3.1 $$\int (d+e x^3)^5 (a+b x^3+c x^6) \, dx$$

Optimal. Leaf size=163 $\frac{1}{16} e^3 x^{16} \left (e (a e+5 b d)+10 c d^2\right )+\frac{5}{13} d e^2 x^{13} \left (e (a e+2 b d)+2 c d^2\right )+\frac{1}{2} d^2 e x^{10} \left (2 e (a e+b d)+c d^2\right )+\frac{1}{7} d^3 x^7 \left (5 e (2 a e+b d)+c d^2\right )+\frac{1}{4} d^4 x^4 (5 a e+b d)+a d^5 x+\frac{1}{19} e^4 x^{19} (b e+5 c d)+\frac{1}{22} c e^5 x^{22}$

[Out]

a*d^5*x + (d^4*(b*d + 5*a*e)*x^4)/4 + (d^3*(c*d^2 + 5*e*(b*d + 2*a*e))*x^7)/7 + (d^2*e*(c*d^2 + 2*e*(b*d + a*e
))*x^10)/2 + (5*d*e^2*(2*c*d^2 + e*(2*b*d + a*e))*x^13)/13 + (e^3*(10*c*d^2 + e*(5*b*d + a*e))*x^16)/16 + (e^4
*(5*c*d + b*e)*x^19)/19 + (c*e^5*x^22)/22

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Rubi [A]  time = 0.185171, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.045, Rules used = {1407} $\frac{1}{16} e^3 x^{16} \left (e (a e+5 b d)+10 c d^2\right )+\frac{5}{13} d e^2 x^{13} \left (e (a e+2 b d)+2 c d^2\right )+\frac{1}{2} d^2 e x^{10} \left (2 e (a e+b d)+c d^2\right )+\frac{1}{7} d^3 x^7 \left (5 e (2 a e+b d)+c d^2\right )+\frac{1}{4} d^4 x^4 (5 a e+b d)+a d^5 x+\frac{1}{19} e^4 x^{19} (b e+5 c d)+\frac{1}{22} c e^5 x^{22}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x^3)^5*(a + b*x^3 + c*x^6),x]

[Out]

a*d^5*x + (d^4*(b*d + 5*a*e)*x^4)/4 + (d^3*(c*d^2 + 5*e*(b*d + 2*a*e))*x^7)/7 + (d^2*e*(c*d^2 + 2*e*(b*d + a*e
))*x^10)/2 + (5*d*e^2*(2*c*d^2 + e*(2*b*d + a*e))*x^13)/13 + (e^3*(10*c*d^2 + e*(5*b*d + a*e))*x^16)/16 + (e^4
*(5*c*d + b*e)*x^19)/19 + (c*e^5*x^22)/22

Rule 1407

Int[((d_) + (e_.)*(x_)^(n_))^(q_.)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x^n)^q*(a + b*x^n + c*x^(2*n)), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[q, 0]

Rubi steps

\begin{align*} \int \left (d+e x^3\right )^5 \left (a+b x^3+c x^6\right ) \, dx &=\int \left (a d^5+d^4 (b d+5 a e) x^3+d^3 \left (c d^2+5 e (b d+2 a e)\right ) x^6+5 d^2 e \left (c d^2+2 e (b d+a e)\right ) x^9+5 d e^2 \left (2 c d^2+e (2 b d+a e)\right ) x^{12}+e^3 \left (10 c d^2+e (5 b d+a e)\right ) x^{15}+e^4 (5 c d+b e) x^{18}+c e^5 x^{21}\right ) \, dx\\ &=a d^5 x+\frac{1}{4} d^4 (b d+5 a e) x^4+\frac{1}{7} d^3 \left (c d^2+5 e (b d+2 a e)\right ) x^7+\frac{1}{2} d^2 e \left (c d^2+2 e (b d+a e)\right ) x^{10}+\frac{5}{13} d e^2 \left (2 c d^2+e (2 b d+a e)\right ) x^{13}+\frac{1}{16} e^3 \left (10 c d^2+e (5 b d+a e)\right ) x^{16}+\frac{1}{19} e^4 (5 c d+b e) x^{19}+\frac{1}{22} c e^5 x^{22}\\ \end{align*}

Mathematica [A]  time = 0.0481313, size = 164, normalized size = 1.01 $\frac{1}{16} e^3 x^{16} \left (a e^2+5 b d e+10 c d^2\right )+\frac{5}{13} d e^2 x^{13} \left (a e^2+2 b d e+2 c d^2\right )+\frac{1}{2} d^2 e x^{10} \left (2 a e^2+2 b d e+c d^2\right )+\frac{1}{7} d^3 x^7 \left (10 a e^2+5 b d e+c d^2\right )+\frac{1}{4} d^4 x^4 (5 a e+b d)+a d^5 x+\frac{1}{19} e^4 x^{19} (b e+5 c d)+\frac{1}{22} c e^5 x^{22}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x^3)^5*(a + b*x^3 + c*x^6),x]

[Out]

a*d^5*x + (d^4*(b*d + 5*a*e)*x^4)/4 + (d^3*(c*d^2 + 5*b*d*e + 10*a*e^2)*x^7)/7 + (d^2*e*(c*d^2 + 2*b*d*e + 2*a
*e^2)*x^10)/2 + (5*d*e^2*(2*c*d^2 + 2*b*d*e + a*e^2)*x^13)/13 + (e^3*(10*c*d^2 + 5*b*d*e + a*e^2)*x^16)/16 + (
e^4*(5*c*d + b*e)*x^19)/19 + (c*e^5*x^22)/22

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Maple [A]  time = 0.001, size = 169, normalized size = 1. \begin{align*}{\frac{c{e}^{5}{x}^{22}}{22}}+{\frac{ \left ({e}^{5}b+5\,d{e}^{4}c \right ){x}^{19}}{19}}+{\frac{ \left ({e}^{5}a+5\,d{e}^{4}b+10\,{d}^{2}{e}^{3}c \right ){x}^{16}}{16}}+{\frac{ \left ( 5\,d{e}^{4}a+10\,{d}^{2}{e}^{3}b+10\,{d}^{3}{e}^{2}c \right ){x}^{13}}{13}}+{\frac{ \left ( 10\,{d}^{2}{e}^{3}a+10\,{d}^{3}{e}^{2}b+5\,{d}^{4}ec \right ){x}^{10}}{10}}+{\frac{ \left ( 10\,{d}^{3}{e}^{2}a+5\,{d}^{4}eb+{d}^{5}c \right ){x}^{7}}{7}}+{\frac{ \left ( 5\,{d}^{4}ea+{d}^{5}b \right ){x}^{4}}{4}}+a{d}^{5}x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^3+d)^5*(c*x^6+b*x^3+a),x)

[Out]

1/22*c*e^5*x^22+1/19*(b*e^5+5*c*d*e^4)*x^19+1/16*(a*e^5+5*b*d*e^4+10*c*d^2*e^3)*x^16+1/13*(5*a*d*e^4+10*b*d^2*
e^3+10*c*d^3*e^2)*x^13+1/10*(10*a*d^2*e^3+10*b*d^3*e^2+5*c*d^4*e)*x^10+1/7*(10*a*d^3*e^2+5*b*d^4*e+c*d^5)*x^7+
1/4*(5*a*d^4*e+b*d^5)*x^4+a*d^5*x

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Maxima [A]  time = 1.00563, size = 224, normalized size = 1.37 \begin{align*} \frac{1}{22} \, c e^{5} x^{22} + \frac{1}{19} \,{\left (5 \, c d e^{4} + b e^{5}\right )} x^{19} + \frac{1}{16} \,{\left (10 \, c d^{2} e^{3} + 5 \, b d e^{4} + a e^{5}\right )} x^{16} + \frac{5}{13} \,{\left (2 \, c d^{3} e^{2} + 2 \, b d^{2} e^{3} + a d e^{4}\right )} x^{13} + \frac{1}{2} \,{\left (c d^{4} e + 2 \, b d^{3} e^{2} + 2 \, a d^{2} e^{3}\right )} x^{10} + \frac{1}{7} \,{\left (c d^{5} + 5 \, b d^{4} e + 10 \, a d^{3} e^{2}\right )} x^{7} + a d^{5} x + \frac{1}{4} \,{\left (b d^{5} + 5 \, a d^{4} e\right )} x^{4} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^3+d)^5*(c*x^6+b*x^3+a),x, algorithm="maxima")

[Out]

1/22*c*e^5*x^22 + 1/19*(5*c*d*e^4 + b*e^5)*x^19 + 1/16*(10*c*d^2*e^3 + 5*b*d*e^4 + a*e^5)*x^16 + 5/13*(2*c*d^3
*e^2 + 2*b*d^2*e^3 + a*d*e^4)*x^13 + 1/2*(c*d^4*e + 2*b*d^3*e^2 + 2*a*d^2*e^3)*x^10 + 1/7*(c*d^5 + 5*b*d^4*e +
10*a*d^3*e^2)*x^7 + a*d^5*x + 1/4*(b*d^5 + 5*a*d^4*e)*x^4

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Fricas [A]  time = 1.13949, size = 448, normalized size = 2.75 \begin{align*} \frac{1}{22} x^{22} e^{5} c + \frac{5}{19} x^{19} e^{4} d c + \frac{1}{19} x^{19} e^{5} b + \frac{5}{8} x^{16} e^{3} d^{2} c + \frac{5}{16} x^{16} e^{4} d b + \frac{1}{16} x^{16} e^{5} a + \frac{10}{13} x^{13} e^{2} d^{3} c + \frac{10}{13} x^{13} e^{3} d^{2} b + \frac{5}{13} x^{13} e^{4} d a + \frac{1}{2} x^{10} e d^{4} c + x^{10} e^{2} d^{3} b + x^{10} e^{3} d^{2} a + \frac{1}{7} x^{7} d^{5} c + \frac{5}{7} x^{7} e d^{4} b + \frac{10}{7} x^{7} e^{2} d^{3} a + \frac{1}{4} x^{4} d^{5} b + \frac{5}{4} x^{4} e d^{4} a + x d^{5} a \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^3+d)^5*(c*x^6+b*x^3+a),x, algorithm="fricas")

[Out]

1/22*x^22*e^5*c + 5/19*x^19*e^4*d*c + 1/19*x^19*e^5*b + 5/8*x^16*e^3*d^2*c + 5/16*x^16*e^4*d*b + 1/16*x^16*e^5
*a + 10/13*x^13*e^2*d^3*c + 10/13*x^13*e^3*d^2*b + 5/13*x^13*e^4*d*a + 1/2*x^10*e*d^4*c + x^10*e^2*d^3*b + x^1
0*e^3*d^2*a + 1/7*x^7*d^5*c + 5/7*x^7*e*d^4*b + 10/7*x^7*e^2*d^3*a + 1/4*x^4*d^5*b + 5/4*x^4*e*d^4*a + x*d^5*a

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Sympy [A]  time = 0.095765, size = 187, normalized size = 1.15 \begin{align*} a d^{5} x + \frac{c e^{5} x^{22}}{22} + x^{19} \left (\frac{b e^{5}}{19} + \frac{5 c d e^{4}}{19}\right ) + x^{16} \left (\frac{a e^{5}}{16} + \frac{5 b d e^{4}}{16} + \frac{5 c d^{2} e^{3}}{8}\right ) + x^{13} \left (\frac{5 a d e^{4}}{13} + \frac{10 b d^{2} e^{3}}{13} + \frac{10 c d^{3} e^{2}}{13}\right ) + x^{10} \left (a d^{2} e^{3} + b d^{3} e^{2} + \frac{c d^{4} e}{2}\right ) + x^{7} \left (\frac{10 a d^{3} e^{2}}{7} + \frac{5 b d^{4} e}{7} + \frac{c d^{5}}{7}\right ) + x^{4} \left (\frac{5 a d^{4} e}{4} + \frac{b d^{5}}{4}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**3+d)**5*(c*x**6+b*x**3+a),x)

[Out]

a*d**5*x + c*e**5*x**22/22 + x**19*(b*e**5/19 + 5*c*d*e**4/19) + x**16*(a*e**5/16 + 5*b*d*e**4/16 + 5*c*d**2*e
**3/8) + x**13*(5*a*d*e**4/13 + 10*b*d**2*e**3/13 + 10*c*d**3*e**2/13) + x**10*(a*d**2*e**3 + b*d**3*e**2 + c*
d**4*e/2) + x**7*(10*a*d**3*e**2/7 + 5*b*d**4*e/7 + c*d**5/7) + x**4*(5*a*d**4*e/4 + b*d**5/4)

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Giac [A]  time = 1.13347, size = 234, normalized size = 1.44 \begin{align*} \frac{1}{22} \, c x^{22} e^{5} + \frac{5}{19} \, c d x^{19} e^{4} + \frac{1}{19} \, b x^{19} e^{5} + \frac{5}{8} \, c d^{2} x^{16} e^{3} + \frac{5}{16} \, b d x^{16} e^{4} + \frac{1}{16} \, a x^{16} e^{5} + \frac{10}{13} \, c d^{3} x^{13} e^{2} + \frac{10}{13} \, b d^{2} x^{13} e^{3} + \frac{5}{13} \, a d x^{13} e^{4} + \frac{1}{2} \, c d^{4} x^{10} e + b d^{3} x^{10} e^{2} + a d^{2} x^{10} e^{3} + \frac{1}{7} \, c d^{5} x^{7} + \frac{5}{7} \, b d^{4} x^{7} e + \frac{10}{7} \, a d^{3} x^{7} e^{2} + \frac{1}{4} \, b d^{5} x^{4} + \frac{5}{4} \, a d^{4} x^{4} e + a d^{5} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^3+d)^5*(c*x^6+b*x^3+a),x, algorithm="giac")

[Out]

1/22*c*x^22*e^5 + 5/19*c*d*x^19*e^4 + 1/19*b*x^19*e^5 + 5/8*c*d^2*x^16*e^3 + 5/16*b*d*x^16*e^4 + 1/16*a*x^16*e
^5 + 10/13*c*d^3*x^13*e^2 + 10/13*b*d^2*x^13*e^3 + 5/13*a*d*x^13*e^4 + 1/2*c*d^4*x^10*e + b*d^3*x^10*e^2 + a*d
^2*x^10*e^3 + 1/7*c*d^5*x^7 + 5/7*b*d^4*x^7*e + 10/7*a*d^3*x^7*e^2 + 1/4*b*d^5*x^4 + 5/4*a*d^4*x^4*e + a*d^5*x