3.3.64 \(\int (a+b x)^n (c+d x^2) \, dx\)

Optimal. Leaf size=70 \[ \frac {\left (a^2 d+b^2 c\right ) (a+b x)^{n+1}}{b^3 (n+1)}-\frac {2 a d (a+b x)^{n+2}}{b^3 (n+2)}+\frac {d (a+b x)^{n+3}}{b^3 (n+3)} \]

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Rubi [A]  time = 0.03, antiderivative size = 70, 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 = {697} \begin {gather*} \frac {\left (a^2 d+b^2 c\right ) (a+b x)^{n+1}}{b^3 (n+1)}-\frac {2 a d (a+b x)^{n+2}}{b^3 (n+2)}+\frac {d (a+b x)^{n+3}}{b^3 (n+3)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^n*(c + d*x^2),x]

[Out]

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

Rule 697

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rubi steps

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

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Mathematica [A]  time = 0.04, size = 65, normalized size = 0.93 \begin {gather*} \frac {(a+b x)^{n+1} \left (2 a^2 d-2 a b d (n+1) x+b^2 (n+2) \left (c (n+3)+d (n+1) x^2\right )\right )}{b^3 (n+1) (n+2) (n+3)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^n*(c + d*x^2),x]

[Out]

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

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IntegrateAlgebraic [F]  time = 0.04, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x)^n \left (c+d x^2\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)^n*(c + d*x^2),x]

[Out]

Defer[IntegrateAlgebraic][(a + b*x)^n*(c + d*x^2), x]

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fricas [B]  time = 0.41, size = 148, normalized size = 2.11 \begin {gather*} \frac {{\left (a b^{2} c n^{2} + 5 \, a b^{2} c n + 6 \, a b^{2} c + 2 \, a^{3} d + {\left (b^{3} d n^{2} + 3 \, b^{3} d n + 2 \, b^{3} d\right )} x^{3} + {\left (a b^{2} d n^{2} + a b^{2} d n\right )} x^{2} + {\left (b^{3} c n^{2} + 6 \, b^{3} c + {\left (5 \, b^{3} c - 2 \, a^{2} b d\right )} n\right )} x\right )} {\left (b x + a\right )}^{n}}{b^{3} n^{3} + 6 \, b^{3} n^{2} + 11 \, b^{3} n + 6 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*b^2*c*n^2 + 5*a*b^2*c*n + 6*a*b^2*c + 2*a^3*d + (b^3*d*n^2 + 3*b^3*d*n + 2*b^3*d)*x^3 + (a*b^2*d*n^2 + a*b^
2*d*n)*x^2 + (b^3*c*n^2 + 6*b^3*c + (5*b^3*c - 2*a^2*b*d)*n)*x)*(b*x + a)^n/(b^3*n^3 + 6*b^3*n^2 + 11*b^3*n +
6*b^3)

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giac [B]  time = 0.16, size = 237, normalized size = 3.39 \begin {gather*} \frac {{\left (b x + a\right )}^{n} b^{3} d n^{2} x^{3} + {\left (b x + a\right )}^{n} a b^{2} d n^{2} x^{2} + 3 \, {\left (b x + a\right )}^{n} b^{3} d n x^{3} + {\left (b x + a\right )}^{n} b^{3} c n^{2} x + {\left (b x + a\right )}^{n} a b^{2} d n x^{2} + 2 \, {\left (b x + a\right )}^{n} b^{3} d x^{3} + {\left (b x + a\right )}^{n} a b^{2} c n^{2} + 5 \, {\left (b x + a\right )}^{n} b^{3} c n x - 2 \, {\left (b x + a\right )}^{n} a^{2} b d n x + 5 \, {\left (b x + a\right )}^{n} a b^{2} c n + 6 \, {\left (b x + a\right )}^{n} b^{3} c x + 6 \, {\left (b x + a\right )}^{n} a b^{2} c + 2 \, {\left (b x + a\right )}^{n} a^{3} d}{b^{3} n^{3} + 6 \, b^{3} n^{2} + 11 \, b^{3} n + 6 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b*x + a)^n*b^3*d*n^2*x^3 + (b*x + a)^n*a*b^2*d*n^2*x^2 + 3*(b*x + a)^n*b^3*d*n*x^3 + (b*x + a)^n*b^3*c*n^2*x
+ (b*x + a)^n*a*b^2*d*n*x^2 + 2*(b*x + a)^n*b^3*d*x^3 + (b*x + a)^n*a*b^2*c*n^2 + 5*(b*x + a)^n*b^3*c*n*x - 2
*(b*x + a)^n*a^2*b*d*n*x + 5*(b*x + a)^n*a*b^2*c*n + 6*(b*x + a)^n*b^3*c*x + 6*(b*x + a)^n*a*b^2*c + 2*(b*x +
a)^n*a^3*d)/(b^3*n^3 + 6*b^3*n^2 + 11*b^3*n + 6*b^3)

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maple [A]  time = 0.00, size = 100, normalized size = 1.43 \begin {gather*} \frac {\left (b^{2} d \,n^{2} x^{2}+3 b^{2} d n \,x^{2}-2 a b d n x +b^{2} c \,n^{2}+2 d \,x^{2} b^{2}-2 a d x b +5 b^{2} c n +2 a^{2} d +6 b^{2} c \right ) \left (b x +a \right )^{n +1}}{\left (n^{3}+6 n^{2}+11 n +6\right ) b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^n*(d*x^2+c),x)

[Out]

(b*x+a)^(n+1)*(b^2*d*n^2*x^2+3*b^2*d*n*x^2-2*a*b*d*n*x+b^2*c*n^2+2*b^2*d*x^2-2*a*b*d*x+5*b^2*c*n+2*a^2*d+6*b^2
*c)/b^3/(n^3+6*n^2+11*n+6)

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maxima [A]  time = 0.46, size = 89, normalized size = 1.27 \begin {gather*} \frac {{\left (b x + a\right )}^{n + 1} c}{b {\left (n + 1\right )}} + \frac {{\left ({\left (n^{2} + 3 \, n + 2\right )} b^{3} x^{3} + {\left (n^{2} + n\right )} a b^{2} x^{2} - 2 \, a^{2} b n x + 2 \, a^{3}\right )} {\left (b x + a\right )}^{n} d}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x + a)^(n + 1)*c/(b*(n + 1)) + ((n^2 + 3*n + 2)*b^3*x^3 + (n^2 + n)*a*b^2*x^2 - 2*a^2*b*n*x + 2*a^3)*(b*x +
a)^n*d/((n^3 + 6*n^2 + 11*n + 6)*b^3)

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mupad [B]  time = 2.63, size = 163, normalized size = 2.33 \begin {gather*} {\left (a+b\,x\right )}^n\,\left (\frac {d\,x^3\,\left (n^2+3\,n+2\right )}{n^3+6\,n^2+11\,n+6}+\frac {x\,\left (-2\,d\,a^2\,b\,n+c\,b^3\,n^2+5\,c\,b^3\,n+6\,c\,b^3\right )}{b^3\,\left (n^3+6\,n^2+11\,n+6\right )}+\frac {a\,\left (2\,d\,a^2+c\,b^2\,n^2+5\,c\,b^2\,n+6\,c\,b^2\right )}{b^3\,\left (n^3+6\,n^2+11\,n+6\right )}+\frac {a\,d\,n\,x^2\,\left (n+1\right )}{b\,\left (n^3+6\,n^2+11\,n+6\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^2)*(a + b*x)^n,x)

[Out]

(a + b*x)^n*((d*x^3*(3*n + n^2 + 2))/(11*n + 6*n^2 + n^3 + 6) + (x*(6*b^3*c + b^3*c*n^2 + 5*b^3*c*n - 2*a^2*b*
d*n))/(b^3*(11*n + 6*n^2 + n^3 + 6)) + (a*(2*a^2*d + 6*b^2*c + b^2*c*n^2 + 5*b^2*c*n))/(b^3*(11*n + 6*n^2 + n^
3 + 6)) + (a*d*n*x^2*(n + 1))/(b*(11*n + 6*n^2 + n^3 + 6)))

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sympy [A]  time = 2.07, size = 952, normalized size = 13.60 \begin {gather*} \begin {cases} a^{n} \left (c x + \frac {d x^{3}}{3}\right ) & \text {for}\: b = 0 \\\frac {2 a^{2} d \log {\left (\frac {a}{b} + x \right )}}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} + \frac {3 a^{2} d}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} + \frac {4 a b d x \log {\left (\frac {a}{b} + x \right )}}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} + \frac {4 a b d x}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} - \frac {b^{2} c}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} + \frac {2 b^{2} d x^{2} \log {\left (\frac {a}{b} + x \right )}}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} & \text {for}\: n = -3 \\- \frac {2 a^{2} d \log {\left (\frac {a}{b} + x \right )}}{a b^{3} + b^{4} x} - \frac {2 a^{2} d}{a b^{3} + b^{4} x} - \frac {2 a b d x \log {\left (\frac {a}{b} + x \right )}}{a b^{3} + b^{4} x} - \frac {b^{2} c}{a b^{3} + b^{4} x} + \frac {b^{2} d x^{2}}{a b^{3} + b^{4} x} & \text {for}\: n = -2 \\\frac {a^{2} d \log {\left (\frac {a}{b} + x \right )}}{b^{3}} - \frac {a d x}{b^{2}} + \frac {c \log {\left (\frac {a}{b} + x \right )}}{b} + \frac {d x^{2}}{2 b} & \text {for}\: n = -1 \\\frac {2 a^{3} d \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} - \frac {2 a^{2} b d n x \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {a b^{2} c n^{2} \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {5 a b^{2} c n \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {6 a b^{2} c \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {a b^{2} d n^{2} x^{2} \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {a b^{2} d n x^{2} \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {b^{3} c n^{2} x \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {5 b^{3} c n x \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {6 b^{3} c x \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {b^{3} d n^{2} x^{3} \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {3 b^{3} d n x^{3} \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {2 b^{3} d x^{3} \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**n*(c*x + d*x**3/3), Eq(b, 0)), (2*a**2*d*log(a/b + x)/(2*a**2*b**3 + 4*a*b**4*x + 2*b**5*x**2) +
3*a**2*d/(2*a**2*b**3 + 4*a*b**4*x + 2*b**5*x**2) + 4*a*b*d*x*log(a/b + x)/(2*a**2*b**3 + 4*a*b**4*x + 2*b**5
*x**2) + 4*a*b*d*x/(2*a**2*b**3 + 4*a*b**4*x + 2*b**5*x**2) - b**2*c/(2*a**2*b**3 + 4*a*b**4*x + 2*b**5*x**2)
+ 2*b**2*d*x**2*log(a/b + x)/(2*a**2*b**3 + 4*a*b**4*x + 2*b**5*x**2), Eq(n, -3)), (-2*a**2*d*log(a/b + x)/(a*
b**3 + b**4*x) - 2*a**2*d/(a*b**3 + b**4*x) - 2*a*b*d*x*log(a/b + x)/(a*b**3 + b**4*x) - b**2*c/(a*b**3 + b**4
*x) + b**2*d*x**2/(a*b**3 + b**4*x), Eq(n, -2)), (a**2*d*log(a/b + x)/b**3 - a*d*x/b**2 + c*log(a/b + x)/b + d
*x**2/(2*b), Eq(n, -1)), (2*a**3*d*(a + b*x)**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) - 2*a**2*b*d*n*
x*(a + b*x)**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) + a*b**2*c*n**2*(a + b*x)**n/(b**3*n**3 + 6*b**3
*n**2 + 11*b**3*n + 6*b**3) + 5*a*b**2*c*n*(a + b*x)**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) + 6*a*b
**2*c*(a + b*x)**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) + a*b**2*d*n**2*x**2*(a + b*x)**n/(b**3*n**3
+ 6*b**3*n**2 + 11*b**3*n + 6*b**3) + a*b**2*d*n*x**2*(a + b*x)**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b
**3) + b**3*c*n**2*x*(a + b*x)**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) + 5*b**3*c*n*x*(a + b*x)**n/(
b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) + 6*b**3*c*x*(a + b*x)**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n +
6*b**3) + b**3*d*n**2*x**3*(a + b*x)**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) + 3*b**3*d*n*x**3*(a +
b*x)**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) + 2*b**3*d*x**3*(a + b*x)**n/(b**3*n**3 + 6*b**3*n**2
+ 11*b**3*n + 6*b**3), True))

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