3) - (b*f*Sqrt[a + b*x^4])/(6*a*x^2) + (2*b^2*c*Sqrt[a + b*
x^4])/(15*a^2*x) - (2*b^(5/2)*c*x*Sqrt[a + b*x^4])/(15*a^2*(Sqrt[a] + Sqrt[b]*x^
2)) + (b^2*d*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]])/(16*a^(3/2)) + (2*b^(9/4)*c*(Sqrt
[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticE[2*ArcTa
n[(b^(1/4)*x)/a^(1/4)], 1/2])/(15*a^(7/4)*Sqrt[a + b*x^4]) - (b^(7/4)*(7*Sqrt[b]
*c + 5*Sqrt[a]*e)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^
2)^2]*EllipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(105*a^(7/4)*Sqrt[a + b*x^4
])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x**3+e*x**2+d*x+c)*(b*x**4+a)**(1/2)/x**10,x)

[Out]

Timed out

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Mathematica [C]  time = 0.872723, size = 305, normalized size = 0.72 \[ \frac{\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \left (315 \sqrt{a} b^2 d x^9 \sqrt{a+b x^4} \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )-\left (a+b x^4\right ) \left (10 a^2 \left (56 c+63 d x+72 e x^2+84 f x^3\right )+a b x^4 (224 c+15 x (21 d+8 x (4 e+7 f x)))-672 b^2 c x^8\right )\right )-672 \sqrt{a} b^{5/2} c x^9 \sqrt{\frac{b x^4}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )+96 \sqrt{a} b^2 x^9 \sqrt{\frac{b x^4}{a}+1} \left (7 \sqrt{b} c+5 i \sqrt{a} e\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )}{5040 a^2 x^9 \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[((c + d*x + e*x^2 + f*x^3)*Sqrt[a + b*x^4])/x^10,x]

[Out]

(Sqrt[(I*Sqrt[b])/Sqrt[a]]*(-((a + b*x^4)*(-672*b^2*c*x^8 + 10*a^2*(56*c + 63*d*
x + 72*e*x^2 + 84*f*x^3) + a*b*x^4*(224*c + 15*x*(21*d + 8*x*(4*e + 7*f*x))))) +
315*Sqrt[a]*b^2*d*x^9*Sqrt[a + b*x^4]*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]]) - 672*S
qrt[a]*b^(5/2)*c*x^9*Sqrt[1 + (b*x^4)/a]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sq
rt[a]]*x], -1] + 96*Sqrt[a]*b^2*(7*Sqrt[b]*c + (5*I)*Sqrt[a]*e)*x^9*Sqrt[1 + (b*
x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1])/(5040*a^2*Sqrt[(I
*Sqrt[b])/Sqrt[a]]*x^9*Sqrt[a + b*x^4])

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Maple [C]  time = 0.028, size = 429, normalized size = 1. \[ -{\frac{c}{9\,{x}^{9}}\sqrt{b{x}^{4}+a}}-{\frac{2\,bc}{45\,a{x}^{5}}\sqrt{b{x}^{4}+a}}+{\frac{2\,{b}^{2}c}{15\,{a}^{2}x}\sqrt{b{x}^{4}+a}}-{{\frac{2\,i}{15}}c{b}^{{\frac{5}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}+{{\frac{2\,i}{15}}c{b}^{{\frac{5}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticE} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{\frac{d}{8\,a{x}^{8}} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{2}}}}+{\frac{bd}{16\,{a}^{2}{x}^{4}} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{2}}}}+{\frac{{b}^{2}d}{16}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{4}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{{b}^{2}d}{16\,{a}^{2}}\sqrt{b{x}^{4}+a}}-{\frac{e}{7\,{x}^{7}}\sqrt{b{x}^{4}+a}}-{\frac{2\,be}{21\,a{x}^{3}}\sqrt{b{x}^{4}+a}}-{\frac{2\,{b}^{2}e}{21\,a}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{\frac{f}{6\,a{x}^{6}} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(1/2)/x^10,x)

[Out]

-1/9*c/x^9*(b*x^4+a)^(1/2)-2/45*b*c*(b*x^4+a)^(1/2)/a/x^5+2/15*b^2*c*(b*x^4+a)^(
1/2)/a^2/x-2/15*I*c/a^(3/2)*b^(5/2)/(I/a^(1/2)*b^(1/2))^(1/2)*(1-I/a^(1/2)*b^(1/
2)*x^2)^(1/2)*(1+I/a^(1/2)*b^(1/2)*x^2)^(1/2)/(b*x^4+a)^(1/2)*EllipticF(x*(I/a^(
1/2)*b^(1/2))^(1/2),I)+2/15*I*c/a^(3/2)*b^(5/2)/(I/a^(1/2)*b^(1/2))^(1/2)*(1-I/a
^(1/2)*b^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*b^(1/2)*x^2)^(1/2)/(b*x^4+a)^(1/2)*Ellipt
icE(x*(I/a^(1/2)*b^(1/2))^(1/2),I)-1/8*d/a/x^8*(b*x^4+a)^(3/2)+1/16*d*b/a^2/x^4*
(b*x^4+a)^(3/2)+1/16*d*b^2/a^(3/2)*ln((2*a+2*a^(1/2)*(b*x^4+a)^(1/2))/x^2)-1/16*
d*b^2/a^2*(b*x^4+a)^(1/2)-1/7*e/x^7*(b*x^4+a)^(1/2)-2/21*b*e*(b*x^4+a)^(1/2)/a/x
^3-2/21*e/a*b^2/(I/a^(1/2)*b^(1/2))^(1/2)*(1-I/a^(1/2)*b^(1/2)*x^2)^(1/2)*(1+I/a
^(1/2)*b^(1/2)*x^2)^(1/2)/(b*x^4+a)^(1/2)*EllipticF(x*(I/a^(1/2)*b^(1/2))^(1/2),
I)-1/6*f/a/x^6*(b*x^4+a)^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x^{4} + a}{\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{10}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^4 + a)*(f*x^3 + e*x^2 + d*x + c)/x^10,x, algorithm="maxima")

[Out]

integrate(sqrt(b*x^4 + a)*(f*x^3 + e*x^2 + d*x + c)/x^10, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{b x^{4} + a}{\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{10}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^4 + a)*(f*x^3 + e*x^2 + d*x + c)/x^10,x, algorithm="fricas")

[Out]

integral(sqrt(b*x^4 + a)*(f*x^3 + e*x^2 + d*x + c)/x^10, x)

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Sympy [A]  time = 14.0266, size = 246, normalized size = 0.58 \[ \frac{\sqrt{a} c \Gamma \left (- \frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{9}{4}, - \frac{1}{2} \\ - \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{9} \Gamma \left (- \frac{5}{4}\right )} + \frac{\sqrt{a} e \Gamma \left (- \frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{7}{4}, - \frac{1}{2} \\ - \frac{3}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{7} \Gamma \left (- \frac{3}{4}\right )} - \frac{a d}{8 \sqrt{b} x^{10} \sqrt{\frac{a}{b x^{4}} + 1}} - \frac{3 \sqrt{b} d}{16 x^{6} \sqrt{\frac{a}{b x^{4}} + 1}} - \frac{\sqrt{b} f \sqrt{\frac{a}{b x^{4}} + 1}}{6 x^{4}} - \frac{b^{\frac{3}{2}} d}{16 a x^{2} \sqrt{\frac{a}{b x^{4}} + 1}} - \frac{b^{\frac{3}{2}} f \sqrt{\frac{a}{b x^{4}} + 1}}{6 a} + \frac{b^{2} d \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{16 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x**3+e*x**2+d*x+c)*(b*x**4+a)**(1/2)/x**10,x)

[Out]

sqrt(a)*c*gamma(-9/4)*hyper((-9/4, -1/2), (-5/4,), b*x**4*exp_polar(I*pi)/a)/(4*
x**9*gamma(-5/4)) + sqrt(a)*e*gamma(-7/4)*hyper((-7/4, -1/2), (-3/4,), b*x**4*ex
p_polar(I*pi)/a)/(4*x**7*gamma(-3/4)) - a*d/(8*sqrt(b)*x**10*sqrt(a/(b*x**4) + 1
)) - 3*sqrt(b)*d/(16*x**6*sqrt(a/(b*x**4) + 1)) - sqrt(b)*f*sqrt(a/(b*x**4) + 1)
/(6*x**4) - b**(3/2)*d/(16*a*x**2*sqrt(a/(b*x**4) + 1)) - b**(3/2)*f*sqrt(a/(b*x
**4) + 1)/(6*a) + b**2*d*asinh(sqrt(a)/(sqrt(b)*x**2))/(16*a**(3/2))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x^{4} + a}{\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{10}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^4 + a)*(f*x^3 + e*x^2 + d*x + c)/x^10,x, algorithm="giac")

[Out]

integrate(sqrt(b*x^4 + a)*(f*x^3 + e*x^2 + d*x + c)/x^10, x)