3.1.4 \(\int (d+e x^3)^2 (a+b x^3+c x^6) \, dx\)

Optimal. Leaf size=73 \[ \frac {1}{7} x^7 \left (e (a e+2 b d)+c d^2\right )+\frac {1}{4} d x^4 (2 a e+b d)+a d^2 x+\frac {1}{10} e x^{10} (b e+2 c d)+\frac {1}{13} c e^2 x^{13} \]

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Rubi [A]  time = 0.06, antiderivative size = 73, 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} \begin {gather*} \frac {1}{7} x^7 \left (e (a e+2 b d)+c d^2\right )+\frac {1}{4} d x^4 (2 a e+b d)+a d^2 x+\frac {1}{10} e x^{10} (b e+2 c d)+\frac {1}{13} c e^2 x^{13} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x^3)^2*(a + b*x^3 + c*x^6),x]

[Out]

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

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

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Mathematica [A]  time = 0.02, size = 73, normalized size = 1.00 \begin {gather*} \frac {1}{7} x^7 \left (a e^2+2 b d e+c d^2\right )+\frac {1}{4} d x^4 (2 a e+b d)+a d^2 x+\frac {1}{10} e x^{10} (b e+2 c d)+\frac {1}{13} c e^2 x^{13} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x^3)^2*(a + b*x^3 + c*x^6),x]

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d+e x^3\right )^2 \left (a+b x^3+c x^6\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(d + e*x^3)^2*(a + b*x^3 + c*x^6),x]

[Out]

IntegrateAlgebraic[(d + e*x^3)^2*(a + b*x^3 + c*x^6), x]

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fricas [A]  time = 0.91, size = 76, normalized size = 1.04 \begin {gather*} \frac {1}{13} x^{13} e^{2} c + \frac {1}{5} x^{10} e d c + \frac {1}{10} x^{10} e^{2} b + \frac {1}{7} x^{7} d^{2} c + \frac {2}{7} x^{7} e d b + \frac {1}{7} x^{7} e^{2} a + \frac {1}{4} x^{4} d^{2} b + \frac {1}{2} x^{4} e d a + x d^{2} a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^3+d)^2*(c*x^6+b*x^3+a),x, algorithm="fricas")

[Out]

1/13*x^13*e^2*c + 1/5*x^10*e*d*c + 1/10*x^10*e^2*b + 1/7*x^7*d^2*c + 2/7*x^7*e*d*b + 1/7*x^7*e^2*a + 1/4*x^4*d
^2*b + 1/2*x^4*e*d*a + x*d^2*a

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giac [A]  time = 0.33, size = 76, normalized size = 1.04 \begin {gather*} \frac {1}{13} \, c x^{13} e^{2} + \frac {1}{5} \, c d x^{10} e + \frac {1}{10} \, b x^{10} e^{2} + \frac {1}{7} \, c d^{2} x^{7} + \frac {2}{7} \, b d x^{7} e + \frac {1}{7} \, a x^{7} e^{2} + \frac {1}{4} \, b d^{2} x^{4} + \frac {1}{2} \, a d x^{4} e + a d^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^3+d)^2*(c*x^6+b*x^3+a),x, algorithm="giac")

[Out]

1/13*c*x^13*e^2 + 1/5*c*d*x^10*e + 1/10*b*x^10*e^2 + 1/7*c*d^2*x^7 + 2/7*b*d*x^7*e + 1/7*a*x^7*e^2 + 1/4*b*d^2
*x^4 + 1/2*a*d*x^4*e + a*d^2*x

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maple [A]  time = 0.00, size = 70, normalized size = 0.96 \begin {gather*} \frac {c \,e^{2} x^{13}}{13}+\frac {\left (b \,e^{2}+2 c d e \right ) x^{10}}{10}+\frac {\left (a \,e^{2}+2 d e b +c \,d^{2}\right ) x^{7}}{7}+a \,d^{2} x +\frac {\left (2 d e a +b \,d^{2}\right ) x^{4}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^3+d)^2*(c*x^6+b*x^3+a),x)

[Out]

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

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maxima [A]  time = 0.75, size = 69, normalized size = 0.95 \begin {gather*} \frac {1}{13} \, c e^{2} x^{13} + \frac {1}{10} \, {\left (2 \, c d e + b e^{2}\right )} x^{10} + \frac {1}{7} \, {\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{7} + \frac {1}{4} \, {\left (b d^{2} + 2 \, a d e\right )} x^{4} + a d^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^3+d)^2*(c*x^6+b*x^3+a),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.04, size = 70, normalized size = 0.96 \begin {gather*} x^7\,\left (\frac {c\,d^2}{7}+\frac {2\,b\,d\,e}{7}+\frac {a\,e^2}{7}\right )+x^4\,\left (\frac {b\,d^2}{4}+\frac {a\,e\,d}{2}\right )+x^{10}\,\left (\frac {b\,e^2}{10}+\frac {c\,d\,e}{5}\right )+\frac {c\,e^2\,x^{13}}{13}+a\,d^2\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x^3)^2*(a + b*x^3 + c*x^6),x)

[Out]

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

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sympy [A]  time = 0.08, size = 75, normalized size = 1.03 \begin {gather*} a d^{2} x + \frac {c e^{2} x^{13}}{13} + x^{10} \left (\frac {b e^{2}}{10} + \frac {c d e}{5}\right ) + x^{7} \left (\frac {a e^{2}}{7} + \frac {2 b d e}{7} + \frac {c d^{2}}{7}\right ) + x^{4} \left (\frac {a d e}{2} + \frac {b d^{2}}{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**3+d)**2*(c*x**6+b*x**3+a),x)

[Out]

a*d**2*x + c*e**2*x**13/13 + x**10*(b*e**2/10 + c*d*e/5) + x**7*(a*e**2/7 + 2*b*d*e/7 + c*d**2/7) + x**4*(a*d*
e/2 + b*d**2/4)

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