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

Optimal. Leaf size=42 \[ \frac {1}{4} x^4 (a e+b d)+a d x+\frac {1}{7} x^7 (b e+c d)+\frac {1}{10} c e x^{10} \]

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Rubi [A]  time = 0.03, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1407} \begin {gather*} \frac {1}{4} x^4 (a e+b d)+a d x+\frac {1}{7} x^7 (b e+c d)+\frac {1}{10} c e x^{10} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.01, size = 42, normalized size = 1.00 \begin {gather*} \frac {1}{4} x^4 (a e+b d)+a d x+\frac {1}{7} x^7 (b e+c d)+\frac {1}{10} c e x^{10} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

[Out]

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

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fricas [A]  time = 1.03, size = 40, normalized size = 0.95 \begin {gather*} \frac {1}{10} x^{10} e c + \frac {1}{7} x^{7} d c + \frac {1}{7} x^{7} e b + \frac {1}{4} x^{4} d b + \frac {1}{4} x^{4} e a + x d a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.33, size = 43, normalized size = 1.02 \begin {gather*} \frac {1}{10} \, c x^{10} e + \frac {1}{7} \, c d x^{7} + \frac {1}{7} \, b x^{7} e + \frac {1}{4} \, b d x^{4} + \frac {1}{4} \, a x^{4} e + a d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 37, normalized size = 0.88 \begin {gather*} \frac {c e \,x^{10}}{10}+\frac {\left (b e +c d \right ) x^{7}}{7}+\frac {\left (a e +b d \right ) x^{4}}{4}+a d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.55, size = 36, normalized size = 0.86 \begin {gather*} \frac {1}{10} \, c e x^{10} + \frac {1}{7} \, {\left (c d + b e\right )} x^{7} + \frac {1}{4} \, {\left (b d + a e\right )} x^{4} + a d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.04, size = 38, normalized size = 0.90 \begin {gather*} \frac {c\,e\,x^{10}}{10}+\left (\frac {b\,e}{7}+\frac {c\,d}{7}\right )\,x^7+\left (\frac {a\,e}{4}+\frac {b\,d}{4}\right )\,x^4+a\,d\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.07, size = 39, normalized size = 0.93 \begin {gather*} a d x + \frac {c e x^{10}}{10} + x^{7} \left (\frac {b e}{7} + \frac {c d}{7}\right ) + x^{4} \left (\frac {a e}{4} + \frac {b d}{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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