3.508 \(\int \frac{\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^6} \, dx\)

Optimal. Leaf size=387 \[ \frac{2 \sqrt [4]{a} b^{3/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (5 \sqrt{a} e+9 \sqrt{b} c\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 \sqrt{a+b x^4}}+\frac{12 b^{3/2} c x \sqrt{a+b x^4}}{5 \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{12 \sqrt [4]{a} b^{5/4} c \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 \sqrt{a+b x^4}}-\frac{1}{60} \left (a+b x^4\right )^{3/2} \left (\frac{12 c}{x^5}+\frac{15 d}{x^4}+\frac{20 e}{x^3}+\frac{30 f}{x^2}\right )-\frac{2 b \sqrt{a+b x^4} \left (9 c-5 e x^2\right )}{15 x}+\frac{3}{4} b \sqrt{a+b x^4} \left (d+f x^2\right )-\frac{3}{4} \sqrt{a} b d \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )+\frac{3}{4} a \sqrt{b} f \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right ) \]

[Out]

(12*b^(3/2)*c*x*Sqrt[a + b*x^4])/(5*(Sqrt[a] + Sqrt[b]*x^2)) - (2*b*(9*c - 5*e*x
^2)*Sqrt[a + b*x^4])/(15*x) + (3*b*(d + f*x^2)*Sqrt[a + b*x^4])/4 - (((12*c)/x^5
+ (15*d)/x^4 + (20*e)/x^3 + (30*f)/x^2)*(a + b*x^4)^(3/2))/60 + (3*a*Sqrt[b]*f*
ArcTanh[(Sqrt[b]*x^2)/Sqrt[a + b*x^4]])/4 - (3*Sqrt[a]*b*d*ArcTanh[Sqrt[a + b*x^
4]/Sqrt[a]])/4 - (12*a^(1/4)*b^(5/4)*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])/(5*Sqr
t[a + b*x^4]) + (2*a^(1/4)*b^(3/4)*(9*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])/(15*Sqrt[a + b*x^4])

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Rubi [A]  time = 0.809648, antiderivative size = 387, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 15, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ \frac{2 \sqrt [4]{a} b^{3/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (5 \sqrt{a} e+9 \sqrt{b} c\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 \sqrt{a+b x^4}}+\frac{12 b^{3/2} c x \sqrt{a+b x^4}}{5 \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{12 \sqrt [4]{a} b^{5/4} c \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 \sqrt{a+b x^4}}-\frac{1}{60} \left (a+b x^4\right )^{3/2} \left (\frac{12 c}{x^5}+\frac{15 d}{x^4}+\frac{20 e}{x^3}+\frac{30 f}{x^2}\right )-\frac{2 b \sqrt{a+b x^4} \left (9 c-5 e x^2\right )}{15 x}+\frac{3}{4} b \sqrt{a+b x^4} \left (d+f x^2\right )-\frac{3}{4} \sqrt{a} b d \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )+\frac{3}{4} a \sqrt{b} f \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[((c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2))/x^6,x]

[Out]

(12*b^(3/2)*c*x*Sqrt[a + b*x^4])/(5*(Sqrt[a] + Sqrt[b]*x^2)) - (2*b*(9*c - 5*e*x
^2)*Sqrt[a + b*x^4])/(15*x) + (3*b*(d + f*x^2)*Sqrt[a + b*x^4])/4 - (((12*c)/x^5
+ (15*d)/x^4 + (20*e)/x^3 + (30*f)/x^2)*(a + b*x^4)^(3/2))/60 + (3*a*Sqrt[b]*f*
ArcTanh[(Sqrt[b]*x^2)/Sqrt[a + b*x^4]])/4 - (3*Sqrt[a]*b*d*ArcTanh[Sqrt[a + b*x^
4]/Sqrt[a]])/4 - (12*a^(1/4)*b^(5/4)*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])/(5*Sqr
t[a + b*x^4]) + (2*a^(1/4)*b^(3/4)*(9*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])/(15*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**6,x)

[Out]

Timed out

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Mathematica [C]  time = 1.51494, size = 331, normalized size = 0.86 \[ \frac{144 \sqrt{a} b^{3/2} c x^5 \sqrt{\frac{b x^4}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )-\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \left (\left (a+b x^4\right ) \left (12 a c+5 a x \left (3 d+4 e x+6 f x^2\right )+84 b c x^4-5 b x^5 (6 d+x (4 e+3 f x))\right )+45 \sqrt{a} b d x^5 \sqrt{a+b x^4} \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )-45 a \sqrt{b} f x^5 \sqrt{a+b x^4} \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )\right )-16 i \sqrt{a} b x^5 \sqrt{\frac{b x^4}{a}+1} \left (5 \sqrt{a} e-9 i \sqrt{b} c\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )}{60 x^5 \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[((c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2))/x^6,x]

[Out]

(-(Sqrt[(I*Sqrt[b])/Sqrt[a]]*((a + b*x^4)*(12*a*c + 84*b*c*x^4 + 5*a*x*(3*d + 4*
e*x + 6*f*x^2) - 5*b*x^5*(6*d + x*(4*e + 3*f*x))) - 45*a*Sqrt[b]*f*x^5*Sqrt[a +
b*x^4]*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a + b*x^4]] + 45*Sqrt[a]*b*d*x^5*Sqrt[a + b*x^
4]*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]])) + 144*Sqrt[a]*b^(3/2)*c*x^5*Sqrt[1 + (b*x^
4)/a]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1] - (16*I)*Sqrt[a]*b*(
(-9*I)*Sqrt[b]*c + 5*Sqrt[a]*e)*x^5*Sqrt[1 + (b*x^4)/a]*EllipticF[I*ArcSinh[Sqrt
[(I*Sqrt[b])/Sqrt[a]]*x], -1])/(60*Sqrt[(I*Sqrt[b])/Sqrt[a]]*x^5*Sqrt[a + b*x^4]
)

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Maple [C]  time = 0.024, size = 409, normalized size = 1.1 \[ -{\frac{ac}{5\,{x}^{5}}\sqrt{b{x}^{4}+a}}-{\frac{7\,bc}{5\,x}\sqrt{b{x}^{4}+a}}+{{\frac{12\,i}{5}}c{b}^{{\frac{3}{2}}}\sqrt{a}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{{\frac{12\,i}{5}}c{b}^{{\frac{3}{2}}}\sqrt{a}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticE} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}+{\frac{bd}{2}\sqrt{b{x}^{4}+a}}-{\frac{ad}{4\,{x}^{4}}\sqrt{b{x}^{4}+a}}-{\frac{3\,bd}{4}\sqrt{a}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{4}+a} \right ) } \right ) }-{\frac{ae}{3\,{x}^{3}}\sqrt{b{x}^{4}+a}}+{\frac{bex}{3}\sqrt{b{x}^{4}+a}}+{\frac{4\,bea}{3}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}+{\frac{bf{x}^{2}}{4}\sqrt{b{x}^{4}+a}}+{\frac{3\,af}{4}\sqrt{b}\ln \left ( \sqrt{b}{x}^{2}+\sqrt{b{x}^{4}+a} \right ) }-{\frac{af}{2\,{x}^{2}}\sqrt{b{x}^{4}+a}} \]

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^6,x)

[Out]

-1/5*c*a*(b*x^4+a)^(1/2)/x^5-7/5*c*b*(b*x^4+a)^(1/2)/x+12/5*I*c*b^(3/2)*a^(1/2)/
(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)-12/5*I*c*b^(3
/2)*a^(1/2)/(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
/2*d*b*(b*x^4+a)^(1/2)-1/4*d*a/x^4*(b*x^4+a)^(1/2)-3/4*d*a^(1/2)*b*ln((2*a+2*a^(
1/2)*(b*x^4+a)^(1/2))/x^2)-1/3*e*a*(b*x^4+a)^(1/2)/x^3+1/3*e*b*x*(b*x^4+a)^(1/2)
+4/3*e*a*b/(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/
4*f*b*x^2*(b*x^4+a)^(1/2)+3/4*f*b^(1/2)*a*ln(b^(1/2)*x^2+(b*x^4+a)^(1/2))-1/2*f*
a/x^2*(b*x^4+a)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{3}{2}}{\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{6}}\,{d x} \]

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^6,x, algorithm="maxima")

[Out]

integrate((b*x^4 + a)^(3/2)*(f*x^3 + e*x^2 + d*x + c)/x^6, x)

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

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^6,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^6, x)

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Sympy [A]  time = 16.0485, size = 386, normalized size = 1. \[ \frac{a^{\frac{3}{2}} c \Gamma \left (- \frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, - \frac{1}{2} \\ - \frac{1}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{5} \Gamma \left (- \frac{1}{4}\right )} + \frac{a^{\frac{3}{2}} e \Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, - \frac{1}{2} \\ \frac{1}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{3} \Gamma \left (\frac{1}{4}\right )} - \frac{a^{\frac{3}{2}} f}{2 x^{2} \sqrt{1 + \frac{b x^{4}}{a}}} + \frac{\sqrt{a} b c \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4} \\ \frac{3}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 x \Gamma \left (\frac{3}{4}\right )} - \frac{3 \sqrt{a} b d \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{4} + \frac{\sqrt{a} b e x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} + \frac{\sqrt{a} b f x^{2} \sqrt{1 + \frac{b x^{4}}{a}}}{4} - \frac{\sqrt{a} b f x^{2}}{2 \sqrt{1 + \frac{b x^{4}}{a}}} - \frac{a \sqrt{b} d \sqrt{\frac{a}{b x^{4}} + 1}}{4 x^{2}} + \frac{a \sqrt{b} d}{2 x^{2} \sqrt{\frac{a}{b x^{4}} + 1}} + \frac{3 a \sqrt{b} f \operatorname{asinh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{4} + \frac{b^{\frac{3}{2}} d x^{2}}{2 \sqrt{\frac{a}{b x^{4}} + 1}} \]

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**6,x)

[Out]

a**(3/2)*c*gamma(-5/4)*hyper((-5/4, -1/2), (-1/4,), b*x**4*exp_polar(I*pi)/a)/(4
*x**5*gamma(-1/4)) + a**(3/2)*e*gamma(-3/4)*hyper((-3/4, -1/2), (1/4,), b*x**4*e
xp_polar(I*pi)/a)/(4*x**3*gamma(1/4)) - a**(3/2)*f/(2*x**2*sqrt(1 + b*x**4/a)) +
sqrt(a)*b*c*gamma(-1/4)*hyper((-1/2, -1/4), (3/4,), b*x**4*exp_polar(I*pi)/a)/(
4*x*gamma(3/4)) - 3*sqrt(a)*b*d*asinh(sqrt(a)/(sqrt(b)*x**2))/4 + sqrt(a)*b*e*x*
gamma(1/4)*hyper((-1/2, 1/4), (5/4,), b*x**4*exp_polar(I*pi)/a)/(4*gamma(5/4)) +
sqrt(a)*b*f*x**2*sqrt(1 + b*x**4/a)/4 - sqrt(a)*b*f*x**2/(2*sqrt(1 + b*x**4/a))
- a*sqrt(b)*d*sqrt(a/(b*x**4) + 1)/(4*x**2) + a*sqrt(b)*d/(2*x**2*sqrt(a/(b*x**
4) + 1)) + 3*a*sqrt(b)*f*asinh(sqrt(b)*x**2/sqrt(a))/4 + b**(3/2)*d*x**2/(2*sqrt
(a/(b*x**4) + 1))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{3}{2}}{\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{6}}\,{d x} \]

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^6,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(3/2)*(f*x^3 + e*x^2 + d*x + c)/x^6, x)