3.1997 \(\int (a+b x) (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\)

Optimal. Leaf size=27 \[ \frac{\left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 b} \]

[Out]

(a^2 + 2*a*b*x + b^2*x^2)^(7/2)/(7*b)

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Rubi [A]  time = 0.008406, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {629} \[ \frac{\left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(a^2 + 2*a*b*x + b^2*x^2)^(7/2)/(7*b)

Rule 629

Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d*(a + b*x + c*x^2)^(p +
1))/(b*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps

\begin{align*} \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx &=\frac{\left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 b}\\ \end{align*}

Mathematica [A]  time = 0.0089287, size = 25, normalized size = 0.93 \[ \frac{(a+b x)^6 \sqrt{(a+b x)^2}}{7 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

((a + b*x)^6*Sqrt[(a + b*x)^2])/(7*b)

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Maple [B]  time = 0.004, size = 82, normalized size = 3. \begin{align*}{\frac{x \left ({b}^{6}{x}^{6}+7\,a{b}^{5}{x}^{5}+21\,{a}^{2}{b}^{4}{x}^{4}+35\,{a}^{3}{b}^{3}{x}^{3}+35\,{a}^{4}{b}^{2}{x}^{2}+21\,{a}^{5}bx+7\,{a}^{6} \right ) }{7\, \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/7*x*(b^6*x^6+7*a*b^5*x^5+21*a^2*b^4*x^4+35*a^3*b^3*x^3+35*a^4*b^2*x^2+21*a^5*b*x+7*a^6)*((b*x+a)^2)^(5/2)/(b
*x+a)^5

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Maxima [A]  time = 0.958892, size = 31, normalized size = 1.15 \begin{align*} \frac{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{7}{2}}}{7 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

1/7*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)/b

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Fricas [B]  time = 1.46799, size = 128, normalized size = 4.74 \begin{align*} \frac{1}{7} \, b^{6} x^{7} + a b^{5} x^{6} + 3 \, a^{2} b^{4} x^{5} + 5 \, a^{3} b^{3} x^{4} + 5 \, a^{4} b^{2} x^{3} + 3 \, a^{5} b x^{2} + a^{6} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

1/7*b^6*x^7 + a*b^5*x^6 + 3*a^2*b^4*x^5 + 5*a^3*b^3*x^4 + 5*a^4*b^2*x^3 + 3*a^5*b*x^2 + a^6*x

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Sympy [A]  time = 5.3187, size = 226, normalized size = 8.37 \begin{align*} \begin{cases} \frac{a^{6} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{7 b} + \frac{6 a^{5} x \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{7} + \frac{15 a^{4} b x^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{7} + \frac{20 a^{3} b^{2} x^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{7} + \frac{15 a^{2} b^{3} x^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{7} + \frac{6 a b^{4} x^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{7} + \frac{b^{5} x^{6} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{7} & \text{for}\: b \neq 0 \\a x \left (a^{2}\right )^{\frac{5}{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Piecewise((a**6*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(7*b) + 6*a**5*x*sqrt(a**2 + 2*a*b*x + b**2*x**2)/7 + 15*a**4
*b*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)/7 + 20*a**3*b**2*x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2)/7 + 15*a**2*b*
*3*x**4*sqrt(a**2 + 2*a*b*x + b**2*x**2)/7 + 6*a*b**4*x**5*sqrt(a**2 + 2*a*b*x + b**2*x**2)/7 + b**5*x**6*sqrt
(a**2 + 2*a*b*x + b**2*x**2)/7, Ne(b, 0)), (a*x*(a**2)**(5/2), True))

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Giac [B]  time = 1.10554, size = 162, normalized size = 6. \begin{align*} \frac{1}{7} \, b^{6} x^{7} \mathrm{sgn}\left (b x + a\right ) + a b^{5} x^{6} \mathrm{sgn}\left (b x + a\right ) + 3 \, a^{2} b^{4} x^{5} \mathrm{sgn}\left (b x + a\right ) + 5 \, a^{3} b^{3} x^{4} \mathrm{sgn}\left (b x + a\right ) + 5 \, a^{4} b^{2} x^{3} \mathrm{sgn}\left (b x + a\right ) + 3 \, a^{5} b x^{2} \mathrm{sgn}\left (b x + a\right ) + a^{6} x \mathrm{sgn}\left (b x + a\right ) + \frac{a^{7} \mathrm{sgn}\left (b x + a\right )}{7 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

1/7*b^6*x^7*sgn(b*x + a) + a*b^5*x^6*sgn(b*x + a) + 3*a^2*b^4*x^5*sgn(b*x + a) + 5*a^3*b^3*x^4*sgn(b*x + a) +
5*a^4*b^2*x^3*sgn(b*x + a) + 3*a^5*b*x^2*sgn(b*x + a) + a^6*x*sgn(b*x + a) + 1/7*a^7*sgn(b*x + a)/b