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

Optimal. Leaf size=103 \[ \frac {1}{10} e x^{10} \left (e (a e+3 b d)+3 c d^2\right )+\frac {1}{7} d x^7 \left (3 e (a e+b d)+c d^2\right )+\frac {1}{4} d^2 x^4 (3 a e+b d)+a d^3 x+\frac {1}{13} e^2 x^{13} (b e+3 c d)+\frac {1}{16} c e^3 x^{16} \]

[Out]

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

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Rubi [A]  time = 0.10, antiderivative size = 103, normalized size of antiderivative = 1.00, 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} \[ \frac {1}{10} e x^{10} \left (e (a e+3 b d)+3 c d^2\right )+\frac {1}{7} d x^7 \left (3 e (a e+b d)+c d^2\right )+\frac {1}{4} d^2 x^4 (3 a e+b d)+a d^3 x+\frac {1}{13} e^2 x^{13} (b e+3 c d)+\frac {1}{16} c e^3 x^{16} \]

Antiderivative was successfully verified.

[In]

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

[Out]

a*d^3*x + (d^2*(b*d + 3*a*e)*x^4)/4 + (d*(c*d^2 + 3*e*(b*d + a*e))*x^7)/7 + (e*(3*c*d^2 + e*(3*b*d + a*e))*x^1
0)/10 + (e^2*(3*c*d + b*e)*x^13)/13 + (c*e^3*x^16)/16

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

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Mathematica [A]  time = 0.03, size = 104, normalized size = 1.01 \[ \frac {1}{10} e x^{10} \left (a e^2+3 b d e+3 c d^2\right )+\frac {1}{7} d x^7 \left (3 a e^2+3 b d e+c d^2\right )+\frac {1}{4} d^2 x^4 (3 a e+b d)+a d^3 x+\frac {1}{13} e^2 x^{13} (b e+3 c d)+\frac {1}{16} c e^3 x^{16} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.80, size = 112, normalized size = 1.09 \[ \frac {1}{16} x^{16} e^{3} c + \frac {3}{13} x^{13} e^{2} d c + \frac {1}{13} x^{13} e^{3} b + \frac {3}{10} x^{10} e d^{2} c + \frac {3}{10} x^{10} e^{2} d b + \frac {1}{10} x^{10} e^{3} a + \frac {1}{7} x^{7} d^{3} c + \frac {3}{7} x^{7} e d^{2} b + \frac {3}{7} x^{7} e^{2} d a + \frac {1}{4} x^{4} d^{3} b + \frac {3}{4} x^{4} e d^{2} a + x d^{3} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.30, size = 109, normalized size = 1.06 \[ \frac {1}{16} \, c x^{16} e^{3} + \frac {3}{13} \, c d x^{13} e^{2} + \frac {1}{13} \, b x^{13} e^{3} + \frac {3}{10} \, c d^{2} x^{10} e + \frac {3}{10} \, b d x^{10} e^{2} + \frac {1}{10} \, a x^{10} e^{3} + \frac {1}{7} \, c d^{3} x^{7} + \frac {3}{7} \, b d^{2} x^{7} e + \frac {3}{7} \, a d x^{7} e^{2} + \frac {1}{4} \, b d^{3} x^{4} + \frac {3}{4} \, a d^{2} x^{4} e + a d^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 103, normalized size = 1.00 \[ \frac {c \,e^{3} x^{16}}{16}+\frac {\left (e^{3} b +3 c d \,e^{2}\right ) x^{13}}{13}+\frac {\left (e^{3} a +3 b d \,e^{2}+3 d^{2} e c \right ) x^{10}}{10}+\frac {\left (3 a d \,e^{2}+3 b \,d^{2} e +c \,d^{3}\right ) x^{7}}{7}+a \,d^{3} x +\frac {\left (3 a \,d^{2} e +b \,d^{3}\right ) x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.65, size = 102, normalized size = 0.99 \[ \frac {1}{16} \, c e^{3} x^{16} + \frac {1}{13} \, {\left (3 \, c d e^{2} + b e^{3}\right )} x^{13} + \frac {1}{10} \, {\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} x^{10} + \frac {1}{7} \, {\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} x^{7} + a d^{3} x + \frac {1}{4} \, {\left (b d^{3} + 3 \, a d^{2} e\right )} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.04, size = 102, normalized size = 0.99 \[ x^4\,\left (\frac {b\,d^3}{4}+\frac {3\,a\,e\,d^2}{4}\right )+x^{13}\,\left (\frac {b\,e^3}{13}+\frac {3\,c\,d\,e^2}{13}\right )+x^7\,\left (\frac {c\,d^3}{7}+\frac {3\,b\,d^2\,e}{7}+\frac {3\,a\,d\,e^2}{7}\right )+x^{10}\,\left (\frac {3\,c\,d^2\,e}{10}+\frac {3\,b\,d\,e^2}{10}+\frac {a\,e^3}{10}\right )+\frac {c\,e^3\,x^{16}}{16}+a\,d^3\,x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.09, size = 117, normalized size = 1.14 \[ a d^{3} x + \frac {c e^{3} x^{16}}{16} + x^{13} \left (\frac {b e^{3}}{13} + \frac {3 c d e^{2}}{13}\right ) + x^{10} \left (\frac {a e^{3}}{10} + \frac {3 b d e^{2}}{10} + \frac {3 c d^{2} e}{10}\right ) + x^{7} \left (\frac {3 a d e^{2}}{7} + \frac {3 b d^{2} e}{7} + \frac {c d^{3}}{7}\right ) + x^{4} \left (\frac {3 a d^{2} e}{4} + \frac {b d^{3}}{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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