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

Optimal. Leaf size=427 \[ \frac{2 a^{9/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (77 \sqrt{b} c-15 \sqrt{a} e\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{1155 b^{5/4} \sqrt{a+b x^4}}-\frac{4 a^{9/4} c \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 b^{3/4} \sqrt{a+b x^4}}-\frac{a^3 f \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{32 b^{3/2}}+\frac{4 a^2 c x \sqrt{a+b x^4}}{15 \sqrt{b} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{4 a^2 e x \sqrt{a+b x^4}}{77 b}-\frac{a^2 f x^2 \sqrt{a+b x^4}}{32 b}+\frac{2 a x^3 \sqrt{a+b x^4} \left (77 c+45 e x^2\right )}{1155}+\frac{1}{99} x^3 \left (a+b x^4\right )^{3/2} \left (11 c+9 e x^2\right )+\frac{\left (a+b x^4\right )^{5/2} \left (6 d+5 f x^2\right )}{60 b}-\frac{a f x^2 \left (a+b x^4\right )^{3/2}}{48 b} \]

[Out]

(4*a^2*e*x*Sqrt[a + b*x^4])/(77*b) - (a^2*f*x^2*Sqrt[a + b*x^4])/(32*b) + (4*a^2
*c*x*Sqrt[a + b*x^4])/(15*Sqrt[b]*(Sqrt[a] + Sqrt[b]*x^2)) + (2*a*x^3*(77*c + 45
*e*x^2)*Sqrt[a + b*x^4])/1155 - (a*f*x^2*(a + b*x^4)^(3/2))/(48*b) + (x^3*(11*c
+ 9*e*x^2)*(a + b*x^4)^(3/2))/99 + ((6*d + 5*f*x^2)*(a + b*x^4)^(5/2))/(60*b) -
(a^3*f*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a + b*x^4]])/(32*b^(3/2)) - (4*a^(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*b^(3/4)*Sqrt[a + b*x^4]) + (2*a^(9/4)*(77*Sqrt
[b]*c - 15*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])/(1155*b^(5/4)*Sqrt[a +
b*x^4])

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Rubi [A]  time = 1.02365, antiderivative size = 427, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.367 \[ \frac{2 a^{9/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (77 \sqrt{b} c-15 \sqrt{a} e\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{1155 b^{5/4} \sqrt{a+b x^4}}-\frac{4 a^{9/4} c \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 b^{3/4} \sqrt{a+b x^4}}-\frac{a^3 f \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{32 b^{3/2}}+\frac{4 a^2 c x \sqrt{a+b x^4}}{15 \sqrt{b} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{4 a^2 e x \sqrt{a+b x^4}}{77 b}-\frac{a^2 f x^2 \sqrt{a+b x^4}}{32 b}+\frac{2 a x^3 \sqrt{a+b x^4} \left (77 c+45 e x^2\right )}{1155}+\frac{1}{99} x^3 \left (a+b x^4\right )^{3/2} \left (11 c+9 e x^2\right )+\frac{\left (a+b x^4\right )^{5/2} \left (6 d+5 f x^2\right )}{60 b}-\frac{a f x^2 \left (a+b x^4\right )^{3/2}}{48 b} \]

Antiderivative was successfully verified.

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

[Out]

(4*a^2*e*x*Sqrt[a + b*x^4])/(77*b) - (a^2*f*x^2*Sqrt[a + b*x^4])/(32*b) + (4*a^2
*c*x*Sqrt[a + b*x^4])/(15*Sqrt[b]*(Sqrt[a] + Sqrt[b]*x^2)) + (2*a*x^3*(77*c + 45
*e*x^2)*Sqrt[a + b*x^4])/1155 - (a*f*x^2*(a + b*x^4)^(3/2))/(48*b) + (x^3*(11*c
+ 9*e*x^2)*(a + b*x^4)^(3/2))/99 + ((6*d + 5*f*x^2)*(a + b*x^4)^(5/2))/(60*b) -
(a^3*f*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a + b*x^4]])/(32*b^(3/2)) - (4*a^(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*b^(3/4)*Sqrt[a + b*x^4]) + (2*a^(9/4)*(77*Sqrt
[b]*c - 15*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])/(1155*b^(5/4)*Sqrt[a +
b*x^4])

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Rubi in Sympy [A]  time = 91.5168, size = 394, normalized size = 0.92 \[ - \frac{4 a^{\frac{9}{4}} c \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{15 b^{\frac{3}{4}} \sqrt{a + b x^{4}}} - \frac{2 a^{\frac{9}{4}} \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) \left (15 \sqrt{a} e - 77 \sqrt{b} c\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{1155 b^{\frac{5}{4}} \sqrt{a + b x^{4}}} - \frac{a^{3} f \operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a + b x^{4}}} \right )}}{32 b^{\frac{3}{2}}} + \frac{4 a^{2} e x \sqrt{a + b x^{4}}}{77 b} - \frac{a^{2} f x^{2} \sqrt{a + b x^{4}}}{32 b} + \frac{4 a^{2} c x \sqrt{a + b x^{4}}}{15 \sqrt{b} \left (\sqrt{a} + \sqrt{b} x^{2}\right )} + \frac{2 a x^{3} \sqrt{a + b x^{4}} \left (77 c + 45 e x^{2}\right )}{1155} - \frac{a f x^{2} \left (a + b x^{4}\right )^{\frac{3}{2}}}{48 b} + \frac{x^{3} \left (a + b x^{4}\right )^{\frac{3}{2}} \left (11 c + 9 e x^{2}\right )}{99} + \frac{\left (a + b x^{4}\right )^{\frac{5}{2}} \left (6 d + 5 f x^{2}\right )}{60 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*a**(9/4)*c*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)*e - 77*sqrt(b)*c)*elliptic_f(2*atan(b**(1/4)*x/a**(1/4)), 1/2
)/(1155*b**(5/4)*sqrt(a + b*x**4)) - a**3*f*atanh(sqrt(b)*x**2/sqrt(a + b*x**4))
/(32*b**(3/2)) + 4*a**2*e*x*sqrt(a + b*x**4)/(77*b) - a**2*f*x**2*sqrt(a + b*x**
4)/(32*b) + 4*a**2*c*x*sqrt(a + b*x**4)/(15*sqrt(b)*(sqrt(a) + sqrt(b)*x**2)) +
2*a*x**3*sqrt(a + b*x**4)*(77*c + 45*e*x**2)/1155 - a*f*x**2*(a + b*x**4)**(3/2)
/(48*b) + x**3*(a + b*x**4)**(3/2)*(11*c + 9*e*x**2)/99 + (a + b*x**4)**(5/2)*(6
*d + 5*f*x**2)/(60*b)

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Mathematica [C]  time = 0.806616, size = 325, normalized size = 0.76 \[ -\frac{a^3 f \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{32 b^{3/2}}+\frac{4 i a^3 e \sqrt{\frac{b x^4}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )}{77 b \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \sqrt{a+b x^4}}+\frac{\sqrt{a+b x^4} \left (9 a^2 (1232 d+5 x (128 e+77 f x))+2 a b x^3 (13552 c+3 x (3696 d+5 x (624 e+539 f x)))+56 b^2 x^7 \left (220 c+3 x \left (66 d+60 e x+55 f x^2\right )\right )\right )}{110880 b}+\frac{4 i a^2 c \sqrt{\frac{b x^4}{a}+1} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )\right )}{15 \left (\frac{i \sqrt{b}}{\sqrt{a}}\right )^{3/2} \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x^4]*(9*a^2*(1232*d + 5*x*(128*e + 77*f*x)) + 56*b^2*x^7*(220*c + 3*
x*(66*d + 60*e*x + 55*f*x^2)) + 2*a*b*x^3*(13552*c + 3*x*(3696*d + 5*x*(624*e +
539*f*x)))))/(110880*b) - (a^3*f*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a + b*x^4]])/(32*b^(
3/2)) + (((4*I)/15)*a^2*c*Sqrt[1 + (b*x^4)/a]*(EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[
b])/Sqrt[a]]*x], -1] - EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1]))/(
((I*Sqrt[b])/Sqrt[a])^(3/2)*Sqrt[a + b*x^4]) + (((4*I)/77)*a^3*e*Sqrt[1 + (b*x^4
)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1])/(Sqrt[(I*Sqrt[b])/Sq
rt[a]]*b*Sqrt[a + b*x^4])

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Maple [C]  time = 0.017, size = 413, normalized size = 1. \[{\frac{d}{10\,b} \left ( b{x}^{4}+a \right ) ^{{\frac{5}{2}}}}+{\frac{bc{x}^{7}}{9}\sqrt{b{x}^{4}+a}}+{\frac{11\,ac{x}^{3}}{45}\sqrt{b{x}^{4}+a}}+{{\frac{4\,i}{15}}c{a}^{{\frac{5}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}{\frac{1}{\sqrt{b}}}}-{{\frac{4\,i}{15}}c{a}^{{\frac{5}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticE} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}{\frac{1}{\sqrt{b}}}}+{\frac{be{x}^{9}}{11}\sqrt{b{x}^{4}+a}}+{\frac{13\,ae{x}^{5}}{77}\sqrt{b{x}^{4}+a}}+{\frac{4\,{a}^{2}ex}{77\,b}\sqrt{b{x}^{4}+a}}-{\frac{4\,{a}^{3}e}{77\,b}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}+{\frac{{x}^{2}{a}^{2}f}{32\,b}\sqrt{b{x}^{4}+a}}-{\frac{{a}^{3}f}{32}\ln \left ( \sqrt{b}{x}^{2}+\sqrt{b{x}^{4}+a} \right ){b}^{-{\frac{3}{2}}}}+{\frac{bf{x}^{10}}{12}\sqrt{b{x}^{4}+a}}+{\frac{7\,af{x}^{6}}{48}\sqrt{b{x}^{4}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*d/b*(b*x^4+a)^(5/2)+1/9*c*b*x^7*(b*x^4+a)^(1/2)+11/45*c*a*x^3*(b*x^4+a)^(1/
2)+4/15*I*c*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*c*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)+1/11*e*b*x^9*(b*x^4+a)^(1/2)+13/77*e*a*x^5*(b*x^4+a)^
(1/2)+4/77*a^2*e*x*(b*x^4+a)^(1/2)/b-4/77*e/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)*Elli
pticF(x*(I/a^(1/2)*b^(1/2))^(1/2),I)+1/32*a^2*f*x^2*(b*x^4+a)^(1/2)/b-1/32*f*a^3
/b^(3/2)*ln(b^(1/2)*x^2+(b*x^4+a)^(1/2))+1/12*f*b*x^10*(b*x^4+a)^(1/2)+7/48*f*a*
x^6*(b*x^4+a)^(1/2)

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

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^2,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b f x^{9} + b e x^{8} + b d x^{7} + b c x^{6} + a f x^{5} + a e x^{4} + a d x^{3} + a c x^{2}\right )} \sqrt{b x^{4} + a}, x\right ) \]

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^2,x, algorithm="fricas")

[Out]

integral((b*f*x^9 + b*e*x^8 + b*d*x^7 + b*c*x^6 + a*f*x^5 + a*e*x^4 + a*d*x^3 +
a*c*x^2)*sqrt(b*x^4 + a), x)

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Sympy [A]  time = 22.1989, size = 398, normalized size = 0.93 \[ \frac{a^{\frac{5}{2}} f x^{2}}{32 b \sqrt{1 + \frac{b x^{4}}{a}}} + \frac{a^{\frac{3}{2}} c x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{7}{4}\right )} + \frac{a^{\frac{3}{2}} e x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{9}{4}\right )} + \frac{17 a^{\frac{3}{2}} f x^{6}}{96 \sqrt{1 + \frac{b x^{4}}{a}}} + \frac{\sqrt{a} b c x^{7} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{11}{4}\right )} + \frac{\sqrt{a} b e x^{9} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{13}{4}\right )} + \frac{11 \sqrt{a} b f x^{10}}{48 \sqrt{1 + \frac{b x^{4}}{a}}} - \frac{a^{3} f \operatorname{asinh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{32 b^{\frac{3}{2}}} + a d \left (\begin{cases} \frac{\sqrt{a} x^{4}}{4} & \text{for}\: b = 0 \\\frac{\left (a + b x^{4}\right )^{\frac{3}{2}}}{6 b} & \text{otherwise} \end{cases}\right ) + b d \left (\begin{cases} - \frac{a^{2} \sqrt{a + b x^{4}}}{15 b^{2}} + \frac{a x^{4} \sqrt{a + b x^{4}}}{30 b} + \frac{x^{8} \sqrt{a + b x^{4}}}{10} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} x^{8}}{8} & \text{otherwise} \end{cases}\right ) + \frac{b^{2} f x^{14}}{12 \sqrt{a} \sqrt{1 + \frac{b x^{4}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(5/2)*f*x**2/(32*b*sqrt(1 + b*x**4/a)) + a**(3/2)*c*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)*e*x**5*gam
ma(5/4)*hyper((-1/2, 5/4), (9/4,), b*x**4*exp_polar(I*pi)/a)/(4*gamma(9/4)) + 17
*a**(3/2)*f*x**6/(96*sqrt(1 + b*x**4/a)) + sqrt(a)*b*c*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*e*x**9*
gamma(9/4)*hyper((-1/2, 9/4), (13/4,), b*x**4*exp_polar(I*pi)/a)/(4*gamma(13/4))
+ 11*sqrt(a)*b*f*x**10/(48*sqrt(1 + b*x**4/a)) - a**3*f*asinh(sqrt(b)*x**2/sqrt
(a))/(32*b**(3/2)) + a*d*Piecewise((sqrt(a)*x**4/4, Eq(b, 0)), ((a + b*x**4)**(3
/2)/(6*b), True)) + b*d*Piecewise((-a**2*sqrt(a + b*x**4)/(15*b**2) + a*x**4*sqr
t(a + b*x**4)/(30*b) + x**8*sqrt(a + b*x**4)/10, Ne(b, 0)), (sqrt(a)*x**8/8, Tru
e)) + b**2*f*x**14/(12*sqrt(a)*sqrt(1 + b*x**4/a))

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

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^2,x, algorithm="giac")

[Out]

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