3.2.76 \(\int (a+b x)^n (c+d x^3) \, dx\) [176]

Optimal. Leaf size=94 \[ \frac {\left (b^3 c-a^3 d\right ) (a+b x)^{1+n}}{b^4 (1+n)}+\frac {3 a^2 d (a+b x)^{2+n}}{b^4 (2+n)}-\frac {3 a d (a+b x)^{3+n}}{b^4 (3+n)}+\frac {d (a+b x)^{4+n}}{b^4 (4+n)} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 94, 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 = {1864} \begin {gather*} \frac {\left (b^3 c-a^3 d\right ) (a+b x)^{n+1}}{b^4 (n+1)}+\frac {3 a^2 d (a+b x)^{n+2}}{b^4 (n+2)}-\frac {3 a d (a+b x)^{n+3}}{b^4 (n+3)}+\frac {d (a+b x)^{n+4}}{b^4 (n+4)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^n*(c + d*x^3),x]

[Out]

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

Rule 1864

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

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

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Mathematica [A]
time = 0.09, size = 95, normalized size = 1.01 \begin {gather*} \frac {(a+b x)^{1+n} \left (-6 a^3 d+6 a^2 b d (1+n) x-3 a b^2 d \left (2+3 n+n^2\right ) x^2+b^3 \left (6+5 n+n^2\right ) \left (c (4+n)+d (1+n) x^3\right )\right )}{b^4 (1+n) (2+n) (3+n) (4+n)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^n*(c + d*x^3),x]

[Out]

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

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Maple [A]
time = 0.20, size = 167, normalized size = 1.78

method	result	size
gosper	\(-\frac {\left (b x +a \right )^{1+n} \left (-b^{3} d \,n^{3} x^{3}-6 b^{3} d \,n^{2} x^{3}+3 a \,b^{2} d \,n^{2} x^{2}-11 b^{3} d n \,x^{3}+9 a \,b^{2} d n \,x^{2}-b^{3} c \,n^{3}-6 d \,x^{3} b^{3}-6 a^{2} b d n x +6 a d \,x^{2} b^{2}-9 b^{3} c \,n^{2}-6 a^{2} d x b -26 b^{3} c n +6 a^{3} d -24 b^{3} c \right )}{b^{4} \left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right )}\)	\(167\)
risch	\(-\frac {\left (-b^{4} d \,n^{3} x^{4}-a \,b^{3} d \,n^{3} x^{3}-6 b^{4} d \,n^{2} x^{4}-3 a \,b^{3} d \,n^{2} x^{3}-11 b^{4} d n \,x^{4}+3 a^{2} b^{2} d \,n^{2} x^{2}-2 a \,b^{3} d n \,x^{3}-b^{4} c \,n^{3} x -6 d \,x^{4} b^{4}+3 a^{2} b^{2} d n \,x^{2}-a \,b^{3} c \,n^{3}-9 b^{4} c \,n^{2} x -6 a^{3} b d n x -9 a \,b^{3} c \,n^{2}-26 b^{4} c n x -26 a \,b^{3} c n -24 b^{4} c x +6 a^{4} d -24 a \,b^{3} c \right ) \left (b x +a \right )^{n}}{\left (3+n \right ) \left (4+n \right ) \left (2+n \right ) \left (1+n \right ) b^{4}}\)	\(227\)
norman	\(\frac {d \,x^{4} {\mathrm e}^{n \ln \left (b x +a \right )}}{4+n}+\frac {\left (b^{3} c \,n^{3}+9 b^{3} c \,n^{2}+6 a^{3} d n +26 b^{3} c n +24 b^{3} c \right ) x \,{\mathrm e}^{n \ln \left (b x +a \right )}}{b^{3} \left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right )}+\frac {n a d \,x^{3} {\mathrm e}^{n \ln \left (b x +a \right )}}{b \left (n^{2}+7 n +12\right )}-\frac {a \left (-b^{3} c \,n^{3}-9 b^{3} c \,n^{2}-26 b^{3} c n +6 a^{3} d -24 b^{3} c \right ) {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{4} \left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right )}-\frac {3 d \,a^{2} n \,x^{2} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{2} \left (n^{3}+9 n^{2}+26 n +24\right )}\)	\(232\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 122, normalized size = 1.30 \begin {gather*} \frac {{\left (b x + a\right )}^{n + 1} c}{b {\left (n + 1\right )}} + \frac {{\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{4} x^{4} + {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a b^{3} x^{3} - 3 \, {\left (n^{2} + n\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b n x - 6 \, a^{4}\right )} {\left (b x + a\right )}^{n} d}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 222 vs. \(2 (94) = 188\).
time = 0.35, size = 222, normalized size = 2.36 \begin {gather*} \frac {{\left (a b^{3} c n^{3} + 9 \, a b^{3} c n^{2} + 26 \, a b^{3} c n + 24 \, a b^{3} c - 6 \, a^{4} d + {\left (b^{4} d n^{3} + 6 \, b^{4} d n^{2} + 11 \, b^{4} d n + 6 \, b^{4} d\right )} x^{4} + {\left (a b^{3} d n^{3} + 3 \, a b^{3} d n^{2} + 2 \, a b^{3} d n\right )} x^{3} - 3 \, {\left (a^{2} b^{2} d n^{2} + a^{2} b^{2} d n\right )} x^{2} + {\left (b^{4} c n^{3} + 9 \, b^{4} c n^{2} + 24 \, b^{4} c + 2 \, {\left (13 \, b^{4} c + 3 \, a^{3} b d\right )} n\right )} x\right )} {\left (b x + a\right )}^{n}}{b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*b^3*c*n^3 + 9*a*b^3*c*n^2 + 26*a*b^3*c*n + 24*a*b^3*c - 6*a^4*d + (b^4*d*n^3 + 6*b^4*d*n^2 + 11*b^4*d*n + 6
*b^4*d)*x^4 + (a*b^3*d*n^3 + 3*a*b^3*d*n^2 + 2*a*b^3*d*n)*x^3 - 3*(a^2*b^2*d*n^2 + a^2*b^2*d*n)*x^2 + (b^4*c*n
^3 + 9*b^4*c*n^2 + 24*b^4*c + 2*(13*b^4*c + 3*a^3*b*d)*n)*x)*(b*x + a)^n/(b^4*n^4 + 10*b^4*n^3 + 35*b^4*n^2 +
50*b^4*n + 24*b^4)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1906 vs. \(2 (83) = 166\).
time = 0.78, size = 1906, normalized size = 20.28 \begin {gather*} \begin {cases} a^{n} \left (c x + \frac {d x^{4}}{4}\right ) & \text {for}\: b = 0 \\\frac {6 a^{3} d \log {\left (\frac {a}{b} + x \right )}}{6 a^{3} b^{4} + 18 a^{2} b^{5} x + 18 a b^{6} x^{2} + 6 b^{7} x^{3}} + \frac {11 a^{3} d}{6 a^{3} b^{4} + 18 a^{2} b^{5} x + 18 a b^{6} x^{2} + 6 b^{7} x^{3}} + \frac {18 a^{2} b d x \log {\left (\frac {a}{b} + x \right )}}{6 a^{3} b^{4} + 18 a^{2} b^{5} x + 18 a b^{6} x^{2} + 6 b^{7} x^{3}} + \frac {27 a^{2} b d x}{6 a^{3} b^{4} + 18 a^{2} b^{5} x + 18 a b^{6} x^{2} + 6 b^{7} x^{3}} + \frac {18 a b^{2} d x^{2} \log {\left (\frac {a}{b} + x \right )}}{6 a^{3} b^{4} + 18 a^{2} b^{5} x + 18 a b^{6} x^{2} + 6 b^{7} x^{3}} + \frac {18 a b^{2} d x^{2}}{6 a^{3} b^{4} + 18 a^{2} b^{5} x + 18 a b^{6} x^{2} + 6 b^{7} x^{3}} - \frac {2 b^{3} c}{6 a^{3} b^{4} + 18 a^{2} b^{5} x + 18 a b^{6} x^{2} + 6 b^{7} x^{3}} + \frac {6 b^{3} d x^{3} \log {\left (\frac {a}{b} + x \right )}}{6 a^{3} b^{4} + 18 a^{2} b^{5} x + 18 a b^{6} x^{2} + 6 b^{7} x^{3}} & \text {for}\: n = -4 \\- \frac {6 a^{3} d \log {\left (\frac {a}{b} + x \right )}}{2 a^{2} b^{4} + 4 a b^{5} x + 2 b^{6} x^{2}} - \frac {9 a^{3} d}{2 a^{2} b^{4} + 4 a b^{5} x + 2 b^{6} x^{2}} - \frac {12 a^{2} b d x \log {\left (\frac {a}{b} + x \right )}}{2 a^{2} b^{4} + 4 a b^{5} x + 2 b^{6} x^{2}} - \frac {12 a^{2} b d x}{2 a^{2} b^{4} + 4 a b^{5} x + 2 b^{6} x^{2}} - \frac {6 a b^{2} d x^{2} \log {\left (\frac {a}{b} + x \right )}}{2 a^{2} b^{4} + 4 a b^{5} x + 2 b^{6} x^{2}} - \frac {b^{3} c}{2 a^{2} b^{4} + 4 a b^{5} x + 2 b^{6} x^{2}} + \frac {2 b^{3} d x^{3}}{2 a^{2} b^{4} + 4 a b^{5} x + 2 b^{6} x^{2}} & \text {for}\: n = -3 \\\frac {6 a^{3} d \log {\left (\frac {a}{b} + x \right )}}{2 a b^{4} + 2 b^{5} x} + \frac {6 a^{3} d}{2 a b^{4} + 2 b^{5} x} + \frac {6 a^{2} b d x \log {\left (\frac {a}{b} + x \right )}}{2 a b^{4} + 2 b^{5} x} - \frac {3 a b^{2} d x^{2}}{2 a b^{4} + 2 b^{5} x} - \frac {2 b^{3} c}{2 a b^{4} + 2 b^{5} x} + \frac {b^{3} d x^{3}}{2 a b^{4} + 2 b^{5} x} & \text {for}\: n = -2 \\- \frac {a^{3} d \log {\left (\frac {a}{b} + x \right )}}{b^{4}} + \frac {a^{2} d x}{b^{3}} - \frac {a d x^{2}}{2 b^{2}} + \frac {c \log {\left (\frac {a}{b} + x \right )}}{b} + \frac {d x^{3}}{3 b} & \text {for}\: n = -1 \\- \frac {6 a^{4} d \left (a + b x\right )^{n}}{b^{4} n^{4} + 10 b^{4} n^{3} + 35 b^{4} n^{2} + 50 b^{4} n + 24 b^{4}} + \frac {6 a^{3} b d n x \left (a + b x\right )^{n}}{b^{4} n^{4} + 10 b^{4} n^{3} + 35 b^{4} n^{2} + 50 b^{4} n + 24 b^{4}} - \frac {3 a^{2} b^{2} d n^{2} x^{2} \left (a + b x\right )^{n}}{b^{4} n^{4} + 10 b^{4} n^{3} + 35 b^{4} n^{2} + 50 b^{4} n + 24 b^{4}} - \frac {3 a^{2} b^{2} d n x^{2} \left (a + b x\right )^{n}}{b^{4} n^{4} + 10 b^{4} n^{3} + 35 b^{4} n^{2} + 50 b^{4} n + 24 b^{4}} + \frac {a b^{3} c n^{3} \left (a + b x\right )^{n}}{b^{4} n^{4} + 10 b^{4} n^{3} + 35 b^{4} n^{2} + 50 b^{4} n + 24 b^{4}} + \frac {9 a b^{3} c n^{2} \left (a + b x\right )^{n}}{b^{4} n^{4} + 10 b^{4} n^{3} + 35 b^{4} n^{2} + 50 b^{4} n + 24 b^{4}} + \frac {26 a b^{3} c n \left (a + b x\right )^{n}}{b^{4} n^{4} + 10 b^{4} n^{3} + 35 b^{4} n^{2} + 50 b^{4} n + 24 b^{4}} + \frac {24 a b^{3} c \left (a + b x\right )^{n}}{b^{4} n^{4} + 10 b^{4} n^{3} + 35 b^{4} n^{2} + 50 b^{4} n + 24 b^{4}} + \frac {a b^{3} d n^{3} x^{3} \left (a + b x\right )^{n}}{b^{4} n^{4} + 10 b^{4} n^{3} + 35 b^{4} n^{2} + 50 b^{4} n + 24 b^{4}} + \frac {3 a b^{3} d n^{2} x^{3} \left (a + b x\right )^{n}}{b^{4} n^{4} + 10 b^{4} n^{3} + 35 b^{4} n^{2} + 50 b^{4} n + 24 b^{4}} + \frac {2 a b^{3} d n x^{3} \left (a + b x\right )^{n}}{b^{4} n^{4} + 10 b^{4} n^{3} + 35 b^{4} n^{2} + 50 b^{4} n + 24 b^{4}} + \frac {b^{4} c n^{3} x \left (a + b x\right )^{n}}{b^{4} n^{4} + 10 b^{4} n^{3} + 35 b^{4} n^{2} + 50 b^{4} n + 24 b^{4}} + \frac {9 b^{4} c n^{2} x \left (a + b x\right )^{n}}{b^{4} n^{4} + 10 b^{4} n^{3} + 35 b^{4} n^{2} + 50 b^{4} n + 24 b^{4}} + \frac {26 b^{4} c n x \left (a + b x\right )^{n}}{b^{4} n^{4} + 10 b^{4} n^{3} + 35 b^{4} n^{2} + 50 b^{4} n + 24 b^{4}} + \frac {24 b^{4} c x \left (a + b x\right )^{n}}{b^{4} n^{4} + 10 b^{4} n^{3} + 35 b^{4} n^{2} + 50 b^{4} n + 24 b^{4}} + \frac {b^{4} d n^{3} x^{4} \left (a + b x\right )^{n}}{b^{4} n^{4} + 10 b^{4} n^{3} + 35 b^{4} n^{2} + 50 b^{4} n + 24 b^{4}} + \frac {6 b^{4} d n^{2} x^{4} \left (a + b x\right )^{n}}{b^{4} n^{4} + 10 b^{4} n^{3} + 35 b^{4} n^{2} + 50 b^{4} n + 24 b^{4}} + \frac {11 b^{4} d n x^{4} \left (a + b x\right )^{n}}{b^{4} n^{4} + 10 b^{4} n^{3} + 35 b^{4} n^{2} + 50 b^{4} n + 24 b^{4}} + \frac {6 b^{4} d x^{4} \left (a + b x\right )^{n}}{b^{4} n^{4} + 10 b^{4} n^{3} + 35 b^{4} n^{2} + 50 b^{4} n + 24 b^{4}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**n*(c*x + d*x**4/4), Eq(b, 0)), (6*a**3*d*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/(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*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*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*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*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/(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*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(n, -4)), (-6*a**3*d*log(a/b + x)/(2*a**2*b**4 + 4*a*b*
*5*x + 2*b**6*x**2) - 9*a**3*d/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 12*a**2*b*d*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*x/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 6*a*b**2*d*x**2*log
(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - b**3*c/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 2*b**3*
d*x**3/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2), Eq(n, -3)), (6*a**3*d*log(a/b + x)/(2*a*b**4 + 2*b**5*x) + 6*
a**3*d/(2*a*b**4 + 2*b**5*x) + 6*a**2*b*d*x*log(a/b + x)/(2*a*b**4 + 2*b**5*x) - 3*a*b**2*d*x**2/(2*a*b**4 + 2
*b**5*x) - 2*b**3*c/(2*a*b**4 + 2*b**5*x) + b**3*d*x**3/(2*a*b**4 + 2*b**5*x), Eq(n, -2)), (-a**3*d*log(a/b +
x)/b**4 + a**2*d*x/b**3 - a*d*x**2/(2*b**2) + c*log(a/b + x)/b + d*x**3/(3*b), Eq(n, -1)), (-6*a**4*d*(a + b*x
)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 6*a**3*b*d*n*x*(a + b*x)**n/(b**4*n**4
+ 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) - 3*a**2*b**2*d*n**2*x**2*(a + b*x)**n/(b**4*n**4 + 10*b*
*4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) - 3*a**2*b**2*d*n*x**2*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 +
35*b**4*n**2 + 50*b**4*n + 24*b**4) + a*b**3*c*n**3*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50
*b**4*n + 24*b**4) + 9*a*b**3*c*n**2*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b*
*4) + 26*a*b**3*c*n*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 24*a*b**3*c
*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + a*b**3*d*n**3*x**3*(a + b*x)**
n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 3*a*b**3*d*n**2*x**3*(a + b*x)**n/(b**4*n*
*4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 2*a*b**3*d*n*x**3*(a + b*x)**n/(b**4*n**4 + 10*b**4*
n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + b**4*c*n**3*x*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n*
*2 + 50*b**4*n + 24*b**4) + 9*b**4*c*n**2*x*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n
+ 24*b**4) + 26*b**4*c*n*x*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 24*b
**4*c*x*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + b**4*d*n**3*x**4*(a + b
*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 6*b**4*d*n**2*x**4*(a + b*x)**n/(b**4
*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 11*b**4*d*n*x**4*(a + b*x)**n/(b**4*n**4 + 10*b**
4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 6*b**4*d*x**4*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*
n**2 + 50*b**4*n + 24*b**4), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 361 vs. \(2 (94) = 188\).
time = 3.99, size = 361, normalized size = 3.84 \begin {gather*} \frac {{\left (b x + a\right )}^{n} b^{4} d n^{3} x^{4} + {\left (b x + a\right )}^{n} a b^{3} d n^{3} x^{3} + 6 \, {\left (b x + a\right )}^{n} b^{4} d n^{2} x^{4} + 3 \, {\left (b x + a\right )}^{n} a b^{3} d n^{2} x^{3} + 11 \, {\left (b x + a\right )}^{n} b^{4} d n x^{4} + {\left (b x + a\right )}^{n} b^{4} c n^{3} x - 3 \, {\left (b x + a\right )}^{n} a^{2} b^{2} d n^{2} x^{2} + 2 \, {\left (b x + a\right )}^{n} a b^{3} d n x^{3} + 6 \, {\left (b x + a\right )}^{n} b^{4} d x^{4} + {\left (b x + a\right )}^{n} a b^{3} c n^{3} + 9 \, {\left (b x + a\right )}^{n} b^{4} c n^{2} x - 3 \, {\left (b x + a\right )}^{n} a^{2} b^{2} d n x^{2} + 9 \, {\left (b x + a\right )}^{n} a b^{3} c n^{2} + 26 \, {\left (b x + a\right )}^{n} b^{4} c n x + 6 \, {\left (b x + a\right )}^{n} a^{3} b d n x + 26 \, {\left (b x + a\right )}^{n} a b^{3} c n + 24 \, {\left (b x + a\right )}^{n} b^{4} c x + 24 \, {\left (b x + a\right )}^{n} a b^{3} c - 6 \, {\left (b x + a\right )}^{n} a^{4} d}{b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b*x + a)^n*b^4*d*n^3*x^4 + (b*x + a)^n*a*b^3*d*n^3*x^3 + 6*(b*x + a)^n*b^4*d*n^2*x^4 + 3*(b*x + a)^n*a*b^3*d
*n^2*x^3 + 11*(b*x + a)^n*b^4*d*n*x^4 + (b*x + a)^n*b^4*c*n^3*x - 3*(b*x + a)^n*a^2*b^2*d*n^2*x^2 + 2*(b*x + a
)^n*a*b^3*d*n*x^3 + 6*(b*x + a)^n*b^4*d*x^4 + (b*x + a)^n*a*b^3*c*n^3 + 9*(b*x + a)^n*b^4*c*n^2*x - 3*(b*x + a
)^n*a^2*b^2*d*n*x^2 + 9*(b*x + a)^n*a*b^3*c*n^2 + 26*(b*x + a)^n*b^4*c*n*x + 6*(b*x + a)^n*a^3*b*d*n*x + 26*(b
*x + a)^n*a*b^3*c*n + 24*(b*x + a)^n*b^4*c*x + 24*(b*x + a)^n*a*b^3*c - 6*(b*x + a)^n*a^4*d)/(b^4*n^4 + 10*b^4
*n^3 + 35*b^4*n^2 + 50*b^4*n + 24*b^4)

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Mupad [B]
time = 2.95, size = 247, normalized size = 2.63 \begin {gather*} {\left (a+b\,x\right )}^n\,\left (\frac {x\,\left (6\,d\,a^3\,b\,n+c\,b^4\,n^3+9\,c\,b^4\,n^2+26\,c\,b^4\,n+24\,c\,b^4\right )}{b^4\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {a\,\left (-6\,d\,a^3+c\,b^3\,n^3+9\,c\,b^3\,n^2+26\,c\,b^3\,n+24\,c\,b^3\right )}{b^4\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {d\,x^4\,\left (n^3+6\,n^2+11\,n+6\right )}{n^4+10\,n^3+35\,n^2+50\,n+24}-\frac {3\,a^2\,d\,n\,x^2\,\left (n+1\right )}{b^2\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {a\,d\,n\,x^3\,\left (n^2+3\,n+2\right )}{b\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^3)*(a + b*x)^n,x)

[Out]

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

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