3.2.48 \(\int (a+b x^4) (c+d x^4)^2 \, dx\) [148]

Optimal. Leaf size=50 \[ a c^2 x+\frac {1}{5} c (b c+2 a d) x^5+\frac {1}{9} d (2 b c+a d) x^9+\frac {1}{13} b d^2 x^{13} \]

[Out]

a*c^2*x+1/5*c*(2*a*d+b*c)*x^5+1/9*d*(a*d+2*b*c)*x^9+1/13*b*d^2*x^13

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Rubi [A]
time = 0.02, 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 = {380} \begin {gather*} \frac {1}{9} d x^9 (a d+2 b c)+\frac {1}{5} c x^5 (2 a d+b c)+a c^2 x+\frac {1}{13} b d^2 x^{13} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)*(c + d*x^4)^2,x]

[Out]

a*c^2*x + (c*(b*c + 2*a*d)*x^5)/5 + (d*(2*b*c + a*d)*x^9)/9 + (b*d^2*x^13)/13

Rule 380

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^4\right ) \left (c+d x^4\right )^2 \, dx &=\int \left (a c^2+c (b c+2 a d) x^4+d (2 b c+a d) x^8+b d^2 x^{12}\right ) \, dx\\ &=a c^2 x+\frac {1}{5} c (b c+2 a d) x^5+\frac {1}{9} d (2 b c+a d) x^9+\frac {1}{13} b d^2 x^{13}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 50, normalized size = 1.00 \begin {gather*} a c^2 x+\frac {1}{5} c (b c+2 a d) x^5+\frac {1}{9} d (2 b c+a d) x^9+\frac {1}{13} b d^2 x^{13} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)*(c + d*x^4)^2,x]

[Out]

a*c^2*x + (c*(b*c + 2*a*d)*x^5)/5 + (d*(2*b*c + a*d)*x^9)/9 + (b*d^2*x^13)/13

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Maple [A]
time = 0.27, size = 49, normalized size = 0.98

method	result	size
default	\(\frac {b \,d^{2} x^{13}}{13}+\frac {\left (a \,d^{2}+2 b c d \right ) x^{9}}{9}+\frac {\left (2 a c d +b \,c^{2}\right ) x^{5}}{5}+a \,c^{2} x\)	\(49\)
norman	\(\frac {b \,d^{2} x^{13}}{13}+\left (\frac {1}{9} a \,d^{2}+\frac {2}{9} b c d \right ) x^{9}+\left (\frac {2}{5} a c d +\frac {1}{5} b \,c^{2}\right ) x^{5}+a \,c^{2} x\)	\(49\)
gosper	\(\frac {1}{13} b \,d^{2} x^{13}+\frac {1}{9} x^{9} a \,d^{2}+\frac {2}{9} x^{9} b c d +\frac {2}{5} x^{5} a c d +\frac {1}{5} x^{5} b \,c^{2}+a \,c^{2} x\)	\(51\)
risch	\(\frac {1}{13} b \,d^{2} x^{13}+\frac {1}{9} x^{9} a \,d^{2}+\frac {2}{9} x^{9} b c d +\frac {2}{5} x^{5} a c d +\frac {1}{5} x^{5} b \,c^{2}+a \,c^{2} x\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^4+a)*(d*x^4+c)^2,x,method=_RETURNVERBOSE)

[Out]

1/13*b*d^2*x^13+1/9*(a*d^2+2*b*c*d)*x^9+1/5*(2*a*c*d+b*c^2)*x^5+a*c^2*x

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Maxima [A]
time = 0.27, size = 48, normalized size = 0.96 \begin {gather*} \frac {1}{13} \, b d^{2} x^{13} + \frac {1}{9} \, {\left (2 \, b c d + a d^{2}\right )} x^{9} + \frac {1}{5} \, {\left (b c^{2} + 2 \, a c d\right )} x^{5} + a c^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)*(d*x^4+c)^2,x, algorithm="maxima")

[Out]

1/13*b*d^2*x^13 + 1/9*(2*b*c*d + a*d^2)*x^9 + 1/5*(b*c^2 + 2*a*c*d)*x^5 + a*c^2*x

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Fricas [A]
time = 3.43, size = 48, normalized size = 0.96 \begin {gather*} \frac {1}{13} \, b d^{2} x^{13} + \frac {1}{9} \, {\left (2 \, b c d + a d^{2}\right )} x^{9} + \frac {1}{5} \, {\left (b c^{2} + 2 \, a c d\right )} x^{5} + a c^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)*(d*x^4+c)^2,x, algorithm="fricas")

[Out]

1/13*b*d^2*x^13 + 1/9*(2*b*c*d + a*d^2)*x^9 + 1/5*(b*c^2 + 2*a*c*d)*x^5 + a*c^2*x

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Sympy [A]
time = 0.01, size = 53, normalized size = 1.06 \begin {gather*} a c^{2} x + \frac {b d^{2} x^{13}}{13} + x^{9} \left (\frac {a d^{2}}{9} + \frac {2 b c d}{9}\right ) + x^{5} \cdot \left (\frac {2 a c d}{5} + \frac {b c^{2}}{5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**4+a)*(d*x**4+c)**2,x)

[Out]

a*c**2*x + b*d**2*x**13/13 + x**9*(a*d**2/9 + 2*b*c*d/9) + x**5*(2*a*c*d/5 + b*c**2/5)

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Giac [A]
time = 0.50, size = 50, normalized size = 1.00 \begin {gather*} \frac {1}{13} \, b d^{2} x^{13} + \frac {2}{9} \, b c d x^{9} + \frac {1}{9} \, a d^{2} x^{9} + \frac {1}{5} \, b c^{2} x^{5} + \frac {2}{5} \, a c d x^{5} + a c^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)*(d*x^4+c)^2,x, algorithm="giac")

[Out]

1/13*b*d^2*x^13 + 2/9*b*c*d*x^9 + 1/9*a*d^2*x^9 + 1/5*b*c^2*x^5 + 2/5*a*c*d*x^5 + a*c^2*x

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Mupad [B]
time = 0.05, size = 48, normalized size = 0.96 \begin {gather*} x^5\,\left (\frac {b\,c^2}{5}+\frac {2\,a\,d\,c}{5}\right )+x^9\,\left (\frac {a\,d^2}{9}+\frac {2\,b\,c\,d}{9}\right )+\frac {b\,d^2\,x^{13}}{13}+a\,c^2\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^4)*(c + d*x^4)^2,x)

[Out]

x^5*((b*c^2)/5 + (2*a*c*d)/5) + x^9*((a*d^2)/9 + (2*b*c*d)/9) + (b*d^2*x^13)/13 + a*c^2*x

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