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

Optimal. Leaf size=645 \[ -\frac{x^2 \left (-6 a^2 c^2 e-12 a b^2 c e+b^4 (-e)+4 b c^3 d\right )}{2 e^2}-\frac{2 x \left (-6 a^2 b c e-2 a b^3 e+2 a c^3 d+3 b^2 c^2 d\right )}{e^2}+\frac{\log \left (d+e x^3\right ) \left (-4 c e \left (b^3 d-a^3 e\right )+6 a^2 b^2 e^2-12 a b c^2 d e+c^4 d^2\right )}{3 e^3}-\frac{\left (a \sqrt [3]{e}+b \sqrt [3]{d}\right ) \tan ^{-1}\left (\frac{\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right ) \left (-e \left (a^3 (-e)-3 a^2 b \sqrt [3]{d} e^{2/3}+3 a b^2 d^{2/3} \sqrt [3]{e}+b^3 d\right )+6 c^2 \left (b d^{5/3} \sqrt [3]{e}-a d^{4/3} e^{2/3}\right )-12 a b c d e+4 c^3 d^2\right )}{\sqrt{3} d^{2/3} e^{8/3}}-\frac{\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (\sqrt [3]{e} \left (a^4 e^2-12 a^2 b c d e-4 a b^3 d e+4 a c^3 d^2+6 b^2 c^2 d^2\right )+\sqrt [3]{d} \left (-4 b \left (a^3 e^2+c^3 d^2\right )+6 a^2 c^2 d e+12 a b^2 c d e+b^4 d e\right )\right )}{6 d^{2/3} e^{8/3}}+\frac{\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (\sqrt [3]{e} \left (a^4 e^2-12 a^2 b c d e-4 a b^3 d e+4 a c^3 d^2+6 b^2 c^2 d^2\right )+\sqrt [3]{d} \left (-4 b \left (a^3 e^2+c^3 d^2\right )+6 a^2 c^2 d e+12 a b^2 c d e+b^4 d e\right )\right )}{3 d^{2/3} e^{8/3}}-\frac{c x^3 \left (-12 a b c e-4 b^3 e+c^3 d\right )}{3 e^2}+\frac{c^2 x^4 \left (2 a c+3 b^2\right )}{2 e}+\frac{4 b c^3 x^5}{5 e}+\frac{c^4 x^6}{6 e} \]

[Out]

(-2*(3*b^2*c^2*d + 2*a*c^3*d - 2*a*b^3*e - 6*a^2*b*c*e)*x)/e^2 - ((4*b*c^3*d - b
^4*e - 12*a*b^2*c*e - 6*a^2*c^2*e)*x^2)/(2*e^2) - (c*(c^3*d - 4*b^3*e - 12*a*b*c
*e)*x^3)/(3*e^2) + (c^2*(3*b^2 + 2*a*c)*x^4)/(2*e) + (4*b*c^3*x^5)/(5*e) + (c^4*
x^6)/(6*e) - ((b*d^(1/3) + a*e^(1/3))*(4*c^3*d^2 + 6*c^2*(b*d^(5/3)*e^(1/3) - a*
d^(4/3)*e^(2/3)) - 12*a*b*c*d*e - e*(b^3*d + 3*a*b^2*d^(2/3)*e^(1/3) - 3*a^2*b*d
^(1/3)*e^(2/3) - a^3*e))*ArcTan[(d^(1/3) - 2*e^(1/3)*x)/(Sqrt[3]*d^(1/3))])/(Sqr
t[3]*d^(2/3)*e^(8/3)) + ((e^(1/3)*(6*b^2*c^2*d^2 + 4*a*c^3*d^2 - 4*a*b^3*d*e - 1
2*a^2*b*c*d*e + a^4*e^2) + d^(1/3)*(b^4*d*e + 12*a*b^2*c*d*e + 6*a^2*c^2*d*e - 4
*b*(c^3*d^2 + a^3*e^2)))*Log[d^(1/3) + e^(1/3)*x])/(3*d^(2/3)*e^(8/3)) - ((e^(1/
3)*(6*b^2*c^2*d^2 + 4*a*c^3*d^2 - 4*a*b^3*d*e - 12*a^2*b*c*d*e + a^4*e^2) + d^(1
/3)*(b^4*d*e + 12*a*b^2*c*d*e + 6*a^2*c^2*d*e - 4*b*(c^3*d^2 + a^3*e^2)))*Log[d^
(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/(6*d^(2/3)*e^(8/3)) + ((c^4*d^2 - 12*a
*b*c^2*d*e + 6*a^2*b^2*e^2 - 4*c*e*(b^3*d - a^3*e))*Log[d + e*x^3])/(3*e^3)

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Rubi [A]  time = 2.20409, antiderivative size = 643, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ -\frac{x^2 \left (-6 a^2 c^2 e-12 a b^2 c e+b^4 (-e)+4 b c^3 d\right )}{2 e^2}-\frac{2 x \left (-6 a^2 b c e-2 a b^3 e+2 a c^3 d+3 b^2 c^2 d\right )}{e^2}+\frac{\log \left (d+e x^3\right ) \left (-4 c e \left (b^3 d-a^3 e\right )+6 a^2 b^2 e^2-12 a b c^2 d e+c^4 d^2\right )}{3 e^3}-\frac{\left (a \sqrt [3]{e}+b \sqrt [3]{d}\right ) \tan ^{-1}\left (\frac{\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right ) \left (-e \left (a^3 (-e)-3 a^2 b \sqrt [3]{d} e^{2/3}+3 a b^2 d^{2/3} \sqrt [3]{e}+b^3 d\right )+6 c^2 \left (b d^{5/3} \sqrt [3]{e}-a d^{4/3} e^{2/3}\right )-12 a b c d e+4 c^3 d^2\right )}{\sqrt{3} d^{2/3} e^{8/3}}-\frac{\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (a^4 e^2-12 a^2 b c d e+\frac{\sqrt [3]{d} \left (-4 b \left (a^3 e^2+c^3 d^2\right )+6 a^2 c^2 d e+12 a b^2 c d e+b^4 d e\right )}{\sqrt [3]{e}}-4 a b^3 d e+4 a c^3 d^2+6 b^2 c^2 d^2\right )}{6 d^{2/3} e^{7/3}}+\frac{\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (\sqrt [3]{e} \left (a^4 e^2-12 a^2 b c d e-4 a b^3 d e+4 a c^3 d^2+6 b^2 c^2 d^2\right )+\sqrt [3]{d} \left (-4 b \left (a^3 e^2+c^3 d^2\right )+6 a^2 c^2 d e+12 a b^2 c d e+b^4 d e\right )\right )}{3 d^{2/3} e^{8/3}}-\frac{c x^3 \left (-12 a b c e-4 b^3 e+c^3 d\right )}{3 e^2}+\frac{c^2 x^4 \left (2 a c+3 b^2\right )}{2 e}+\frac{4 b c^3 x^5}{5 e}+\frac{c^4 x^6}{6 e} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(3*b^2*c^2*d + 2*a*c^3*d - 2*a*b^3*e - 6*a^2*b*c*e)*x)/e^2 - ((4*b*c^3*d - b
^4*e - 12*a*b^2*c*e - 6*a^2*c^2*e)*x^2)/(2*e^2) - (c*(c^3*d - 4*b^3*e - 12*a*b*c
*e)*x^3)/(3*e^2) + (c^2*(3*b^2 + 2*a*c)*x^4)/(2*e) + (4*b*c^3*x^5)/(5*e) + (c^4*
x^6)/(6*e) - ((b*d^(1/3) + a*e^(1/3))*(4*c^3*d^2 + 6*c^2*(b*d^(5/3)*e^(1/3) - a*
d^(4/3)*e^(2/3)) - 12*a*b*c*d*e - e*(b^3*d + 3*a*b^2*d^(2/3)*e^(1/3) - 3*a^2*b*d
^(1/3)*e^(2/3) - a^3*e))*ArcTan[(d^(1/3) - 2*e^(1/3)*x)/(Sqrt[3]*d^(1/3))])/(Sqr
t[3]*d^(2/3)*e^(8/3)) + ((e^(1/3)*(6*b^2*c^2*d^2 + 4*a*c^3*d^2 - 4*a*b^3*d*e - 1
2*a^2*b*c*d*e + a^4*e^2) + d^(1/3)*(b^4*d*e + 12*a*b^2*c*d*e + 6*a^2*c^2*d*e - 4
*b*(c^3*d^2 + a^3*e^2)))*Log[d^(1/3) + e^(1/3)*x])/(3*d^(2/3)*e^(8/3)) - ((6*b^2
*c^2*d^2 + 4*a*c^3*d^2 - 4*a*b^3*d*e - 12*a^2*b*c*d*e + a^4*e^2 + (d^(1/3)*(b^4*
d*e + 12*a*b^2*c*d*e + 6*a^2*c^2*d*e - 4*b*(c^3*d^2 + a^3*e^2)))/e^(1/3))*Log[d^
(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/(6*d^(2/3)*e^(7/3)) + ((c^4*d^2 - 12*a
*b*c^2*d*e + 6*a^2*b^2*e^2 - 4*c*e*(b^3*d - a^3*e))*Log[d + e*x^3])/(3*e^3)

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

[Out]

Timed out

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Mathematica [A]  time = 0.752682, size = 678, normalized size = 1.05 \[ \frac{15 e^{2/3} x^2 \left (6 a^2 c^2 e+12 a b^2 c e+b^4 e-4 b c^3 d\right )+60 e^{2/3} x \left (6 a^2 b c e+2 a b^3 e-2 a c^3 d-3 b^2 c^2 d\right )+\frac{10 \log \left (d+e x^3\right ) \left (4 c e \left (a^3 e-b^3 d\right )+6 a^2 b^2 e^2-12 a b c^2 d e+c^4 d^2\right )}{\sqrt [3]{e}}+\frac{10 \sqrt{3} \left (a \sqrt [3]{e}+b \sqrt [3]{d}\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt{3}}\right ) \left (e \left (a^3 (-e)-3 a^2 b \sqrt [3]{d} e^{2/3}+3 a b^2 d^{2/3} \sqrt [3]{e}+b^3 d\right )+c^2 \left (6 a d^{4/3} e^{2/3}-6 b d^{5/3} \sqrt [3]{e}\right )+12 a b c d e-4 c^3 d^2\right )}{d^{2/3}}-\frac{5 \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (a^4 e^{7/3}+6 a^2 c^2 d^{4/3} e-4 b \left (a^3 \sqrt [3]{d} e^2+3 a^2 c d e^{4/3}+c^3 d^{7/3}\right )-4 a b^3 d e^{4/3}+6 b^2 \left (2 a c d^{4/3} e+c^2 d^2 \sqrt [3]{e}\right )+4 a c^3 d^2 \sqrt [3]{e}+b^4 d^{4/3} e\right )}{d^{2/3}}+\frac{10 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (a^4 e^{7/3}+6 a^2 c^2 d^{4/3} e-4 b \left (a^3 \sqrt [3]{d} e^2+3 a^2 c d e^{4/3}+c^3 d^{7/3}\right )-4 a b^3 d e^{4/3}+6 b^2 \left (2 a c d^{4/3} e+c^2 d^2 \sqrt [3]{e}\right )+4 a c^3 d^2 \sqrt [3]{e}+b^4 d^{4/3} e\right )}{d^{2/3}}+10 c e^{2/3} x^3 \left (12 a b c e+4 b^3 e-c^3 d\right )+15 c^2 e^{5/3} x^4 \left (2 a c+3 b^2\right )+24 b c^3 e^{5/3} x^5+5 c^4 e^{5/3} x^6}{30 e^{8/3}} \]

Antiderivative was successfully verified.

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

[Out]

(60*e^(2/3)*(-3*b^2*c^2*d - 2*a*c^3*d + 2*a*b^3*e + 6*a^2*b*c*e)*x + 15*e^(2/3)*
(-4*b*c^3*d + b^4*e + 12*a*b^2*c*e + 6*a^2*c^2*e)*x^2 + 10*c*e^(2/3)*(-(c^3*d) +
4*b^3*e + 12*a*b*c*e)*x^3 + 15*c^2*(3*b^2 + 2*a*c)*e^(5/3)*x^4 + 24*b*c^3*e^(5/
3)*x^5 + 5*c^4*e^(5/3)*x^6 + (10*Sqrt[3]*(b*d^(1/3) + a*e^(1/3))*(-4*c^3*d^2 + c
^2*(-6*b*d^(5/3)*e^(1/3) + 6*a*d^(4/3)*e^(2/3)) + 12*a*b*c*d*e + e*(b^3*d + 3*a*
b^2*d^(2/3)*e^(1/3) - 3*a^2*b*d^(1/3)*e^(2/3) - a^3*e))*ArcTan[(1 - (2*e^(1/3)*x
)/d^(1/3))/Sqrt[3]])/d^(2/3) + (10*(4*a*c^3*d^2*e^(1/3) + b^4*d^(4/3)*e + 6*a^2*
c^2*d^(4/3)*e - 4*a*b^3*d*e^(4/3) + a^4*e^(7/3) + 6*b^2*(c^2*d^2*e^(1/3) + 2*a*c
*d^(4/3)*e) - 4*b*(c^3*d^(7/3) + 3*a^2*c*d*e^(4/3) + a^3*d^(1/3)*e^2))*Log[d^(1/
3) + e^(1/3)*x])/d^(2/3) - (5*(4*a*c^3*d^2*e^(1/3) + b^4*d^(4/3)*e + 6*a^2*c^2*d
^(4/3)*e - 4*a*b^3*d*e^(4/3) + a^4*e^(7/3) + 6*b^2*(c^2*d^2*e^(1/3) + 2*a*c*d^(4
/3)*e) - 4*b*(c^3*d^(7/3) + 3*a^2*c*d*e^(4/3) + a^3*d^(1/3)*e^2))*Log[d^(2/3) -
d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/d^(2/3) + (10*(c^4*d^2 - 12*a*b*c^2*d*e + 6*a^
2*b^2*e^2 + 4*c*e*(-(b^3*d) + a^3*e))*Log[d + e*x^3])/e^(1/3))/(30*e^(8/3))

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Maple [B]  time = 0.01, size = 1339, normalized size = 2.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/e^2/(d/e)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1))*a^2*b*c*d-4/e
^2*3^(1/2)/(d/e)^(1/3)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1))*a*b^2*c*d-4/3/e^3
/(d/e)^(1/3)*ln(x+(d/e)^(1/3))*b*c^3*d^2-1/e^2/(d/e)^(1/3)*ln(x^2-x*(d/e)^(1/3)+
(d/e)^(2/3))*a^2*c^2*d+2/3/e^3/(d/e)^(1/3)*ln(x^2-x*(d/e)^(1/3)+(d/e)^(2/3))*b*c
^3*d^2+4/3/e*3^(1/2)/(d/e)^(1/3)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1))*a^3*b-1
/3/e^2*3^(1/2)/(d/e)^(1/3)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1))*b^4*d-4/3/e^2
/(d/e)^(2/3)*ln(x+(d/e)^(1/3))*a*b^3*d+4/3/e^3/(d/e)^(2/3)*ln(x+(d/e)^(1/3))*a*c
^3*d^2+2/e^2/(d/e)^(1/3)*ln(x+(d/e)^(1/3))*a^2*c^2*d-2/3/e^3/(d/e)^(2/3)*ln(x^2-
x*(d/e)^(1/3)+(d/e)^(2/3))*a*c^3*d^2-1/e^3/(d/e)^(2/3)*ln(x^2-x*(d/e)^(1/3)+(d/e
)^(2/3))*b^2*c^2*d^2+2/3/e^2/(d/e)^(2/3)*ln(x^2-x*(d/e)^(1/3)+(d/e)^(2/3))*a*b^3
*d-4/e^2*ln(e*x^3+d)*a*b*c^2*d+2/e^3/(d/e)^(2/3)*ln(x+(d/e)^(1/3))*b^2*c^2*d^2+2
/3/e/(d/e)^(1/3)*ln(x^2-x*(d/e)^(1/3)+(d/e)^(2/3))*a^3*b+1/3/e/(d/e)^(2/3)*3^(1/
2)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1))*a^4+1/e*x^4*a*c^3+4/e*a*b^3*x+3/2/e*x
^4*b^2*c^2+4/3/e*x^3*b^3*c-4/3/e/(d/e)^(1/3)*ln(x+(d/e)^(1/3))*a^3*b-1/6/e^2/(d/
e)^(1/3)*ln(x^2-x*(d/e)^(1/3)+(d/e)^(2/3))*b^4*d-4/3/e^2*ln(e*x^3+d)*b^3*c*d+1/3
/e^2/(d/e)^(1/3)*ln(x+(d/e)^(1/3))*b^4*d-2/e^2*3^(1/2)/(d/e)^(1/3)*arctan(1/3*3^
(1/2)*(2/(d/e)^(1/3)*x-1))*a^2*c^2*d+2/e*ln(e*x^3+d)*a^2*b^2+1/3/e^3*ln(e*x^3+d)
*c^4*d^2+3/e*x^2*a^2*c^2-1/6/e/(d/e)^(2/3)*ln(x^2-x*(d/e)^(1/3)+(d/e)^(2/3))*a^4
+1/3/e/(d/e)^(2/3)*ln(x+(d/e)^(1/3))*a^4+4/3/e*ln(e*x^3+d)*a^3*c-2/e^2*x^2*b*c^3
*d+4/3/e^3*3^(1/2)/(d/e)^(1/3)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1))*b*c^3*d^2
-1/3/e^2*x^3*c^4*d+1/6*c^4*x^6/e-4/e^2*a*c^3*d*x-6/e^2*x*b^2*c^2*d+1/2/e*x^2*b^4
+4/e^2/(d/e)^(1/3)*ln(x+(d/e)^(1/3))*a*b^2*c*d-2/e^2/(d/e)^(1/3)*ln(x^2-x*(d/e)^
(1/3)+(d/e)^(2/3))*a*b^2*c*d+2/e^3/(d/e)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(d/
e)^(1/3)*x-1))*b^2*c^2*d^2+4/3/e^3/(d/e)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(d/
e)^(1/3)*x-1))*a*c^3*d^2+6/e*x^2*a*b^2*c+4/e*x^3*a*b*c^2-4/e^2/(d/e)^(2/3)*ln(x+
(d/e)^(1/3))*a^2*b*c*d+2/e^2/(d/e)^(2/3)*ln(x^2-x*(d/e)^(1/3)+(d/e)^(2/3))*a^2*b
*c*d-4/3/e^2/(d/e)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1))*a*b^3*d
+12/e*a^2*b*c*x+4/5*b*c^3*x^5/e

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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((c*x^2 + b*x + a)^4/(e*x^3 + d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.221811, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done