3.2.46 \(\int (a+b x^4) (c+d x^4)^4 \, dx\) [146]

Optimal. Leaf size=94 \[ a c^4 x+\frac {1}{5} c^3 (b c+4 a d) x^5+\frac {2}{9} c^2 d (2 b c+3 a d) x^9+\frac {2}{13} c d^2 (3 b c+2 a d) x^{13}+\frac {1}{17} d^3 (4 b c+a d) x^{17}+\frac {1}{21} b d^4 x^{21} \]

[Out]

a*c^4*x+1/5*c^3*(4*a*d+b*c)*x^5+2/9*c^2*d*(3*a*d+2*b*c)*x^9+2/13*c*d^2*(2*a*d+3*b*c)*x^13+1/17*d^3*(a*d+4*b*c)
*x^17+1/21*b*d^4*x^21

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Rubi [A]
time = 0.05, antiderivative size = 94, 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}{5} c^3 x^5 (4 a d+b c)+\frac {2}{9} c^2 d x^9 (3 a d+2 b c)+\frac {1}{17} d^3 x^{17} (a d+4 b c)+\frac {2}{13} c d^2 x^{13} (2 a d+3 b c)+a c^4 x+\frac {1}{21} b d^4 x^{21} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)*(c + d*x^4)^4,x]

[Out]

a*c^4*x + (c^3*(b*c + 4*a*d)*x^5)/5 + (2*c^2*d*(2*b*c + 3*a*d)*x^9)/9 + (2*c*d^2*(3*b*c + 2*a*d)*x^13)/13 + (d
^3*(4*b*c + a*d)*x^17)/17 + (b*d^4*x^21)/21

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 )^4 \, dx &=\int \left (a c^4+c^3 (b c+4 a d) x^4+2 c^2 d (2 b c+3 a d) x^8+2 c d^2 (3 b c+2 a d) x^{12}+d^3 (4 b c+a d) x^{16}+b d^4 x^{20}\right ) \, dx\\ &=a c^4 x+\frac {1}{5} c^3 (b c+4 a d) x^5+\frac {2}{9} c^2 d (2 b c+3 a d) x^9+\frac {2}{13} c d^2 (3 b c+2 a d) x^{13}+\frac {1}{17} d^3 (4 b c+a d) x^{17}+\frac {1}{21} b d^4 x^{21}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 94, normalized size = 1.00 \begin {gather*} a c^4 x+\frac {1}{5} c^3 (b c+4 a d) x^5+\frac {2}{9} c^2 d (2 b c+3 a d) x^9+\frac {2}{13} c d^2 (3 b c+2 a d) x^{13}+\frac {1}{17} d^3 (4 b c+a d) x^{17}+\frac {1}{21} b d^4 x^{21} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a*c^4*x + (c^3*(b*c + 4*a*d)*x^5)/5 + (2*c^2*d*(2*b*c + 3*a*d)*x^9)/9 + (2*c*d^2*(3*b*c + 2*a*d)*x^13)/13 + (d
^3*(4*b*c + a*d)*x^17)/17 + (b*d^4*x^21)/21

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Maple [A]
time = 0.28, size = 97, normalized size = 1.03

method	result	size
norman	\(a \,c^{4} x +\left (\frac {4}{5} a \,c^{3} d +\frac {1}{5} b \,c^{4}\right ) x^{5}+\left (\frac {2}{3} a \,c^{2} d^{2}+\frac {4}{9} b \,c^{3} d \right ) x^{9}+\left (\frac {4}{13} a c \,d^{3}+\frac {6}{13} d^{2} b \,c^{2}\right ) x^{13}+\left (\frac {1}{17} a \,d^{4}+\frac {4}{17} b c \,d^{3}\right ) x^{17}+\frac {b \,d^{4} x^{21}}{21}\)	\(95\)
default	\(\frac {b \,d^{4} x^{21}}{21}+\frac {\left (a \,d^{4}+4 b c \,d^{3}\right ) x^{17}}{17}+\frac {\left (4 a c \,d^{3}+6 d^{2} b \,c^{2}\right ) x^{13}}{13}+\frac {\left (6 a \,c^{2} d^{2}+4 b \,c^{3} d \right ) x^{9}}{9}+\frac {\left (4 a \,c^{3} d +b \,c^{4}\right ) x^{5}}{5}+a \,c^{4} x\)	\(97\)
gosper	\(a \,c^{4} x +\frac {4}{5} x^{5} a \,c^{3} d +\frac {1}{5} x^{5} b \,c^{4}+\frac {2}{3} x^{9} a \,c^{2} d^{2}+\frac {4}{9} x^{9} b \,c^{3} d +\frac {4}{13} x^{13} a c \,d^{3}+\frac {6}{13} x^{13} d^{2} b \,c^{2}+\frac {1}{17} x^{17} a \,d^{4}+\frac {4}{17} x^{17} b c \,d^{3}+\frac {1}{21} b \,d^{4} x^{21}\)	\(99\)
risch	\(a \,c^{4} x +\frac {4}{5} x^{5} a \,c^{3} d +\frac {1}{5} x^{5} b \,c^{4}+\frac {2}{3} x^{9} a \,c^{2} d^{2}+\frac {4}{9} x^{9} b \,c^{3} d +\frac {4}{13} x^{13} a c \,d^{3}+\frac {6}{13} x^{13} d^{2} b \,c^{2}+\frac {1}{17} x^{17} a \,d^{4}+\frac {4}{17} x^{17} b c \,d^{3}+\frac {1}{21} b \,d^{4} x^{21}\)	\(99\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/21*b*d^4*x^21+1/17*(a*d^4+4*b*c*d^3)*x^17+1/13*(4*a*c*d^3+6*b*c^2*d^2)*x^13+1/9*(6*a*c^2*d^2+4*b*c^3*d)*x^9+
1/5*(4*a*c^3*d+b*c^4)*x^5+a*c^4*x

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Maxima [A]
time = 0.28, size = 96, normalized size = 1.02 \begin {gather*} \frac {1}{21} \, b d^{4} x^{21} + \frac {1}{17} \, {\left (4 \, b c d^{3} + a d^{4}\right )} x^{17} + \frac {2}{13} \, {\left (3 \, b c^{2} d^{2} + 2 \, a c d^{3}\right )} x^{13} + \frac {2}{9} \, {\left (2 \, b c^{3} d + 3 \, a c^{2} d^{2}\right )} x^{9} + a c^{4} x + \frac {1}{5} \, {\left (b c^{4} + 4 \, a c^{3} d\right )} x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/21*b*d^4*x^21 + 1/17*(4*b*c*d^3 + a*d^4)*x^17 + 2/13*(3*b*c^2*d^2 + 2*a*c*d^3)*x^13 + 2/9*(2*b*c^3*d + 3*a*c
^2*d^2)*x^9 + a*c^4*x + 1/5*(b*c^4 + 4*a*c^3*d)*x^5

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Fricas [A]
time = 2.96, size = 96, normalized size = 1.02 \begin {gather*} \frac {1}{21} \, b d^{4} x^{21} + \frac {1}{17} \, {\left (4 \, b c d^{3} + a d^{4}\right )} x^{17} + \frac {2}{13} \, {\left (3 \, b c^{2} d^{2} + 2 \, a c d^{3}\right )} x^{13} + \frac {2}{9} \, {\left (2 \, b c^{3} d + 3 \, a c^{2} d^{2}\right )} x^{9} + a c^{4} x + \frac {1}{5} \, {\left (b c^{4} + 4 \, a c^{3} d\right )} x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/21*b*d^4*x^21 + 1/17*(4*b*c*d^3 + a*d^4)*x^17 + 2/13*(3*b*c^2*d^2 + 2*a*c*d^3)*x^13 + 2/9*(2*b*c^3*d + 3*a*c
^2*d^2)*x^9 + a*c^4*x + 1/5*(b*c^4 + 4*a*c^3*d)*x^5

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Sympy [A]
time = 0.02, size = 107, normalized size = 1.14 \begin {gather*} a c^{4} x + \frac {b d^{4} x^{21}}{21} + x^{17} \left (\frac {a d^{4}}{17} + \frac {4 b c d^{3}}{17}\right ) + x^{13} \cdot \left (\frac {4 a c d^{3}}{13} + \frac {6 b c^{2} d^{2}}{13}\right ) + x^{9} \cdot \left (\frac {2 a c^{2} d^{2}}{3} + \frac {4 b c^{3} d}{9}\right ) + x^{5} \cdot \left (\frac {4 a c^{3} d}{5} + \frac {b c^{4}}{5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*c**4*x + b*d**4*x**21/21 + x**17*(a*d**4/17 + 4*b*c*d**3/17) + x**13*(4*a*c*d**3/13 + 6*b*c**2*d**2/13) + x*
*9*(2*a*c**2*d**2/3 + 4*b*c**3*d/9) + x**5*(4*a*c**3*d/5 + b*c**4/5)

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Giac [A]
time = 0.54, size = 98, normalized size = 1.04 \begin {gather*} \frac {1}{21} \, b d^{4} x^{21} + \frac {4}{17} \, b c d^{3} x^{17} + \frac {1}{17} \, a d^{4} x^{17} + \frac {6}{13} \, b c^{2} d^{2} x^{13} + \frac {4}{13} \, a c d^{3} x^{13} + \frac {4}{9} \, b c^{3} d x^{9} + \frac {2}{3} \, a c^{2} d^{2} x^{9} + \frac {1}{5} \, b c^{4} x^{5} + \frac {4}{5} \, a c^{3} d x^{5} + a c^{4} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/21*b*d^4*x^21 + 4/17*b*c*d^3*x^17 + 1/17*a*d^4*x^17 + 6/13*b*c^2*d^2*x^13 + 4/13*a*c*d^3*x^13 + 4/9*b*c^3*d*
x^9 + 2/3*a*c^2*d^2*x^9 + 1/5*b*c^4*x^5 + 4/5*a*c^3*d*x^5 + a*c^4*x

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Mupad [B]
time = 1.30, size = 88, normalized size = 0.94 \begin {gather*} x^5\,\left (\frac {b\,c^4}{5}+\frac {4\,a\,d\,c^3}{5}\right )+x^{17}\,\left (\frac {a\,d^4}{17}+\frac {4\,b\,c\,d^3}{17}\right )+\frac {b\,d^4\,x^{21}}{21}+a\,c^4\,x+\frac {2\,c^2\,d\,x^9\,\left (3\,a\,d+2\,b\,c\right )}{9}+\frac {2\,c\,d^2\,x^{13}\,\left (2\,a\,d+3\,b\,c\right )}{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^4)*(c + d*x^4)^4,x)

[Out]

x^5*((b*c^4)/5 + (4*a*c^3*d)/5) + x^17*((a*d^4)/17 + (4*b*c*d^3)/17) + (b*d^4*x^21)/21 + a*c^4*x + (2*c^2*d*x^
9*(3*a*d + 2*b*c))/9 + (2*c*d^2*x^13*(2*a*d + 3*b*c))/13

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