3.13.59 \(\int (a+b x)^4 (c+d x)^3 \, dx\) [1259]

Optimal. Leaf size=92 \[ \frac {(b c-a d)^3 (a+b x)^5}{5 b^4}+\frac {d (b c-a d)^2 (a+b x)^6}{2 b^4}+\frac {3 d^2 (b c-a d) (a+b x)^7}{7 b^4}+\frac {d^3 (a+b x)^8}{8 b^4} \]

[Out]

1/5*(-a*d+b*c)^3*(b*x+a)^5/b^4+1/2*d*(-a*d+b*c)^2*(b*x+a)^6/b^4+3/7*d^2*(-a*d+b*c)*(b*x+a)^7/b^4+1/8*d^3*(b*x+
a)^8/b^4

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Rubi [A]
time = 0.08, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45} \begin {gather*} \frac {3 d^2 (a+b x)^7 (b c-a d)}{7 b^4}+\frac {d (a+b x)^6 (b c-a d)^2}{2 b^4}+\frac {(a+b x)^5 (b c-a d)^3}{5 b^4}+\frac {d^3 (a+b x)^8}{8 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^4*(c + d*x)^3,x]

[Out]

((b*c - a*d)^3*(a + b*x)^5)/(5*b^4) + (d*(b*c - a*d)^2*(a + b*x)^6)/(2*b^4) + (3*d^2*(b*c - a*d)*(a + b*x)^7)/
(7*b^4) + (d^3*(a + b*x)^8)/(8*b^4)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int (a+b x)^4 (c+d x)^3 \, dx &=\int \left (\frac {(b c-a d)^3 (a+b x)^4}{b^3}+\frac {3 d (b c-a d)^2 (a+b x)^5}{b^3}+\frac {3 d^2 (b c-a d) (a+b x)^6}{b^3}+\frac {d^3 (a+b x)^7}{b^3}\right ) \, dx\\ &=\frac {(b c-a d)^3 (a+b x)^5}{5 b^4}+\frac {d (b c-a d)^2 (a+b x)^6}{2 b^4}+\frac {3 d^2 (b c-a d) (a+b x)^7}{7 b^4}+\frac {d^3 (a+b x)^8}{8 b^4}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(217\) vs. \(2(92)=184\).
time = 0.02, size = 217, normalized size = 2.36 \begin {gather*} a^4 c^3 x+\frac {1}{2} a^3 c^2 (4 b c+3 a d) x^2+a^2 c \left (2 b^2 c^2+4 a b c d+a^2 d^2\right ) x^3+\frac {1}{4} a \left (4 b^3 c^3+18 a b^2 c^2 d+12 a^2 b c d^2+a^3 d^3\right ) x^4+\frac {1}{5} b \left (b^3 c^3+12 a b^2 c^2 d+18 a^2 b c d^2+4 a^3 d^3\right ) x^5+\frac {1}{2} b^2 d \left (b^2 c^2+4 a b c d+2 a^2 d^2\right ) x^6+\frac {1}{7} b^3 d^2 (3 b c+4 a d) x^7+\frac {1}{8} b^4 d^3 x^8 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^4*(c + d*x)^3,x]

[Out]

a^4*c^3*x + (a^3*c^2*(4*b*c + 3*a*d)*x^2)/2 + a^2*c*(2*b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^3 + (a*(4*b^3*c^3 + 18
*a*b^2*c^2*d + 12*a^2*b*c*d^2 + a^3*d^3)*x^4)/4 + (b*(b^3*c^3 + 12*a*b^2*c^2*d + 18*a^2*b*c*d^2 + 4*a^3*d^3)*x
^5)/5 + (b^2*d*(b^2*c^2 + 4*a*b*c*d + 2*a^2*d^2)*x^6)/2 + (b^3*d^2*(3*b*c + 4*a*d)*x^7)/7 + (b^4*d^3*x^8)/8

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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(216\) vs. \(2(92)=184\).
time = 3.18, size = 204, normalized size = 2.22 \begin {gather*} x \left (a^4 c^3+\frac {a^3 c^2 x \left (3 a d+4 b c\right )}{2}+a^2 c x^2 \left (a^2 d^2+4 a b c d+2 b^2 c^2\right )+\frac {a x^3 \left (a^3 d^3+12 a^2 b c d^2+18 a b^2 c^2 d+4 b^3 c^3\right )}{4}+\frac {b x^4 \left (4 a^3 d^3+18 a^2 b c d^2+12 a b^2 c^2 d+b^3 c^3\right )}{5}+\frac {b^2 d x^5 \left (2 a^2 d^2+4 a b c d+b^2 c^2\right )}{2}+\frac {b^3 d^2 x^6 \left (4 a d+3 b c\right )}{7}+\frac {b^4 d^3 x^7}{8}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[(a + b*x)^4*(c + d*x)^3,x]')

[Out]

x (a ^ 4 c ^ 3 + a ^ 3 c ^ 2 x (3 a d + 4 b c) / 2 + a ^ 2 c x ^ 2 (a ^ 2 d ^ 2 + 4 a b c d + 2 b ^ 2 c ^ 2) +
a x ^ 3 (a ^ 3 d ^ 3 + 12 a ^ 2 b c d ^ 2 + 18 a b ^ 2 c ^ 2 d + 4 b ^ 3 c ^ 3) / 4 + b x ^ 4 (4 a ^ 3 d ^ 3
+ 18 a ^ 2 b c d ^ 2 + 12 a b ^ 2 c ^ 2 d + b ^ 3 c ^ 3) / 5 + b ^ 2 d x ^ 5 (2 a ^ 2 d ^ 2 + 4 a b c d + b ^
2 c ^ 2) / 2 + b ^ 3 d ^ 2 x ^ 6 (4 a d + 3 b c) / 7 + b ^ 4 d ^ 3 x ^ 7 / 8)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(228\) vs. \(2(84)=168\).
time = 0.13, size = 229, normalized size = 2.49

method	result	size
norman	\(\frac {b^{4} d^{3} x^{8}}{8}+\left (\frac {4}{7} a \,b^{3} d^{3}+\frac {3}{7} b^{4} c \,d^{2}\right ) x^{7}+\left (b^{2} a^{2} d^{3}+2 a \,b^{3} c \,d^{2}+\frac {1}{2} b^{4} c^{2} d \right ) x^{6}+\left (\frac {4}{5} a^{3} b \,d^{3}+\frac {18}{5} b^{2} a^{2} c \,d^{2}+\frac {12}{5} a \,b^{3} c^{2} d +\frac {1}{5} b^{4} c^{3}\right ) x^{5}+\left (\frac {1}{4} a^{4} d^{3}+3 a^{3} b c \,d^{2}+\frac {9}{2} b^{2} a^{2} c^{2} d +a \,b^{3} c^{3}\right ) x^{4}+\left (a^{4} c \,d^{2}+4 a^{3} b \,c^{2} d +2 b^{2} a^{2} c^{3}\right ) x^{3}+\left (\frac {3}{2} a^{4} c^{2} d +2 a^{3} b \,c^{3}\right ) x^{2}+a^{4} c^{3} x\)	\(222\)
default	\(\frac {b^{4} d^{3} x^{8}}{8}+\frac {\left (4 a \,b^{3} d^{3}+3 b^{4} c \,d^{2}\right ) x^{7}}{7}+\frac {\left (6 b^{2} a^{2} d^{3}+12 a \,b^{3} c \,d^{2}+3 b^{4} c^{2} d \right ) x^{6}}{6}+\frac {\left (4 a^{3} b \,d^{3}+18 b^{2} a^{2} c \,d^{2}+12 a \,b^{3} c^{2} d +b^{4} c^{3}\right ) x^{5}}{5}+\frac {\left (a^{4} d^{3}+12 a^{3} b c \,d^{2}+18 b^{2} a^{2} c^{2} d +4 a \,b^{3} c^{3}\right ) x^{4}}{4}+\frac {\left (3 a^{4} c \,d^{2}+12 a^{3} b \,c^{2} d +6 b^{2} a^{2} c^{3}\right ) x^{3}}{3}+\frac {\left (3 a^{4} c^{2} d +4 a^{3} b \,c^{3}\right ) x^{2}}{2}+a^{4} c^{3} x\)	\(229\)
gosper	\(\frac {1}{8} b^{4} d^{3} x^{8}+\frac {4}{7} x^{7} a \,b^{3} d^{3}+\frac {3}{7} x^{7} b^{4} c \,d^{2}+x^{6} b^{2} a^{2} d^{3}+2 x^{6} a \,b^{3} c \,d^{2}+\frac {1}{2} x^{6} b^{4} c^{2} d +\frac {4}{5} x^{5} a^{3} b \,d^{3}+\frac {18}{5} x^{5} b^{2} a^{2} c \,d^{2}+\frac {12}{5} x^{5} a \,b^{3} c^{2} d +\frac {1}{5} x^{5} b^{4} c^{3}+\frac {1}{4} x^{4} a^{4} d^{3}+3 x^{4} a^{3} b c \,d^{2}+\frac {9}{2} x^{4} b^{2} a^{2} c^{2} d +x^{4} a \,b^{3} c^{3}+a^{4} c \,d^{2} x^{3}+4 a^{3} b \,c^{2} d \,x^{3}+2 a^{2} b^{2} c^{3} x^{3}+\frac {3}{2} x^{2} a^{4} c^{2} d +2 x^{2} a^{3} b \,c^{3}+a^{4} c^{3} x\)	\(246\)
risch	\(\frac {1}{8} b^{4} d^{3} x^{8}+\frac {4}{7} x^{7} a \,b^{3} d^{3}+\frac {3}{7} x^{7} b^{4} c \,d^{2}+x^{6} b^{2} a^{2} d^{3}+2 x^{6} a \,b^{3} c \,d^{2}+\frac {1}{2} x^{6} b^{4} c^{2} d +\frac {4}{5} x^{5} a^{3} b \,d^{3}+\frac {18}{5} x^{5} b^{2} a^{2} c \,d^{2}+\frac {12}{5} x^{5} a \,b^{3} c^{2} d +\frac {1}{5} x^{5} b^{4} c^{3}+\frac {1}{4} x^{4} a^{4} d^{3}+3 x^{4} a^{3} b c \,d^{2}+\frac {9}{2} x^{4} b^{2} a^{2} c^{2} d +x^{4} a \,b^{3} c^{3}+a^{4} c \,d^{2} x^{3}+4 a^{3} b \,c^{2} d \,x^{3}+2 a^{2} b^{2} c^{3} x^{3}+\frac {3}{2} x^{2} a^{4} c^{2} d +2 x^{2} a^{3} b \,c^{3}+a^{4} c^{3} x\)	\(246\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^4*(d*x+c)^3,x,method=_RETURNVERBOSE)

[Out]

1/8*b^4*d^3*x^8+1/7*(4*a*b^3*d^3+3*b^4*c*d^2)*x^7+1/6*(6*a^2*b^2*d^3+12*a*b^3*c*d^2+3*b^4*c^2*d)*x^6+1/5*(4*a^
3*b*d^3+18*a^2*b^2*c*d^2+12*a*b^3*c^2*d+b^4*c^3)*x^5+1/4*(a^4*d^3+12*a^3*b*c*d^2+18*a^2*b^2*c^2*d+4*a*b^3*c^3)
*x^4+1/3*(3*a^4*c*d^2+12*a^3*b*c^2*d+6*a^2*b^2*c^3)*x^3+1/2*(3*a^4*c^2*d+4*a^3*b*c^3)*x^2+a^4*c^3*x

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 225 vs. \(2 (84) = 168\).
time = 0.26, size = 225, normalized size = 2.45 \begin {gather*} \frac {1}{8} \, b^{4} d^{3} x^{8} + a^{4} c^{3} x + \frac {1}{7} \, {\left (3 \, b^{4} c d^{2} + 4 \, a b^{3} d^{3}\right )} x^{7} + \frac {1}{2} \, {\left (b^{4} c^{2} d + 4 \, a b^{3} c d^{2} + 2 \, a^{2} b^{2} d^{3}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} c^{3} + 12 \, a b^{3} c^{2} d + 18 \, a^{2} b^{2} c d^{2} + 4 \, a^{3} b d^{3}\right )} x^{5} + \frac {1}{4} \, {\left (4 \, a b^{3} c^{3} + 18 \, a^{2} b^{2} c^{2} d + 12 \, a^{3} b c d^{2} + a^{4} d^{3}\right )} x^{4} + {\left (2 \, a^{2} b^{2} c^{3} + 4 \, a^{3} b c^{2} d + a^{4} c d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{3} b c^{3} + 3 \, a^{4} c^{2} d\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4*(d*x+c)^3,x, algorithm="maxima")

[Out]

1/8*b^4*d^3*x^8 + a^4*c^3*x + 1/7*(3*b^4*c*d^2 + 4*a*b^3*d^3)*x^7 + 1/2*(b^4*c^2*d + 4*a*b^3*c*d^2 + 2*a^2*b^2
*d^3)*x^6 + 1/5*(b^4*c^3 + 12*a*b^3*c^2*d + 18*a^2*b^2*c*d^2 + 4*a^3*b*d^3)*x^5 + 1/4*(4*a*b^3*c^3 + 18*a^2*b^
2*c^2*d + 12*a^3*b*c*d^2 + a^4*d^3)*x^4 + (2*a^2*b^2*c^3 + 4*a^3*b*c^2*d + a^4*c*d^2)*x^3 + 1/2*(4*a^3*b*c^3 +
3*a^4*c^2*d)*x^2

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 225 vs. \(2 (84) = 168\).
time = 0.29, size = 225, normalized size = 2.45 \begin {gather*} \frac {1}{8} \, b^{4} d^{3} x^{8} + a^{4} c^{3} x + \frac {1}{7} \, {\left (3 \, b^{4} c d^{2} + 4 \, a b^{3} d^{3}\right )} x^{7} + \frac {1}{2} \, {\left (b^{4} c^{2} d + 4 \, a b^{3} c d^{2} + 2 \, a^{2} b^{2} d^{3}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} c^{3} + 12 \, a b^{3} c^{2} d + 18 \, a^{2} b^{2} c d^{2} + 4 \, a^{3} b d^{3}\right )} x^{5} + \frac {1}{4} \, {\left (4 \, a b^{3} c^{3} + 18 \, a^{2} b^{2} c^{2} d + 12 \, a^{3} b c d^{2} + a^{4} d^{3}\right )} x^{4} + {\left (2 \, a^{2} b^{2} c^{3} + 4 \, a^{3} b c^{2} d + a^{4} c d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{3} b c^{3} + 3 \, a^{4} c^{2} d\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4*(d*x+c)^3,x, algorithm="fricas")

[Out]

1/8*b^4*d^3*x^8 + a^4*c^3*x + 1/7*(3*b^4*c*d^2 + 4*a*b^3*d^3)*x^7 + 1/2*(b^4*c^2*d + 4*a*b^3*c*d^2 + 2*a^2*b^2
*d^3)*x^6 + 1/5*(b^4*c^3 + 12*a*b^3*c^2*d + 18*a^2*b^2*c*d^2 + 4*a^3*b*d^3)*x^5 + 1/4*(4*a*b^3*c^3 + 18*a^2*b^
2*c^2*d + 12*a^3*b*c*d^2 + a^4*d^3)*x^4 + (2*a^2*b^2*c^3 + 4*a^3*b*c^2*d + a^4*c*d^2)*x^3 + 1/2*(4*a^3*b*c^3 +
3*a^4*c^2*d)*x^2

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (80) = 160\).
time = 0.05, size = 243, normalized size = 2.64 \begin {gather*} a^{4} c^{3} x + \frac {b^{4} d^{3} x^{8}}{8} + x^{7} \cdot \left (\frac {4 a b^{3} d^{3}}{7} + \frac {3 b^{4} c d^{2}}{7}\right ) + x^{6} \left (a^{2} b^{2} d^{3} + 2 a b^{3} c d^{2} + \frac {b^{4} c^{2} d}{2}\right ) + x^{5} \cdot \left (\frac {4 a^{3} b d^{3}}{5} + \frac {18 a^{2} b^{2} c d^{2}}{5} + \frac {12 a b^{3} c^{2} d}{5} + \frac {b^{4} c^{3}}{5}\right ) + x^{4} \left (\frac {a^{4} d^{3}}{4} + 3 a^{3} b c d^{2} + \frac {9 a^{2} b^{2} c^{2} d}{2} + a b^{3} c^{3}\right ) + x^{3} \left (a^{4} c d^{2} + 4 a^{3} b c^{2} d + 2 a^{2} b^{2} c^{3}\right ) + x^{2} \cdot \left (\frac {3 a^{4} c^{2} d}{2} + 2 a^{3} b c^{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**4*(d*x+c)**3,x)

[Out]

a**4*c**3*x + b**4*d**3*x**8/8 + x**7*(4*a*b**3*d**3/7 + 3*b**4*c*d**2/7) + x**6*(a**2*b**2*d**3 + 2*a*b**3*c*
d**2 + b**4*c**2*d/2) + x**5*(4*a**3*b*d**3/5 + 18*a**2*b**2*c*d**2/5 + 12*a*b**3*c**2*d/5 + b**4*c**3/5) + x*
*4*(a**4*d**3/4 + 3*a**3*b*c*d**2 + 9*a**2*b**2*c**2*d/2 + a*b**3*c**3) + x**3*(a**4*c*d**2 + 4*a**3*b*c**2*d
+ 2*a**2*b**2*c**3) + x**2*(3*a**4*c**2*d/2 + 2*a**3*b*c**3)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (84) = 168\).
time = 0.00, size = 267, normalized size = 2.90 \begin {gather*} \frac {1}{8} x^{8} b^{4} d^{3}+\frac {3}{7} x^{7} b^{4} d^{2} c+\frac {4}{7} x^{7} b^{3} a d^{3}+\frac {1}{2} x^{6} b^{4} d c^{2}+2 x^{6} b^{3} a d^{2} c+x^{6} b^{2} a^{2} d^{3}+\frac {1}{5} x^{5} b^{4} c^{3}+\frac {12}{5} x^{5} b^{3} a d c^{2}+\frac {18}{5} x^{5} b^{2} a^{2} d^{2} c+\frac {4}{5} x^{5} b a^{3} d^{3}+x^{4} b^{3} a c^{3}+\frac {9}{2} x^{4} b^{2} a^{2} d c^{2}+3 x^{4} b a^{3} d^{2} c+\frac {1}{4} x^{4} a^{4} d^{3}+2 x^{3} b^{2} a^{2} c^{3}+4 x^{3} b a^{3} d c^{2}+x^{3} a^{4} d^{2} c+2 x^{2} b a^{3} c^{3}+\frac {3}{2} x^{2} a^{4} d c^{2}+x a^{4} c^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4*(d*x+c)^3,x)

[Out]

1/8*b^4*d^3*x^8 + 3/7*b^4*c*d^2*x^7 + 4/7*a*b^3*d^3*x^7 + 1/2*b^4*c^2*d*x^6 + 2*a*b^3*c*d^2*x^6 + a^2*b^2*d^3*
x^6 + 1/5*b^4*c^3*x^5 + 12/5*a*b^3*c^2*d*x^5 + 18/5*a^2*b^2*c*d^2*x^5 + 4/5*a^3*b*d^3*x^5 + a*b^3*c^3*x^4 + 9/
2*a^2*b^2*c^2*d*x^4 + 3*a^3*b*c*d^2*x^4 + 1/4*a^4*d^3*x^4 + 2*a^2*b^2*c^3*x^3 + 4*a^3*b*c^2*d*x^3 + a^4*c*d^2*
x^3 + 2*a^3*b*c^3*x^2 + 3/2*a^4*c^2*d*x^2 + a^4*c^3*x

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Mupad [B]
time = 0.21, size = 208, normalized size = 2.26 \begin {gather*} x^4\,\left (\frac {a^4\,d^3}{4}+3\,a^3\,b\,c\,d^2+\frac {9\,a^2\,b^2\,c^2\,d}{2}+a\,b^3\,c^3\right )+x^5\,\left (\frac {4\,a^3\,b\,d^3}{5}+\frac {18\,a^2\,b^2\,c\,d^2}{5}+\frac {12\,a\,b^3\,c^2\,d}{5}+\frac {b^4\,c^3}{5}\right )+a^4\,c^3\,x+\frac {b^4\,d^3\,x^8}{8}+\frac {a^3\,c^2\,x^2\,\left (3\,a\,d+4\,b\,c\right )}{2}+\frac {b^3\,d^2\,x^7\,\left (4\,a\,d+3\,b\,c\right )}{7}+a^2\,c\,x^3\,\left (a^2\,d^2+4\,a\,b\,c\,d+2\,b^2\,c^2\right )+\frac {b^2\,d\,x^6\,\left (2\,a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^4*(c + d*x)^3,x)

[Out]

x^4*((a^4*d^3)/4 + a*b^3*c^3 + (9*a^2*b^2*c^2*d)/2 + 3*a^3*b*c*d^2) + x^5*((b^4*c^3)/5 + (4*a^3*b*d^3)/5 + (18
*a^2*b^2*c*d^2)/5 + (12*a*b^3*c^2*d)/5) + a^4*c^3*x + (b^4*d^3*x^8)/8 + (a^3*c^2*x^2*(3*a*d + 4*b*c))/2 + (b^3
*d^2*x^7*(4*a*d + 3*b*c))/7 + a^2*c*x^3*(a^2*d^2 + 2*b^2*c^2 + 4*a*b*c*d) + (b^2*d*x^6*(2*a^2*d^2 + b^2*c^2 +
4*a*b*c*d))/2

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