3.3 \(\int (a+b x^3) (c+d x^3)^2 \, dx\)

Optimal. Leaf size=50 \[ \frac {1}{7} d x^7 (a d+2 b c)+\frac {1}{4} c x^4 (2 a d+b c)+a c^2 x+\frac {1}{10} b d^2 x^{10} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {373} \[ \frac {1}{7} d x^7 (a d+2 b c)+\frac {1}{4} c x^4 (2 a d+b c)+a c^2 x+\frac {1}{10} b d^2 x^{10} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)*(c + d*x^3)^2,x]

[Out]

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

Rule 373

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n
)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && IGtQ[q, 0]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 50, normalized size = 1.00 \[ \frac {1}{7} d x^7 (a d+2 b c)+\frac {1}{4} c x^4 (2 a d+b c)+a c^2 x+\frac {1}{10} b d^2 x^{10} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.36, size = 50, normalized size = 1.00 \[ \frac {1}{10} x^{10} d^{2} b + \frac {2}{7} x^{7} d c b + \frac {1}{7} x^{7} d^{2} a + \frac {1}{4} x^{4} c^{2} b + \frac {1}{2} x^{4} d c a + x c^{2} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.19, size = 50, normalized size = 1.00 \[ \frac {1}{10} \, b d^{2} x^{10} + \frac {2}{7} \, b c d x^{7} + \frac {1}{7} \, a d^{2} x^{7} + \frac {1}{4} \, b c^{2} x^{4} + \frac {1}{2} \, a c d x^{4} + a c^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 49, normalized size = 0.98 \[ \frac {b \,d^{2} x^{10}}{10}+\frac {\left (a \,d^{2}+2 b c d \right ) x^{7}}{7}+a \,c^{2} x +\frac {\left (2 a c d +b \,c^{2}\right ) x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.48, size = 48, normalized size = 0.96 \[ \frac {1}{10} \, b d^{2} x^{10} + \frac {1}{7} \, {\left (2 \, b c d + a d^{2}\right )} x^{7} + \frac {1}{4} \, {\left (b c^{2} + 2 \, a c d\right )} x^{4} + a c^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.05, size = 48, normalized size = 0.96 \[ x^4\,\left (\frac {b\,c^2}{4}+\frac {a\,d\,c}{2}\right )+x^7\,\left (\frac {a\,d^2}{7}+\frac {2\,b\,c\,d}{7}\right )+\frac {b\,d^2\,x^{10}}{10}+a\,c^2\,x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)*(c + d*x^3)^2,x)

[Out]

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

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sympy [A]  time = 0.07, size = 51, normalized size = 1.02 \[ a c^{2} x + \frac {b d^{2} x^{10}}{10} + x^{7} \left (\frac {a d^{2}}{7} + \frac {2 b c d}{7}\right ) + x^{4} \left (\frac {a c d}{2} + \frac {b c^{2}}{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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