3.516 \(\int \frac{\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^{14}} \, dx\)

Optimal. Leaf size=474 \[ -\frac{2 b^{11/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (65 \sqrt{a} e+77 \sqrt{b} c\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5005 a^{7/4} \sqrt{a+b x^4}}+\frac{4 b^{13/4} c \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{65 a^{7/4} \sqrt{a+b x^4}}+\frac{b^3 d \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{32 a^{3/2}}-\frac{4 b^{7/2} c x \sqrt{a+b x^4}}{65 a^2 \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{4 b^3 c \sqrt{a+b x^4}}{65 a^2 x}-\frac{4 b^2 c \sqrt{a+b x^4}}{195 a x^5}-\frac{b^2 d \sqrt{a+b x^4}}{32 a x^4}-\frac{4 b^2 e \sqrt{a+b x^4}}{77 a x^3}-\frac{b^2 f \sqrt{a+b x^4}}{10 a x^2}-\frac{\left (a+b x^4\right )^{3/2} \left (\frac{660 c}{x^{13}}+\frac{715 d}{x^{12}}+\frac{780 e}{x^{11}}+\frac{858 f}{x^{10}}\right )}{8580}-\frac{b \sqrt{a+b x^4} \left (\frac{12320 c}{x^9}+\frac{15015 d}{x^8}+\frac{18720 e}{x^7}+\frac{24024 f}{x^6}\right )}{240240} \]

[Out]

-(b*((12320*c)/x^9 + (15015*d)/x^8 + (18720*e)/x^7 + (24024*f)/x^6)*Sqrt[a + b*x
^4])/240240 - (4*b^2*c*Sqrt[a + b*x^4])/(195*a*x^5) - (b^2*d*Sqrt[a + b*x^4])/(3
2*a*x^4) - (4*b^2*e*Sqrt[a + b*x^4])/(77*a*x^3) - (b^2*f*Sqrt[a + b*x^4])/(10*a*
x^2) + (4*b^3*c*Sqrt[a + b*x^4])/(65*a^2*x) - (4*b^(7/2)*c*x*Sqrt[a + b*x^4])/(6
5*a^2*(Sqrt[a] + Sqrt[b]*x^2)) - (((660*c)/x^13 + (715*d)/x^12 + (780*e)/x^11 +
(858*f)/x^10)*(a + b*x^4)^(3/2))/8580 + (b^3*d*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]])
/(32*a^(3/2)) + (4*b^(13/4)*c*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a]
+ Sqrt[b]*x^2)^2]*EllipticE[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(65*a^(7/4)*Sqr
t[a + b*x^4]) - (2*b^(11/4)*(77*Sqrt[b]*c + 65*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])/(5005*a^(7/4)*Sqrt[a + b*x^4])

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Rubi [A]  time = 1.20802, antiderivative size = 474, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 13, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.433 \[ -\frac{2 b^{11/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (65 \sqrt{a} e+77 \sqrt{b} c\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5005 a^{7/4} \sqrt{a+b x^4}}+\frac{4 b^{13/4} c \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{65 a^{7/4} \sqrt{a+b x^4}}+\frac{b^3 d \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{32 a^{3/2}}-\frac{4 b^{7/2} c x \sqrt{a+b x^4}}{65 a^2 \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{4 b^3 c \sqrt{a+b x^4}}{65 a^2 x}-\frac{4 b^2 c \sqrt{a+b x^4}}{195 a x^5}-\frac{b^2 d \sqrt{a+b x^4}}{32 a x^4}-\frac{4 b^2 e \sqrt{a+b x^4}}{77 a x^3}-\frac{b^2 f \sqrt{a+b x^4}}{10 a x^2}-\frac{\left (a+b x^4\right )^{3/2} \left (\frac{660 c}{x^{13}}+\frac{715 d}{x^{12}}+\frac{780 e}{x^{11}}+\frac{858 f}{x^{10}}\right )}{8580}-\frac{b \sqrt{a+b x^4} \left (\frac{12320 c}{x^9}+\frac{15015 d}{x^8}+\frac{18720 e}{x^7}+\frac{24024 f}{x^6}\right )}{240240} \]

Antiderivative was successfully verified.

[In]  Int[((c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2))/x^14,x]

[Out]

-(b*((12320*c)/x^9 + (15015*d)/x^8 + (18720*e)/x^7 + (24024*f)/x^6)*Sqrt[a + b*x
^4])/240240 - (4*b^2*c*Sqrt[a + b*x^4])/(195*a*x^5) - (b^2*d*Sqrt[a + b*x^4])/(3
2*a*x^4) - (4*b^2*e*Sqrt[a + b*x^4])/(77*a*x^3) - (b^2*f*Sqrt[a + b*x^4])/(10*a*
x^2) + (4*b^3*c*Sqrt[a + b*x^4])/(65*a^2*x) - (4*b^(7/2)*c*x*Sqrt[a + b*x^4])/(6
5*a^2*(Sqrt[a] + Sqrt[b]*x^2)) - (((660*c)/x^13 + (715*d)/x^12 + (780*e)/x^11 +
(858*f)/x^10)*(a + b*x^4)^(3/2))/8580 + (b^3*d*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]])
/(32*a^(3/2)) + (4*b^(13/4)*c*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a]
+ Sqrt[b]*x^2)^2]*EllipticE[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(65*a^(7/4)*Sqr
t[a + b*x^4]) - (2*b^(11/4)*(77*Sqrt[b]*c + 65*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])/(5005*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)**(3/2)/x**14,x)

[Out]

Timed out

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Mathematica [C]  time = 1.09918, size = 339, normalized size = 0.72 \[ \frac{\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \left (15015 \sqrt{a} b^3 d x^{13} \sqrt{a+b x^4} \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )-\left (a+b x^4\right ) \left (56 a^3 \left (660 c+13 x \left (55 d+60 e x+66 f x^2\right )\right )+2 a^2 b x^4 (30800 c+13 x (2695 d+48 x (65 e+77 f x)))+a b^2 x^8 (9856 c+39 x (385 d+16 x (40 e+77 f x)))-29568 b^3 c x^{12}\right )\right )-29568 \sqrt{a} b^{7/2} c x^{13} \sqrt{\frac{b x^4}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )+384 \sqrt{a} b^3 x^{13} \sqrt{\frac{b x^4}{a}+1} \left (77 \sqrt{b} c+65 i \sqrt{a} e\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )}{480480 a^2 x^{13} \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[((c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2))/x^14,x]

[Out]

(Sqrt[(I*Sqrt[b])/Sqrt[a]]*(-((a + b*x^4)*(-29568*b^3*c*x^12 + 56*a^3*(660*c + 1
3*x*(55*d + 60*e*x + 66*f*x^2)) + a*b^2*x^8*(9856*c + 39*x*(385*d + 16*x*(40*e +
77*f*x))) + 2*a^2*b*x^4*(30800*c + 13*x*(2695*d + 48*x*(65*e + 77*f*x))))) + 15
015*Sqrt[a]*b^3*d*x^13*Sqrt[a + b*x^4]*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]]) - 29568
*Sqrt[a]*b^(7/2)*c*x^13*Sqrt[1 + (b*x^4)/a]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[b])
/Sqrt[a]]*x], -1] + 384*Sqrt[a]*b^3*(77*Sqrt[b]*c + (65*I)*Sqrt[a]*e)*x^13*Sqrt[
1 + (b*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1])/(480480*a^
2*Sqrt[(I*Sqrt[b])/Sqrt[a]]*x^13*Sqrt[a + b*x^4])

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Maple [C]  time = 0.034, size = 483, normalized size = 1. \[ -{\frac{ac}{13\,{x}^{13}}\sqrt{b{x}^{4}+a}}-{\frac{5\,bc}{39\,{x}^{9}}\sqrt{b{x}^{4}+a}}-{\frac{4\,{b}^{2}c}{195\,a{x}^{5}}\sqrt{b{x}^{4}+a}}+{\frac{4\,{b}^{3}c}{65\,{a}^{2}x}\sqrt{b{x}^{4}+a}}-{{\frac{4\,i}{65}}c{b}^{{\frac{7}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}+{{\frac{4\,i}{65}}c{b}^{{\frac{7}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticE} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{\frac{ad}{12\,{x}^{12}}\sqrt{b{x}^{4}+a}}-{\frac{7\,bd}{48\,{x}^{8}}\sqrt{b{x}^{4}+a}}-{\frac{{b}^{2}d}{32\,a{x}^{4}}\sqrt{b{x}^{4}+a}}+{\frac{{b}^{3}d}{32}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{4}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{ae}{11\,{x}^{11}}\sqrt{b{x}^{4}+a}}-{\frac{13\,be}{77\,{x}^{7}}\sqrt{b{x}^{4}+a}}-{\frac{4\,{b}^{2}e}{77\,a{x}^{3}}\sqrt{b{x}^{4}+a}}-{\frac{4\,e{b}^{3}}{77\,a}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{\frac{f \left ({b}^{2}{x}^{8}+2\,ab{x}^{4}+{a}^{2} \right ) }{10\,{x}^{10}a}\sqrt{b{x}^{4}+a}} \]

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)^(3/2)/x^14,x)

[Out]

-1/13*c*a*(b*x^4+a)^(1/2)/x^13-5/39*c*b*(b*x^4+a)^(1/2)/x^9-4/195*b^2*c*(b*x^4+a
)^(1/2)/a/x^5+4/65*b^3*c*(b*x^4+a)^(1/2)/a^2/x-4/65*I*c*b^(7/2)/a^(3/2)/(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)+4/65*I*c*b^(7/2)/a^(3
/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)*EllipticE(x*(I/a^(1/2)*b^(1/2))^(1/2),I)-1/12*d*a/
x^12*(b*x^4+a)^(1/2)-7/48*d*b/x^8*(b*x^4+a)^(1/2)-1/32*b^2*d*(b*x^4+a)^(1/2)/a/x
^4+1/32*d/a^(3/2)*b^3*ln((2*a+2*a^(1/2)*(b*x^4+a)^(1/2))/x^2)-1/11*e*a*(b*x^4+a)
^(1/2)/x^11-13/77*e*b*(b*x^4+a)^(1/2)/x^7-4/77*b^2*e*(b*x^4+a)^(1/2)/a/x^3-4/77*
e/a*b^3/(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)-1/10*
f*(b*x^4+a)^(1/2)/x^10/a*(b^2*x^8+2*a*b*x^4+a^2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{3}{2}}{\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{14}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/2)*(f*x^3 + e*x^2 + d*x + c)/x^14,x, algorithm="maxima")

[Out]

integrate((b*x^4 + a)^(3/2)*(f*x^3 + e*x^2 + d*x + c)/x^14, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b f x^{7} + b e x^{6} + b d x^{5} + b c x^{4} + a f x^{3} + a e x^{2} + a d x + a c\right )} \sqrt{b x^{4} + a}}{x^{14}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/2)*(f*x^3 + e*x^2 + d*x + c)/x^14,x, algorithm="fricas")

[Out]

integral((b*f*x^7 + b*e*x^6 + b*d*x^5 + b*c*x^4 + a*f*x^3 + a*e*x^2 + a*d*x + a*
c)*sqrt(b*x^4 + a)/x^14, x)

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Sympy [A]  time = 42.4552, size = 403, normalized size = 0.85 \[ \frac{a^{\frac{3}{2}} c \Gamma \left (- \frac{13}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{13}{4}, - \frac{1}{2} \\ - \frac{9}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{13} \Gamma \left (- \frac{9}{4}\right )} + \frac{a^{\frac{3}{2}} e \Gamma \left (- \frac{11}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{11}{4}, - \frac{1}{2} \\ - \frac{7}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{11} \Gamma \left (- \frac{7}{4}\right )} + \frac{\sqrt{a} b c \Gamma \left (- \frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{9}{4}, - \frac{1}{2} \\ - \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{9} \Gamma \left (- \frac{5}{4}\right )} + \frac{\sqrt{a} b e \Gamma \left (- \frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{7}{4}, - \frac{1}{2} \\ - \frac{3}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{7} \Gamma \left (- \frac{3}{4}\right )} - \frac{a^{2} d}{12 \sqrt{b} x^{14} \sqrt{\frac{a}{b x^{4}} + 1}} - \frac{11 a \sqrt{b} d}{48 x^{10} \sqrt{\frac{a}{b x^{4}} + 1}} - \frac{a \sqrt{b} f \sqrt{\frac{a}{b x^{4}} + 1}}{10 x^{8}} - \frac{17 b^{\frac{3}{2}} d}{96 x^{6} \sqrt{\frac{a}{b x^{4}} + 1}} - \frac{b^{\frac{3}{2}} f \sqrt{\frac{a}{b x^{4}} + 1}}{5 x^{4}} - \frac{b^{\frac{5}{2}} d}{32 a x^{2} \sqrt{\frac{a}{b x^{4}} + 1}} - \frac{b^{\frac{5}{2}} f \sqrt{\frac{a}{b x^{4}} + 1}}{10 a} + \frac{b^{3} d \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{32 a^{\frac{3}{2}}} \]

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)**(3/2)/x**14,x)

[Out]

a**(3/2)*c*gamma(-13/4)*hyper((-13/4, -1/2), (-9/4,), b*x**4*exp_polar(I*pi)/a)/
(4*x**13*gamma(-9/4)) + a**(3/2)*e*gamma(-11/4)*hyper((-11/4, -1/2), (-7/4,), b*
x**4*exp_polar(I*pi)/a)/(4*x**11*gamma(-7/4)) + sqrt(a)*b*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)*b*
e*gamma(-7/4)*hyper((-7/4, -1/2), (-3/4,), b*x**4*exp_polar(I*pi)/a)/(4*x**7*gam
ma(-3/4)) - a**2*d/(12*sqrt(b)*x**14*sqrt(a/(b*x**4) + 1)) - 11*a*sqrt(b)*d/(48*
x**10*sqrt(a/(b*x**4) + 1)) - a*sqrt(b)*f*sqrt(a/(b*x**4) + 1)/(10*x**8) - 17*b*
*(3/2)*d/(96*x**6*sqrt(a/(b*x**4) + 1)) - b**(3/2)*f*sqrt(a/(b*x**4) + 1)/(5*x**
4) - b**(5/2)*d/(32*a*x**2*sqrt(a/(b*x**4) + 1)) - b**(5/2)*f*sqrt(a/(b*x**4) +
1)/(10*a) + b**3*d*asinh(sqrt(a)/(sqrt(b)*x**2))/(32*a**(3/2))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{3}{2}}{\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{14}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/2)*(f*x^3 + e*x^2 + d*x + c)/x^14,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(3/2)*(f*x^3 + e*x^2 + d*x + c)/x^14, x)