3.16.55 \(\int (a+b x)^m (c+d x)^3 \, dx\)

Optimal. Leaf size=110 \[ \frac {3 d^2 (b c-a d) (a+b x)^{m+3}}{b^4 (m+3)}+\frac {(b c-a d)^3 (a+b x)^{m+1}}{b^4 (m+1)}+\frac {3 d (b c-a d)^2 (a+b x)^{m+2}}{b^4 (m+2)}+\frac {d^3 (a+b x)^{m+4}}{b^4 (m+4)} \]

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Rubi [A]  time = 0.05, antiderivative size = 110, 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 = {43} \begin {gather*} \frac {3 d^2 (b c-a d) (a+b x)^{m+3}}{b^4 (m+3)}+\frac {(b c-a d)^3 (a+b x)^{m+1}}{b^4 (m+1)}+\frac {3 d (b c-a d)^2 (a+b x)^{m+2}}{b^4 (m+2)}+\frac {d^3 (a+b x)^{m+4}}{b^4 (m+4)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^m*(c + d*x)^3,x]

[Out]

((b*c - a*d)^3*(a + b*x)^(1 + m))/(b^4*(1 + m)) + (3*d*(b*c - a*d)^2*(a + b*x)^(2 + m))/(b^4*(2 + m)) + (3*d^2
*(b*c - a*d)*(a + b*x)^(3 + m))/(b^4*(3 + m)) + (d^3*(a + b*x)^(4 + m))/(b^4*(4 + m))

Rule 43

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)^m (c+d x)^3 \, dx &=\int \left (\frac {(b c-a d)^3 (a+b x)^m}{b^3}+\frac {3 d (b c-a d)^2 (a+b x)^{1+m}}{b^3}+\frac {3 d^2 (b c-a d) (a+b x)^{2+m}}{b^3}+\frac {d^3 (a+b x)^{3+m}}{b^3}\right ) \, dx\\ &=\frac {(b c-a d)^3 (a+b x)^{1+m}}{b^4 (1+m)}+\frac {3 d (b c-a d)^2 (a+b x)^{2+m}}{b^4 (2+m)}+\frac {3 d^2 (b c-a d) (a+b x)^{3+m}}{b^4 (3+m)}+\frac {d^3 (a+b x)^{4+m}}{b^4 (4+m)}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 94, normalized size = 0.85 \begin {gather*} \frac {(a+b x)^{m+1} \left (\frac {3 d^2 (a+b x)^2 (b c-a d)}{m+3}+\frac {3 d (a+b x) (b c-a d)^2}{m+2}+\frac {(b c-a d)^3}{m+1}+\frac {d^3 (a+b x)^3}{m+4}\right )}{b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^m*(c + d*x)^3,x]

[Out]

((a + b*x)^(1 + m)*((b*c - a*d)^3/(1 + m) + (3*d*(b*c - a*d)^2*(a + b*x))/(2 + m) + (3*d^2*(b*c - a*d)*(a + b*
x)^2)/(3 + m) + (d^3*(a + b*x)^3)/(4 + m)))/b^4

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IntegrateAlgebraic [F]  time = 0.05, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x)^m (c+d x)^3 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

Defer[IntegrateAlgebraic][(a + b*x)^m*(c + d*x)^3, x]

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fricas [B]  time = 1.26, size = 497, normalized size = 4.52 \begin {gather*} \frac {{\left (a b^{3} c^{3} m^{3} + 24 \, a b^{3} c^{3} - 36 \, a^{2} b^{2} c^{2} d + 24 \, a^{3} b c d^{2} - 6 \, a^{4} d^{3} + {\left (b^{4} d^{3} m^{3} + 6 \, b^{4} d^{3} m^{2} + 11 \, b^{4} d^{3} m + 6 \, b^{4} d^{3}\right )} x^{4} + {\left (24 \, b^{4} c d^{2} + {\left (3 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} m^{3} + 3 \, {\left (7 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} m^{2} + 2 \, {\left (21 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} m\right )} x^{3} + 3 \, {\left (3 \, a b^{3} c^{3} - a^{2} b^{2} c^{2} d\right )} m^{2} + 3 \, {\left (12 \, b^{4} c^{2} d + {\left (b^{4} c^{2} d + a b^{3} c d^{2}\right )} m^{3} + {\left (8 \, b^{4} c^{2} d + 5 \, a b^{3} c d^{2} - a^{2} b^{2} d^{3}\right )} m^{2} + {\left (19 \, b^{4} c^{2} d + 4 \, a b^{3} c d^{2} - a^{2} b^{2} d^{3}\right )} m\right )} x^{2} + {\left (26 \, a b^{3} c^{3} - 21 \, a^{2} b^{2} c^{2} d + 6 \, a^{3} b c d^{2}\right )} m + {\left (24 \, b^{4} c^{3} + {\left (b^{4} c^{3} + 3 \, a b^{3} c^{2} d\right )} m^{3} + 3 \, {\left (3 \, b^{4} c^{3} + 7 \, a b^{3} c^{2} d - 2 \, a^{2} b^{2} c d^{2}\right )} m^{2} + 2 \, {\left (13 \, b^{4} c^{3} + 18 \, a b^{3} c^{2} d - 12 \, a^{2} b^{2} c d^{2} + 3 \, a^{3} b d^{3}\right )} m\right )} x\right )} {\left (b x + a\right )}^{m}}{b^{4} m^{4} + 10 \, b^{4} m^{3} + 35 \, b^{4} m^{2} + 50 \, b^{4} m + 24 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*b^3*c^3*m^3 + 24*a*b^3*c^3 - 36*a^2*b^2*c^2*d + 24*a^3*b*c*d^2 - 6*a^4*d^3 + (b^4*d^3*m^3 + 6*b^4*d^3*m^2 +
11*b^4*d^3*m + 6*b^4*d^3)*x^4 + (24*b^4*c*d^2 + (3*b^4*c*d^2 + a*b^3*d^3)*m^3 + 3*(7*b^4*c*d^2 + a*b^3*d^3)*m
^2 + 2*(21*b^4*c*d^2 + a*b^3*d^3)*m)*x^3 + 3*(3*a*b^3*c^3 - a^2*b^2*c^2*d)*m^2 + 3*(12*b^4*c^2*d + (b^4*c^2*d
+ a*b^3*c*d^2)*m^3 + (8*b^4*c^2*d + 5*a*b^3*c*d^2 - a^2*b^2*d^3)*m^2 + (19*b^4*c^2*d + 4*a*b^3*c*d^2 - a^2*b^2
*d^3)*m)*x^2 + (26*a*b^3*c^3 - 21*a^2*b^2*c^2*d + 6*a^3*b*c*d^2)*m + (24*b^4*c^3 + (b^4*c^3 + 3*a*b^3*c^2*d)*m
^3 + 3*(3*b^4*c^3 + 7*a*b^3*c^2*d - 2*a^2*b^2*c*d^2)*m^2 + 2*(13*b^4*c^3 + 18*a*b^3*c^2*d - 12*a^2*b^2*c*d^2 +
3*a^3*b*d^3)*m)*x)*(b*x + a)^m/(b^4*m^4 + 10*b^4*m^3 + 35*b^4*m^2 + 50*b^4*m + 24*b^4)

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giac [B]  time = 1.04, size = 833, normalized size = 7.57 \begin {gather*} \frac {{\left (b x + a\right )}^{m} b^{4} d^{3} m^{3} x^{4} + 3 \, {\left (b x + a\right )}^{m} b^{4} c d^{2} m^{3} x^{3} + {\left (b x + a\right )}^{m} a b^{3} d^{3} m^{3} x^{3} + 6 \, {\left (b x + a\right )}^{m} b^{4} d^{3} m^{2} x^{4} + 3 \, {\left (b x + a\right )}^{m} b^{4} c^{2} d m^{3} x^{2} + 3 \, {\left (b x + a\right )}^{m} a b^{3} c d^{2} m^{3} x^{2} + 21 \, {\left (b x + a\right )}^{m} b^{4} c d^{2} m^{2} x^{3} + 3 \, {\left (b x + a\right )}^{m} a b^{3} d^{3} m^{2} x^{3} + 11 \, {\left (b x + a\right )}^{m} b^{4} d^{3} m x^{4} + {\left (b x + a\right )}^{m} b^{4} c^{3} m^{3} x + 3 \, {\left (b x + a\right )}^{m} a b^{3} c^{2} d m^{3} x + 24 \, {\left (b x + a\right )}^{m} b^{4} c^{2} d m^{2} x^{2} + 15 \, {\left (b x + a\right )}^{m} a b^{3} c d^{2} m^{2} x^{2} - 3 \, {\left (b x + a\right )}^{m} a^{2} b^{2} d^{3} m^{2} x^{2} + 42 \, {\left (b x + a\right )}^{m} b^{4} c d^{2} m x^{3} + 2 \, {\left (b x + a\right )}^{m} a b^{3} d^{3} m x^{3} + 6 \, {\left (b x + a\right )}^{m} b^{4} d^{3} x^{4} + {\left (b x + a\right )}^{m} a b^{3} c^{3} m^{3} + 9 \, {\left (b x + a\right )}^{m} b^{4} c^{3} m^{2} x + 21 \, {\left (b x + a\right )}^{m} a b^{3} c^{2} d m^{2} x - 6 \, {\left (b x + a\right )}^{m} a^{2} b^{2} c d^{2} m^{2} x + 57 \, {\left (b x + a\right )}^{m} b^{4} c^{2} d m x^{2} + 12 \, {\left (b x + a\right )}^{m} a b^{3} c d^{2} m x^{2} - 3 \, {\left (b x + a\right )}^{m} a^{2} b^{2} d^{3} m x^{2} + 24 \, {\left (b x + a\right )}^{m} b^{4} c d^{2} x^{3} + 9 \, {\left (b x + a\right )}^{m} a b^{3} c^{3} m^{2} - 3 \, {\left (b x + a\right )}^{m} a^{2} b^{2} c^{2} d m^{2} + 26 \, {\left (b x + a\right )}^{m} b^{4} c^{3} m x + 36 \, {\left (b x + a\right )}^{m} a b^{3} c^{2} d m x - 24 \, {\left (b x + a\right )}^{m} a^{2} b^{2} c d^{2} m x + 6 \, {\left (b x + a\right )}^{m} a^{3} b d^{3} m x + 36 \, {\left (b x + a\right )}^{m} b^{4} c^{2} d x^{2} + 26 \, {\left (b x + a\right )}^{m} a b^{3} c^{3} m - 21 \, {\left (b x + a\right )}^{m} a^{2} b^{2} c^{2} d m + 6 \, {\left (b x + a\right )}^{m} a^{3} b c d^{2} m + 24 \, {\left (b x + a\right )}^{m} b^{4} c^{3} x + 24 \, {\left (b x + a\right )}^{m} a b^{3} c^{3} - 36 \, {\left (b x + a\right )}^{m} a^{2} b^{2} c^{2} d + 24 \, {\left (b x + a\right )}^{m} a^{3} b c d^{2} - 6 \, {\left (b x + a\right )}^{m} a^{4} d^{3}}{b^{4} m^{4} + 10 \, b^{4} m^{3} + 35 \, b^{4} m^{2} + 50 \, b^{4} m + 24 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b*x + a)^m*b^4*d^3*m^3*x^4 + 3*(b*x + a)^m*b^4*c*d^2*m^3*x^3 + (b*x + a)^m*a*b^3*d^3*m^3*x^3 + 6*(b*x + a)^m
*b^4*d^3*m^2*x^4 + 3*(b*x + a)^m*b^4*c^2*d*m^3*x^2 + 3*(b*x + a)^m*a*b^3*c*d^2*m^3*x^2 + 21*(b*x + a)^m*b^4*c*
d^2*m^2*x^3 + 3*(b*x + a)^m*a*b^3*d^3*m^2*x^3 + 11*(b*x + a)^m*b^4*d^3*m*x^4 + (b*x + a)^m*b^4*c^3*m^3*x + 3*(
b*x + a)^m*a*b^3*c^2*d*m^3*x + 24*(b*x + a)^m*b^4*c^2*d*m^2*x^2 + 15*(b*x + a)^m*a*b^3*c*d^2*m^2*x^2 - 3*(b*x
+ a)^m*a^2*b^2*d^3*m^2*x^2 + 42*(b*x + a)^m*b^4*c*d^2*m*x^3 + 2*(b*x + a)^m*a*b^3*d^3*m*x^3 + 6*(b*x + a)^m*b^
4*d^3*x^4 + (b*x + a)^m*a*b^3*c^3*m^3 + 9*(b*x + a)^m*b^4*c^3*m^2*x + 21*(b*x + a)^m*a*b^3*c^2*d*m^2*x - 6*(b*
x + a)^m*a^2*b^2*c*d^2*m^2*x + 57*(b*x + a)^m*b^4*c^2*d*m*x^2 + 12*(b*x + a)^m*a*b^3*c*d^2*m*x^2 - 3*(b*x + a)
^m*a^2*b^2*d^3*m*x^2 + 24*(b*x + a)^m*b^4*c*d^2*x^3 + 9*(b*x + a)^m*a*b^3*c^3*m^2 - 3*(b*x + a)^m*a^2*b^2*c^2*
d*m^2 + 26*(b*x + a)^m*b^4*c^3*m*x + 36*(b*x + a)^m*a*b^3*c^2*d*m*x - 24*(b*x + a)^m*a^2*b^2*c*d^2*m*x + 6*(b*
x + a)^m*a^3*b*d^3*m*x + 36*(b*x + a)^m*b^4*c^2*d*x^2 + 26*(b*x + a)^m*a*b^3*c^3*m - 21*(b*x + a)^m*a^2*b^2*c^
2*d*m + 6*(b*x + a)^m*a^3*b*c*d^2*m + 24*(b*x + a)^m*b^4*c^3*x + 24*(b*x + a)^m*a*b^3*c^3 - 36*(b*x + a)^m*a^2
*b^2*c^2*d + 24*(b*x + a)^m*a^3*b*c*d^2 - 6*(b*x + a)^m*a^4*d^3)/(b^4*m^4 + 10*b^4*m^3 + 35*b^4*m^2 + 50*b^4*m
+ 24*b^4)

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maple [B]  time = 0.01, size = 389, normalized size = 3.54 \begin {gather*} -\frac {\left (-b^{3} d^{3} m^{3} x^{3}-3 b^{3} c \,d^{2} m^{3} x^{2}-6 b^{3} d^{3} m^{2} x^{3}+3 a \,b^{2} d^{3} m^{2} x^{2}-3 b^{3} c^{2} d \,m^{3} x -21 b^{3} c \,d^{2} m^{2} x^{2}-11 b^{3} d^{3} m \,x^{3}+6 a \,b^{2} c \,d^{2} m^{2} x +9 a \,b^{2} d^{3} m \,x^{2}-b^{3} c^{3} m^{3}-24 b^{3} c^{2} d \,m^{2} x -42 b^{3} c \,d^{2} m \,x^{2}-6 b^{3} d^{3} x^{3}-6 a^{2} b \,d^{3} m x +3 a \,b^{2} c^{2} d \,m^{2}+30 a \,b^{2} c \,d^{2} m x +6 a \,b^{2} d^{3} x^{2}-9 b^{3} c^{3} m^{2}-57 b^{3} c^{2} d m x -24 b^{3} c \,d^{2} x^{2}-6 a^{2} b c \,d^{2} m -6 a^{2} b \,d^{3} x +21 a \,b^{2} c^{2} d m +24 a \,b^{2} c \,d^{2} x -26 b^{3} c^{3} m -36 b^{3} c^{2} d x +6 a^{3} d^{3}-24 a^{2} b c \,d^{2}+36 a \,b^{2} c^{2} d -24 b^{3} c^{3}\right ) \left (b x +a \right )^{m +1}}{\left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right ) b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*x+a)^(m+1)*(-b^3*d^3*m^3*x^3-3*b^3*c*d^2*m^3*x^2-6*b^3*d^3*m^2*x^3+3*a*b^2*d^3*m^2*x^2-3*b^3*c^2*d*m^3*x-2
1*b^3*c*d^2*m^2*x^2-11*b^3*d^3*m*x^3+6*a*b^2*c*d^2*m^2*x+9*a*b^2*d^3*m*x^2-b^3*c^3*m^3-24*b^3*c^2*d*m^2*x-42*b
^3*c*d^2*m*x^2-6*b^3*d^3*x^3-6*a^2*b*d^3*m*x+3*a*b^2*c^2*d*m^2+30*a*b^2*c*d^2*m*x+6*a*b^2*d^3*x^2-9*b^3*c^3*m^
2-57*b^3*c^2*d*m*x-24*b^3*c*d^2*x^2-6*a^2*b*c*d^2*m-6*a^2*b*d^3*x+21*a*b^2*c^2*d*m+24*a*b^2*c*d^2*x-26*b^3*c^3
*m-36*b^3*c^2*d*x+6*a^3*d^3-24*a^2*b*c*d^2+36*a*b^2*c^2*d-24*b^3*c^3)/b^4/(m^4+10*m^3+35*m^2+50*m+24)

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maxima [B]  time = 1.17, size = 246, normalized size = 2.24 \begin {gather*} \frac {3 \, {\left (b^{2} {\left (m + 1\right )} x^{2} + a b m x - a^{2}\right )} {\left (b x + a\right )}^{m} c^{2} d}{{\left (m^{2} + 3 \, m + 2\right )} b^{2}} + \frac {{\left (b x + a\right )}^{m + 1} c^{3}}{b {\left (m + 1\right )}} + \frac {3 \, {\left ({\left (m^{2} + 3 \, m + 2\right )} b^{3} x^{3} + {\left (m^{2} + m\right )} a b^{2} x^{2} - 2 \, a^{2} b m x + 2 \, a^{3}\right )} {\left (b x + a\right )}^{m} c d^{2}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} b^{3}} + \frac {{\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} b^{4} x^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} a b^{3} x^{3} - 3 \, {\left (m^{2} + m\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b m x - 6 \, a^{4}\right )} {\left (b x + a\right )}^{m} d^{3}}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(b^2*(m + 1)*x^2 + a*b*m*x - a^2)*(b*x + a)^m*c^2*d/((m^2 + 3*m + 2)*b^2) + (b*x + a)^(m + 1)*c^3/(b*(m + 1)
) + 3*((m^2 + 3*m + 2)*b^3*x^3 + (m^2 + m)*a*b^2*x^2 - 2*a^2*b*m*x + 2*a^3)*(b*x + a)^m*c*d^2/((m^3 + 6*m^2 +
11*m + 6)*b^3) + ((m^3 + 6*m^2 + 11*m + 6)*b^4*x^4 + (m^3 + 3*m^2 + 2*m)*a*b^3*x^3 - 3*(m^2 + m)*a^2*b^2*x^2 +
6*a^3*b*m*x - 6*a^4)*(b*x + a)^m*d^3/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*b^4)

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mupad [B]  time = 0.94, size = 478, normalized size = 4.35 \begin {gather*} \frac {d^3\,x^4\,{\left (a+b\,x\right )}^m\,\left (m^3+6\,m^2+11\,m+6\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}+\frac {a\,{\left (a+b\,x\right )}^m\,\left (-6\,a^3\,d^3+6\,a^2\,b\,c\,d^2\,m+24\,a^2\,b\,c\,d^2-3\,a\,b^2\,c^2\,d\,m^2-21\,a\,b^2\,c^2\,d\,m-36\,a\,b^2\,c^2\,d+b^3\,c^3\,m^3+9\,b^3\,c^3\,m^2+26\,b^3\,c^3\,m+24\,b^3\,c^3\right )}{b^4\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {x\,{\left (a+b\,x\right )}^m\,\left (6\,a^3\,b\,d^3\,m-6\,a^2\,b^2\,c\,d^2\,m^2-24\,a^2\,b^2\,c\,d^2\,m+3\,a\,b^3\,c^2\,d\,m^3+21\,a\,b^3\,c^2\,d\,m^2+36\,a\,b^3\,c^2\,d\,m+b^4\,c^3\,m^3+9\,b^4\,c^3\,m^2+26\,b^4\,c^3\,m+24\,b^4\,c^3\right )}{b^4\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {3\,d\,x^2\,\left (m+1\right )\,{\left (a+b\,x\right )}^m\,\left (-a^2\,d^2\,m+a\,b\,c\,d\,m^2+4\,a\,b\,c\,d\,m+b^2\,c^2\,m^2+7\,b^2\,c^2\,m+12\,b^2\,c^2\right )}{b^2\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {d^2\,x^3\,{\left (a+b\,x\right )}^m\,\left (12\,b\,c+a\,d\,m+3\,b\,c\,m\right )\,\left (m^2+3\,m+2\right )}{b\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^m*(c + d*x)^3,x)

[Out]

(d^3*x^4*(a + b*x)^m*(11*m + 6*m^2 + m^3 + 6))/(50*m + 35*m^2 + 10*m^3 + m^4 + 24) + (a*(a + b*x)^m*(24*b^3*c^
3 - 6*a^3*d^3 + 26*b^3*c^3*m + 9*b^3*c^3*m^2 + b^3*c^3*m^3 - 36*a*b^2*c^2*d + 24*a^2*b*c*d^2 - 21*a*b^2*c^2*d*
m + 6*a^2*b*c*d^2*m - 3*a*b^2*c^2*d*m^2))/(b^4*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) + (x*(a + b*x)^m*(24*b^4*c
^3 + 26*b^4*c^3*m + 9*b^4*c^3*m^2 + b^4*c^3*m^3 + 6*a^3*b*d^3*m + 36*a*b^3*c^2*d*m - 24*a^2*b^2*c*d^2*m + 21*a
*b^3*c^2*d*m^2 + 3*a*b^3*c^2*d*m^3 - 6*a^2*b^2*c*d^2*m^2))/(b^4*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) + (3*d*x^
2*(m + 1)*(a + b*x)^m*(12*b^2*c^2 - a^2*d^2*m + 7*b^2*c^2*m + b^2*c^2*m^2 + 4*a*b*c*d*m + a*b*c*d*m^2))/(b^2*(
50*m + 35*m^2 + 10*m^3 + m^4 + 24)) + (d^2*x^3*(a + b*x)^m*(12*b*c + a*d*m + 3*b*c*m)*(3*m + m^2 + 2))/(b*(50*
m + 35*m^2 + 10*m^3 + m^4 + 24))

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sympy [A]  time = 4.67, size = 4058, normalized size = 36.89

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**m*(c**3*x + 3*c**2*d*x**2/2 + c*d**2*x**3 + d**3*x**4/4), Eq(b, 0)), (6*a**3*d**3*log(a/b + x)/(
6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 11*a**3*d**3/(6*a**3*b**4 + 18*a**2*b**5*x + 18
*a*b**6*x**2 + 6*b**7*x**3) - 6*a**2*b*c*d**2/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) +
18*a**2*b*d**3*x*log(a/b + x)/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 27*a**2*b*d**3*x
/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - 3*a*b**2*c**2*d/(6*a**3*b**4 + 18*a**2*b**5*x
+ 18*a*b**6*x**2 + 6*b**7*x**3) - 18*a*b**2*c*d**2*x/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*
x**3) + 18*a*b**2*d**3*x**2*log(a/b + x)/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 18*a*
b**2*d**3*x**2/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - 2*b**3*c**3/(6*a**3*b**4 + 18*a
**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - 9*b**3*c**2*d*x/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 +
6*b**7*x**3) - 18*b**3*c*d**2*x**2/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 6*b**3*d**3
*x**3*log(a/b + x)/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3), Eq(m, -4)), (-6*a**3*d**3*lo
g(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 9*a**3*d**3/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 6
*a**2*b*c*d**2*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 9*a**2*b*c*d**2/(2*a**2*b**4 + 4*a*b**5
*x + 2*b**6*x**2) - 12*a**2*b*d**3*x*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 12*a**2*b*d**3*x/
(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 3*a*b**2*c**2*d/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 12*a*b**
2*c*d**2*x*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 12*a*b**2*c*d**2*x/(2*a**2*b**4 + 4*a*b**5*
x + 2*b**6*x**2) - 6*a*b**2*d**3*x**2*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - b**3*c**3/(2*a**
2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 6*b**3*c**2*d*x/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 6*b**3*c*d**2*
x**2*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 2*b**3*d**3*x**3/(2*a**2*b**4 + 4*a*b**5*x + 2*b*
*6*x**2), Eq(m, -3)), (6*a**3*d**3*log(a/b + x)/(2*a*b**4 + 2*b**5*x) + 6*a**3*d**3/(2*a*b**4 + 2*b**5*x) - 12
*a**2*b*c*d**2*log(a/b + x)/(2*a*b**4 + 2*b**5*x) - 12*a**2*b*c*d**2/(2*a*b**4 + 2*b**5*x) + 6*a**2*b*d**3*x*l
og(a/b + x)/(2*a*b**4 + 2*b**5*x) + 6*a*b**2*c**2*d*log(a/b + x)/(2*a*b**4 + 2*b**5*x) + 6*a*b**2*c**2*d/(2*a*
b**4 + 2*b**5*x) - 12*a*b**2*c*d**2*x*log(a/b + x)/(2*a*b**4 + 2*b**5*x) - 3*a*b**2*d**3*x**2/(2*a*b**4 + 2*b*
*5*x) - 2*b**3*c**3/(2*a*b**4 + 2*b**5*x) + 6*b**3*c**2*d*x*log(a/b + x)/(2*a*b**4 + 2*b**5*x) + 6*b**3*c*d**2
*x**2/(2*a*b**4 + 2*b**5*x) + b**3*d**3*x**3/(2*a*b**4 + 2*b**5*x), Eq(m, -2)), (-a**3*d**3*log(a/b + x)/b**4
+ 3*a**2*c*d**2*log(a/b + x)/b**3 + a**2*d**3*x/b**3 - 3*a*c**2*d*log(a/b + x)/b**2 - 3*a*c*d**2*x/b**2 - a*d*
*3*x**2/(2*b**2) + c**3*log(a/b + x)/b + 3*c**2*d*x/b + 3*c*d**2*x**2/(2*b) + d**3*x**3/(3*b), Eq(m, -1)), (-6
*a**4*d**3*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 6*a**3*b*c*d**2*m*(a
+ b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 24*a**3*b*c*d**2*(a + b*x)**m/(b*
*4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 6*a**3*b*d**3*m*x*(a + b*x)**m/(b**4*m**4 + 10*
b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) - 3*a**2*b**2*c**2*d*m**2*(a + b*x)**m/(b**4*m**4 + 10*b**4*m*
*3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) - 21*a**2*b**2*c**2*d*m*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b
**4*m**2 + 50*b**4*m + 24*b**4) - 36*a**2*b**2*c**2*d*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 +
50*b**4*m + 24*b**4) - 6*a**2*b**2*c*d**2*m**2*x*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b*
*4*m + 24*b**4) - 24*a**2*b**2*c*d**2*m*x*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m +
24*b**4) - 3*a**2*b**2*d**3*m**2*x**2*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b
**4) - 3*a**2*b**2*d**3*m*x**2*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) +
a*b**3*c**3*m**3*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 9*a*b**3*c**3*
m**2*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 26*a*b**3*c**3*m*(a + b*x)
**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 24*a*b**3*c**3*(a + b*x)**m/(b**4*m**4 +
10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 3*a*b**3*c**2*d*m**3*x*(a + b*x)**m/(b**4*m**4 + 10*b**4
*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 21*a*b**3*c**2*d*m**2*x*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 +
35*b**4*m**2 + 50*b**4*m + 24*b**4) + 36*a*b**3*c**2*d*m*x*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m
**2 + 50*b**4*m + 24*b**4) + 3*a*b**3*c*d**2*m**3*x**2*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 +
50*b**4*m + 24*b**4) + 15*a*b**3*c*d**2*m**2*x**2*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*
b**4*m + 24*b**4) + 12*a*b**3*c*d**2*m*x**2*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m
+ 24*b**4) + a*b**3*d**3*m**3*x**3*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4
) + 3*a*b**3*d**3*m**2*x**3*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 2*a
*b**3*d**3*m*x**3*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + b**4*c**3*m**
3*x*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 9*b**4*c**3*m**2*x*(a + b*x
)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 26*b**4*c**3*m*x*(a + b*x)**m/(b**4*m**
4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 24*b**4*c**3*x*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3
+ 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 3*b**4*c**2*d*m**3*x**2*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b
**4*m**2 + 50*b**4*m + 24*b**4) + 24*b**4*c**2*d*m**2*x**2*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m*
*2 + 50*b**4*m + 24*b**4) + 57*b**4*c**2*d*m*x**2*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b
**4*m + 24*b**4) + 36*b**4*c**2*d*x**2*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*
b**4) + 3*b**4*c*d**2*m**3*x**3*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) +
21*b**4*c*d**2*m**2*x**3*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 42*b*
*4*c*d**2*m*x**3*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 24*b**4*c*d**2
*x**3*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + b**4*d**3*m**3*x**4*(a +
b*x)**m/(b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 6*b**4*d**3*m**2*x**4*(a + b*x)**m/(
b**4*m**4 + 10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 11*b**4*d**3*m*x**4*(a + b*x)**m/(b**4*m**4 +
10*b**4*m**3 + 35*b**4*m**2 + 50*b**4*m + 24*b**4) + 6*b**4*d**3*x**4*(a + b*x)**m/(b**4*m**4 + 10*b**4*m**3
+ 35*b**4*m**2 + 50*b**4*m + 24*b**4), True))

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