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

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

[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))

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Rubi [A]  time = 0.031032, antiderivative size = 70, normalized size of antiderivative = 1., 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} \[ \frac{\left (a^2 d+b^2 c\right ) (a+b x)^{n+1}}{b^3 (n+1)}-\frac{2 a d (a+b x)^{n+2}}{b^3 (n+2)}+\frac{d (a+b x)^{n+3}}{b^3 (n+3)} \]

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*}

Mathematica [A]  time = 0.0470745, size = 65, normalized size = 0.93 \[ \frac{(a+b x)^{n+1} \left (2 a^2 d-2 a b d (n+1) x+b^2 (n+2) \left (c (n+3)+d (n+1) x^2\right )\right )}{b^3 (n+1) (n+2) (n+3)} \]

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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Maple [A]  time = 0.049, size = 100, normalized size = 1.4 \begin{align*}{\frac{ \left ( bx+a \right ) ^{1+n} \left ({b}^{2}d{n}^{2}{x}^{2}+3\,{b}^{2}dn{x}^{2}-2\,abdnx+{b}^{2}c{n}^{2}+2\,d{x}^{2}{b}^{2}-2\,adxb+5\,{b}^{2}cn+2\,{a}^{2}d+6\,{b}^{2}c \right ) }{{b}^{3} \left ({n}^{3}+6\,{n}^{2}+11\,n+6 \right ) }} \end{align*}

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)^(1+n)*(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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.90208, size = 308, normalized size = 4.4 \begin{align*} \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{align*}

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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Sympy [A]  time = 2.31337, size = 978, normalized size = 13.97 \begin{align*} \text{result too large to display} \end{align*}

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**3*d*log(a/b + x)/(a*
*2*b**3 + a*b**4*x) - 2*a**2*b*d*x*log(a/b + x)/(a**2*b**3 + a*b**4*x) + 2*a**2*b*d*x/(a**2*b**3 + a*b**4*x) +
a*b**2*d*x**2/(a**2*b**3 + a*b**4*x) + b**3*c*x/(a**2*b**3 + a*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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Giac [B]  time = 1.20527, size = 320, normalized size = 4.57 \begin{align*} \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{align*}

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)