3.515 \(\int \frac{\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^{13}} \, dx\)

Optimal. Leaf size=449 \[ -\frac{2 b^{9/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (15 \sqrt{b} d-77 \sqrt{a} f\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{1155 a^{5/4} \sqrt{a+b x^4}}-\frac{4 b^{9/4} f \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 a^{3/4} \sqrt{a+b x^4}}+\frac{b^3 c \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{32 a^{3/2}}+\frac{4 b^{5/2} f x \sqrt{a+b x^4}}{15 a \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{b^2 c \sqrt{a+b x^4}}{32 a x^4}-\frac{4 b^2 d \sqrt{a+b x^4}}{77 a x^3}-\frac{b^2 e \sqrt{a+b x^4}}{10 a x^2}-\frac{4 b^2 f \sqrt{a+b x^4}}{15 a x}-\frac{\left (a+b x^4\right )^{3/2} \left (\frac{165 c}{x^{12}}+\frac{180 d}{x^{11}}+\frac{198 e}{x^{10}}+\frac{220 f}{x^9}\right )}{1980}-\frac{b \sqrt{a+b x^4} \left (\frac{1155 c}{x^8}+\frac{1440 d}{x^7}+\frac{1848 e}{x^6}+\frac{2464 f}{x^5}\right )}{18480} \]

[Out]

-(b*((1155*c)/x^8 + (1440*d)/x^7 + (1848*e)/x^6 + (2464*f)/x^5)*Sqrt[a + b*x^4])
/18480 - (b^2*c*Sqrt[a + b*x^4])/(32*a*x^4) - (4*b^2*d*Sqrt[a + b*x^4])/(77*a*x^
3) - (b^2*e*Sqrt[a + b*x^4])/(10*a*x^2) - (4*b^2*f*Sqrt[a + b*x^4])/(15*a*x) + (
4*b^(5/2)*f*x*Sqrt[a + b*x^4])/(15*a*(Sqrt[a] + Sqrt[b]*x^2)) - (((165*c)/x^12 +
(180*d)/x^11 + (198*e)/x^10 + (220*f)/x^9)*(a + b*x^4)^(3/2))/1980 + (b^3*c*Arc
Tanh[Sqrt[a + b*x^4]/Sqrt[a]])/(32*a^(3/2)) - (4*b^(9/4)*f*(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])/(15*a^(3/4)*Sqrt[a + b*x^4]) - (2*b^(9/4)*(15*Sqrt[b]*d - 77*Sqrt[
a]*f)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*Ellipt
icF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(1155*a^(5/4)*Sqrt[a + b*x^4])

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Rubi [A]  time = 1.06846, antiderivative size = 449, 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 \[ -\frac{2 b^{9/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (15 \sqrt{b} d-77 \sqrt{a} f\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{1155 a^{5/4} \sqrt{a+b x^4}}-\frac{4 b^{9/4} f \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 a^{3/4} \sqrt{a+b x^4}}+\frac{b^3 c \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{32 a^{3/2}}+\frac{4 b^{5/2} f x \sqrt{a+b x^4}}{15 a \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{b^2 c \sqrt{a+b x^4}}{32 a x^4}-\frac{4 b^2 d \sqrt{a+b x^4}}{77 a x^3}-\frac{b^2 e \sqrt{a+b x^4}}{10 a x^2}-\frac{4 b^2 f \sqrt{a+b x^4}}{15 a x}-\frac{\left (a+b x^4\right )^{3/2} \left (\frac{165 c}{x^{12}}+\frac{180 d}{x^{11}}+\frac{198 e}{x^{10}}+\frac{220 f}{x^9}\right )}{1980}-\frac{b \sqrt{a+b x^4} \left (\frac{1155 c}{x^8}+\frac{1440 d}{x^7}+\frac{1848 e}{x^6}+\frac{2464 f}{x^5}\right )}{18480} \]

Antiderivative was successfully verified.

[In]  Int[((c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2))/x^13,x]

[Out]

-(b*((1155*c)/x^8 + (1440*d)/x^7 + (1848*e)/x^6 + (2464*f)/x^5)*Sqrt[a + b*x^4])
/18480 - (b^2*c*Sqrt[a + b*x^4])/(32*a*x^4) - (4*b^2*d*Sqrt[a + b*x^4])/(77*a*x^
3) - (b^2*e*Sqrt[a + b*x^4])/(10*a*x^2) - (4*b^2*f*Sqrt[a + b*x^4])/(15*a*x) + (
4*b^(5/2)*f*x*Sqrt[a + b*x^4])/(15*a*(Sqrt[a] + Sqrt[b]*x^2)) - (((165*c)/x^12 +
(180*d)/x^11 + (198*e)/x^10 + (220*f)/x^9)*(a + b*x^4)^(3/2))/1980 + (b^3*c*Arc
Tanh[Sqrt[a + b*x^4]/Sqrt[a]])/(32*a^(3/2)) - (4*b^(9/4)*f*(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])/(15*a^(3/4)*Sqrt[a + b*x^4]) - (2*b^(9/4)*(15*Sqrt[b]*d - 77*Sqrt[
a]*f)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*Ellipt
icF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(1155*a^(5/4)*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**13,x)

[Out]

Timed out

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Mathematica [C]  time = 1.17393, size = 328, normalized size = 0.73 \[ \frac{\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \left (3465 b^3 c x^{12} \sqrt{a+b x^4} \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )-\sqrt{a} \left (a+b x^4\right ) \left (56 a^2 \left (165 c+2 x \left (90 d+99 e x+110 f x^2\right )\right )+2 a b x^4 (8085 c+16 x (585 d+77 x (9 e+11 f x)))+3 b^2 x^8 (1155 c+16 x (120 d+77 x (3 e+8 f x)))\right )\right )-384 \sqrt{a} b^{5/2} x^{12} \sqrt{\frac{b x^4}{a}+1} \left (77 \sqrt{a} f-15 i \sqrt{b} d\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )+29568 a b^{5/2} f x^{12} \sqrt{\frac{b x^4}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )}{110880 a^{3/2} x^{12} \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[((c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2))/x^13,x]

[Out]

(Sqrt[(I*Sqrt[b])/Sqrt[a]]*(-(Sqrt[a]*(a + b*x^4)*(56*a^2*(165*c + 2*x*(90*d + 9
9*e*x + 110*f*x^2)) + 3*b^2*x^8*(1155*c + 16*x*(120*d + 77*x*(3*e + 8*f*x))) + 2
*a*b*x^4*(8085*c + 16*x*(585*d + 77*x*(9*e + 11*f*x))))) + 3465*b^3*c*x^12*Sqrt[
a + b*x^4]*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]]) + 29568*a*b^(5/2)*f*x^12*Sqrt[1 + (
b*x^4)/a]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1] - 384*Sqrt[a]*b^
(5/2)*((-15*I)*Sqrt[b]*d + 77*Sqrt[a]*f)*x^12*Sqrt[1 + (b*x^4)/a]*EllipticF[I*Ar
cSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1])/(110880*a^(3/2)*Sqrt[(I*Sqrt[b])/Sqrt[a
]]*x^12*Sqrt[a + b*x^4])

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Maple [C]  time = 0.03, size = 462, normalized size = 1. \[ -{\frac{ac}{12\,{x}^{12}}\sqrt{b{x}^{4}+a}}-{\frac{7\,bc}{48\,{x}^{8}}\sqrt{b{x}^{4}+a}}-{\frac{{b}^{2}c}{32\,a{x}^{4}}\sqrt{b{x}^{4}+a}}+{\frac{{b}^{3}c}{32}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{4}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{ad}{11\,{x}^{11}}\sqrt{b{x}^{4}+a}}-{\frac{13\,bd}{77\,{x}^{7}}\sqrt{b{x}^{4}+a}}-{\frac{4\,{b}^{2}d}{77\,a{x}^{3}}\sqrt{b{x}^{4}+a}}-{\frac{4\,{b}^{3}d}{77\,a}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{\frac{e \left ({b}^{2}{x}^{8}+2\,ab{x}^{4}+{a}^{2} \right ) }{10\,{x}^{10}a}\sqrt{b{x}^{4}+a}}-{\frac{af}{9\,{x}^{9}}\sqrt{b{x}^{4}+a}}-{\frac{11\,fb}{45\,{x}^{5}}\sqrt{b{x}^{4}+a}}-{\frac{4\,{b}^{2}f}{15\,ax}\sqrt{b{x}^{4}+a}}+{{\frac{4\,i}{15}}f{b}^{{\frac{5}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{{\frac{4\,i}{15}}f{b}^{{\frac{5}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticE} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \]

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^13,x)

[Out]

-1/12*c*a/x^12*(b*x^4+a)^(1/2)-7/48*c*b/x^8*(b*x^4+a)^(1/2)-1/32*b^2*c*(b*x^4+a)
^(1/2)/a/x^4+1/32*c/a^(3/2)*b^3*ln((2*a+2*a^(1/2)*(b*x^4+a)^(1/2))/x^2)-1/11*d*a
*(b*x^4+a)^(1/2)/x^11-13/77*d*b*(b*x^4+a)^(1/2)/x^7-4/77*b^2*d*(b*x^4+a)^(1/2)/a
/x^3-4/77*d/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*e*(b*x^4+a)^(1/2)/x^10/a*(b^2*x^8+2*a*b*x^4+a^2)-1/9*f*a*(b*x^4+a)^(1/
2)/x^9-11/45*f*b*(b*x^4+a)^(1/2)/x^5-4/15*b^2*f*(b*x^4+a)^(1/2)/a/x+4/15*I*f/a^(
1/2)*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)-
4/15*I*f/a^(1/2)*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)*EllipticE(x*(I/a^(1/2)*b^(1/2
))^(1/2),I)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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^13,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b f x^{7} + b e x^{6} + b d x^{5} + b c x^{4} + a f x^{3} + a e x^{2} + a d x + a c\right )} \sqrt{b x^{4} + a}}{x^{13}}, x\right ) \]

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^13,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^13, x)

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Sympy [A]  time = 37.6741, size = 403, normalized size = 0.9 \[ \frac{a^{\frac{3}{2}} d \Gamma \left (- \frac{11}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{11}{4}, - \frac{1}{2} \\ - \frac{7}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{11} \Gamma \left (- \frac{7}{4}\right )} + \frac{a^{\frac{3}{2}} f \Gamma \left (- \frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{9}{4}, - \frac{1}{2} \\ - \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{9} \Gamma \left (- \frac{5}{4}\right )} + \frac{\sqrt{a} b d \Gamma \left (- \frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{7}{4}, - \frac{1}{2} \\ - \frac{3}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{7} \Gamma \left (- \frac{3}{4}\right )} + \frac{\sqrt{a} b f \Gamma \left (- \frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, - \frac{1}{2} \\ - \frac{1}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{5} \Gamma \left (- \frac{1}{4}\right )} - \frac{a^{2} c}{12 \sqrt{b} x^{14} \sqrt{\frac{a}{b x^{4}} + 1}} - \frac{11 a \sqrt{b} c}{48 x^{10} \sqrt{\frac{a}{b x^{4}} + 1}} - \frac{a \sqrt{b} e \sqrt{\frac{a}{b x^{4}} + 1}}{10 x^{8}} - \frac{17 b^{\frac{3}{2}} c}{96 x^{6} \sqrt{\frac{a}{b x^{4}} + 1}} - \frac{b^{\frac{3}{2}} e \sqrt{\frac{a}{b x^{4}} + 1}}{5 x^{4}} - \frac{b^{\frac{5}{2}} c}{32 a x^{2} \sqrt{\frac{a}{b x^{4}} + 1}} - \frac{b^{\frac{5}{2}} e \sqrt{\frac{a}{b x^{4}} + 1}}{10 a} + \frac{b^{3} c \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{32 a^{\frac{3}{2}}} \]

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**13,x)

[Out]

a**(3/2)*d*gamma(-11/4)*hyper((-11/4, -1/2), (-7/4,), b*x**4*exp_polar(I*pi)/a)/
(4*x**11*gamma(-7/4)) + a**(3/2)*f*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*d*gamma(-7/4)*hyper((-7/4
, -1/2), (-3/4,), b*x**4*exp_polar(I*pi)/a)/(4*x**7*gamma(-3/4)) + sqrt(a)*b*f*g
amma(-5/4)*hyper((-5/4, -1/2), (-1/4,), b*x**4*exp_polar(I*pi)/a)/(4*x**5*gamma(
-1/4)) - a**2*c/(12*sqrt(b)*x**14*sqrt(a/(b*x**4) + 1)) - 11*a*sqrt(b)*c/(48*x**
10*sqrt(a/(b*x**4) + 1)) - a*sqrt(b)*e*sqrt(a/(b*x**4) + 1)/(10*x**8) - 17*b**(3
/2)*c/(96*x**6*sqrt(a/(b*x**4) + 1)) - b**(3/2)*e*sqrt(a/(b*x**4) + 1)/(5*x**4)
- b**(5/2)*c/(32*a*x**2*sqrt(a/(b*x**4) + 1)) - b**(5/2)*e*sqrt(a/(b*x**4) + 1)/
(10*a) + b**3*c*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. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{3}{2}}{\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{13}}\,{d x} \]

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^13,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(3/2)*(f*x^3 + e*x^2 + d*x + c)/x^13, x)