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

Optimal. Leaf size=409 \[ \frac{2 a^{9/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (77 \sqrt{b} d-15 \sqrt{a} f\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{1155 b^{5/4} \sqrt{a+b x^4}}-\frac{4 a^{9/4} d \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 b^{3/4} \sqrt{a+b x^4}}+\frac{3 a^2 c \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{16 \sqrt{b}}+\frac{4 a^2 d x \sqrt{a+b x^4}}{15 \sqrt{b} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{4 a^2 f x \sqrt{a+b x^4}}{77 b}+\frac{1}{8} c x^2 \left (a+b x^4\right )^{3/2}+\frac{3}{16} a c x^2 \sqrt{a+b x^4}+\frac{1}{99} x^3 \left (a+b x^4\right )^{3/2} \left (11 d+9 f x^2\right )+\frac{2 a x^3 \sqrt{a+b x^4} \left (77 d+45 f x^2\right )}{1155}+\frac{e \left (a+b x^4\right )^{5/2}}{10 b} \]

[Out]

(4*a^2*f*x*Sqrt[a + b*x^4])/(77*b) + (3*a*c*x^2*Sqrt[a + b*x^4])/16 + (4*a^2*d*x
*Sqrt[a + b*x^4])/(15*Sqrt[b]*(Sqrt[a] + Sqrt[b]*x^2)) + (2*a*x^3*(77*d + 45*f*x
^2)*Sqrt[a + b*x^4])/1155 + (c*x^2*(a + b*x^4)^(3/2))/8 + (x^3*(11*d + 9*f*x^2)*
(a + b*x^4)^(3/2))/99 + (e*(a + b*x^4)^(5/2))/(10*b) + (3*a^2*c*ArcTanh[(Sqrt[b]
*x^2)/Sqrt[a + b*x^4]])/(16*Sqrt[b]) - (4*a^(9/4)*d*(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*b^(3/4)*Sqrt[a + b*x^4]) + (2*a^(9/4)*(77*Sqrt[b]*d - 15*Sqrt[a]*f)*(
Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2*A
rcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(1155*b^(5/4)*Sqrt[a + b*x^4])

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Rubi [A]  time = 0.909908, antiderivative size = 409, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.393 \[ \frac{2 a^{9/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (77 \sqrt{b} d-15 \sqrt{a} f\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{1155 b^{5/4} \sqrt{a+b x^4}}-\frac{4 a^{9/4} d \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 b^{3/4} \sqrt{a+b x^4}}+\frac{3 a^2 c \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{16 \sqrt{b}}+\frac{4 a^2 d x \sqrt{a+b x^4}}{15 \sqrt{b} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{4 a^2 f x \sqrt{a+b x^4}}{77 b}+\frac{1}{8} c x^2 \left (a+b x^4\right )^{3/2}+\frac{3}{16} a c x^2 \sqrt{a+b x^4}+\frac{1}{99} x^3 \left (a+b x^4\right )^{3/2} \left (11 d+9 f x^2\right )+\frac{2 a x^3 \sqrt{a+b x^4} \left (77 d+45 f x^2\right )}{1155}+\frac{e \left (a+b x^4\right )^{5/2}}{10 b} \]

Antiderivative was successfully verified.

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

[Out]

(4*a^2*f*x*Sqrt[a + b*x^4])/(77*b) + (3*a*c*x^2*Sqrt[a + b*x^4])/16 + (4*a^2*d*x
*Sqrt[a + b*x^4])/(15*Sqrt[b]*(Sqrt[a] + Sqrt[b]*x^2)) + (2*a*x^3*(77*d + 45*f*x
^2)*Sqrt[a + b*x^4])/1155 + (c*x^2*(a + b*x^4)^(3/2))/8 + (x^3*(11*d + 9*f*x^2)*
(a + b*x^4)^(3/2))/99 + (e*(a + b*x^4)^(5/2))/(10*b) + (3*a^2*c*ArcTanh[(Sqrt[b]
*x^2)/Sqrt[a + b*x^4]])/(16*Sqrt[b]) - (4*a^(9/4)*d*(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*b^(3/4)*Sqrt[a + b*x^4]) + (2*a^(9/4)*(77*Sqrt[b]*d - 15*Sqrt[a]*f)*(
Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2*A
rcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(1155*b^(5/4)*Sqrt[a + b*x^4])

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Rubi in Sympy [A]  time = 85.4895, size = 382, normalized size = 0.93 \[ - \frac{4 a^{\frac{9}{4}} d \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{15 b^{\frac{3}{4}} \sqrt{a + b x^{4}}} - \frac{2 a^{\frac{9}{4}} \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) \left (15 \sqrt{a} f - 77 \sqrt{b} d\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{1155 b^{\frac{5}{4}} \sqrt{a + b x^{4}}} + \frac{4 a^{2} f x \sqrt{a + b x^{4}}}{77 b} + \frac{3 a^{2} c \operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a + b x^{4}}} \right )}}{16 \sqrt{b}} + \frac{4 a^{2} d x \sqrt{a + b x^{4}}}{15 \sqrt{b} \left (\sqrt{a} + \sqrt{b} x^{2}\right )} + \frac{3 a c x^{2} \sqrt{a + b x^{4}}}{16} + \frac{2 a x^{3} \sqrt{a + b x^{4}} \left (77 d + 45 f x^{2}\right )}{1155} + \frac{c x^{2} \left (a + b x^{4}\right )^{\frac{3}{2}}}{8} + \frac{x^{3} \left (a + b x^{4}\right )^{\frac{3}{2}} \left (11 d + 9 f x^{2}\right )}{99} + \frac{e \left (a + b x^{4}\right )^{\frac{5}{2}}}{10 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*a**(9/4)*d*sqrt((a + b*x**4)/(sqrt(a) + sqrt(b)*x**2)**2)*(sqrt(a) + sqrt(b)*
x**2)*elliptic_e(2*atan(b**(1/4)*x/a**(1/4)), 1/2)/(15*b**(3/4)*sqrt(a + b*x**4)
) - 2*a**(9/4)*sqrt((a + b*x**4)/(sqrt(a) + sqrt(b)*x**2)**2)*(sqrt(a) + sqrt(b)
*x**2)*(15*sqrt(a)*f - 77*sqrt(b)*d)*elliptic_f(2*atan(b**(1/4)*x/a**(1/4)), 1/2
)/(1155*b**(5/4)*sqrt(a + b*x**4)) + 4*a**2*f*x*sqrt(a + b*x**4)/(77*b) + 3*a**2
*c*atanh(sqrt(b)*x**2/sqrt(a + b*x**4))/(16*sqrt(b)) + 4*a**2*d*x*sqrt(a + b*x**
4)/(15*sqrt(b)*(sqrt(a) + sqrt(b)*x**2)) + 3*a*c*x**2*sqrt(a + b*x**4)/16 + 2*a*
x**3*sqrt(a + b*x**4)*(77*d + 45*f*x**2)/1155 + c*x**2*(a + b*x**4)**(3/2)/8 + x
**3*(a + b*x**4)**(3/2)*(11*d + 9*f*x**2)/99 + e*(a + b*x**4)**(5/2)/(10*b)

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Mathematica [C]  time = 0.836326, size = 302, normalized size = 0.74 \[ \frac{192 i a^{5/2} \sqrt{\frac{b x^4}{a}+1} \left (15 \sqrt{a} f+77 i \sqrt{b} d\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )+14784 a^{5/2} \sqrt{b} d \sqrt{\frac{b x^4}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )+\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \left (\left (a+b x^4\right ) \left (72 a^2 (77 e+40 f x)+a b x^2 (17325 c+16 x (847 d+9 x (77 e+65 f x)))+14 b^2 x^6 \left (495 c+4 x \left (110 d+99 e x+90 f x^2\right )\right )\right )+10395 a^2 \sqrt{b} c \sqrt{a+b x^4} \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )\right )}{55440 b \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(I*Sqrt[b])/Sqrt[a]]*((a + b*x^4)*(72*a^2*(77*e + 40*f*x) + 14*b^2*x^6*(49
5*c + 4*x*(110*d + 99*e*x + 90*f*x^2)) + a*b*x^2*(17325*c + 16*x*(847*d + 9*x*(7
7*e + 65*f*x)))) + 10395*a^2*Sqrt[b]*c*Sqrt[a + b*x^4]*ArcTanh[(Sqrt[b]*x^2)/Sqr
t[a + b*x^4]]) + 14784*a^(5/2)*Sqrt[b]*d*Sqrt[1 + (b*x^4)/a]*EllipticE[I*ArcSinh
[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1] + (192*I)*a^(5/2)*((77*I)*Sqrt[b]*d + 15*Sqrt
[a]*f)*Sqrt[1 + (b*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1]
)/(55440*Sqrt[(I*Sqrt[b])/Sqrt[a]]*b*Sqrt[a + b*x^4])

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Maple [C]  time = 0.019, size = 392, normalized size = 1. \[{\frac{bd{x}^{7}}{9}\sqrt{b{x}^{4}+a}}+{\frac{11\,ad{x}^{3}}{45}\sqrt{b{x}^{4}+a}}+{{\frac{4\,i}{15}}d{a}^{{\frac{5}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}{\frac{1}{\sqrt{b}}}}-{{\frac{4\,i}{15}}d{a}^{{\frac{5}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticE} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}{\frac{1}{\sqrt{b}}}}+{\frac{3\,{a}^{2}c}{16}\ln \left ( \sqrt{b}{x}^{2}+\sqrt{b{x}^{4}+a} \right ){\frac{1}{\sqrt{b}}}}+{\frac{bc{x}^{6}}{8}\sqrt{b{x}^{4}+a}}+{\frac{5\,ac{x}^{2}}{16}\sqrt{b{x}^{4}+a}}+{\frac{e}{10\,b} \left ( b{x}^{4}+a \right ) ^{{\frac{5}{2}}}}+{\frac{bf{x}^{9}}{11}\sqrt{b{x}^{4}+a}}+{\frac{13\,af{x}^{5}}{77}\sqrt{b{x}^{4}+a}}+{\frac{4\,{a}^{2}fx}{77\,b}\sqrt{b{x}^{4}+a}}-{\frac{4\,{a}^{3}f}{77\,b}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9*d*b*x^7*(b*x^4+a)^(1/2)+11/45*d*a*x^3*(b*x^4+a)^(1/2)+4/15*I*d*a^(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)/b^(1/2)*EllipticF(x*(I/a^(1/2)*b^(1/2))^(1/2),I)-4/15*I*d*a
^(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)/b^(1/2)*EllipticE(x*(I/a^(1/2)*b^(1/2))^(1/2),I
)+3/16*c*a^2*ln(b^(1/2)*x^2+(b*x^4+a)^(1/2))/b^(1/2)+1/8*c*b*x^6*(b*x^4+a)^(1/2)
+5/16*a*c*x^2*(b*x^4+a)^(1/2)+1/10*e*(b*x^4+a)^(5/2)/b+1/11*f*b*x^9*(b*x^4+a)^(1
/2)+13/77*f*a*x^5*(b*x^4+a)^(1/2)+4/77*a^2*f*x*(b*x^4+a)^(1/2)/b-4/77*f/b*a^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)

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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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b f x^{8} + b e x^{7} + b d x^{6} + b c x^{5} + a f x^{4} + a e x^{3} + a d x^{2} + a c x\right )} \sqrt{b x^{4} + a}, x\right ) \]

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,x, algorithm="fricas")

[Out]

integral((b*f*x^8 + b*e*x^7 + b*d*x^6 + b*c*x^5 + a*f*x^4 + a*e*x^3 + a*d*x^2 +
a*c*x)*sqrt(b*x^4 + a), x)

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Sympy [A]  time = 14.8972, size = 396, normalized size = 0.97 \[ \frac{a^{\frac{3}{2}} c x^{2} \sqrt{1 + \frac{b x^{4}}{a}}}{4} + \frac{a^{\frac{3}{2}} c x^{2}}{16 \sqrt{1 + \frac{b x^{4}}{a}}} + \frac{a^{\frac{3}{2}} d x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{7}{4}\right )} + \frac{a^{\frac{3}{2}} f x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{9}{4}\right )} + \frac{3 \sqrt{a} b c x^{6}}{16 \sqrt{1 + \frac{b x^{4}}{a}}} + \frac{\sqrt{a} b d x^{7} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{11}{4}\right )} + \frac{\sqrt{a} b f x^{9} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{13}{4}\right )} + \frac{3 a^{2} c \operatorname{asinh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{16 \sqrt{b}} + a e \left (\begin{cases} \frac{\sqrt{a} x^{4}}{4} & \text{for}\: b = 0 \\\frac{\left (a + b x^{4}\right )^{\frac{3}{2}}}{6 b} & \text{otherwise} \end{cases}\right ) + b e \left (\begin{cases} - \frac{a^{2} \sqrt{a + b x^{4}}}{15 b^{2}} + \frac{a x^{4} \sqrt{a + b x^{4}}}{30 b} + \frac{x^{8} \sqrt{a + b x^{4}}}{10} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} x^{8}}{8} & \text{otherwise} \end{cases}\right ) + \frac{b^{2} c x^{10}}{8 \sqrt{a} \sqrt{1 + \frac{b x^{4}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(3/2)*c*x**2*sqrt(1 + b*x**4/a)/4 + a**(3/2)*c*x**2/(16*sqrt(1 + b*x**4/a)) +
a**(3/2)*d*x**3*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), b*x**4*exp_polar(I*pi)/a)
/(4*gamma(7/4)) + a**(3/2)*f*x**5*gamma(5/4)*hyper((-1/2, 5/4), (9/4,), b*x**4*e
xp_polar(I*pi)/a)/(4*gamma(9/4)) + 3*sqrt(a)*b*c*x**6/(16*sqrt(1 + b*x**4/a)) +
sqrt(a)*b*d*x**7*gamma(7/4)*hyper((-1/2, 7/4), (11/4,), b*x**4*exp_polar(I*pi)/a
)/(4*gamma(11/4)) + sqrt(a)*b*f*x**9*gamma(9/4)*hyper((-1/2, 9/4), (13/4,), b*x*
*4*exp_polar(I*pi)/a)/(4*gamma(13/4)) + 3*a**2*c*asinh(sqrt(b)*x**2/sqrt(a))/(16
*sqrt(b)) + a*e*Piecewise((sqrt(a)*x**4/4, Eq(b, 0)), ((a + b*x**4)**(3/2)/(6*b)
, True)) + b*e*Piecewise((-a**2*sqrt(a + b*x**4)/(15*b**2) + a*x**4*sqrt(a + b*x
**4)/(30*b) + x**8*sqrt(a + b*x**4)/10, Ne(b, 0)), (sqrt(a)*x**8/8, True)) + b**
2*c*x**10/(8*sqrt(a)*sqrt(1 + b*x**4/a))

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

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,x, algorithm="giac")

[Out]

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