3.1954 \(\int (a+b x) (d+e x)^5 \sqrt{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=146 \[ \frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^8}{8 e^3 (a+b x)}-\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^7 (b d-a e)}{7 e^3 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^6 (b d-a e)^2}{6 e^3 (a+b x)} \]

[Out]

((b*d - a*e)^2*(d + e*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*e^3*(a + b*x)) - (2*b*(b*d - a*e)*(d + e*x)^7*Sqr
t[a^2 + 2*a*b*x + b^2*x^2])/(7*e^3*(a + b*x)) + (b^2*(d + e*x)^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*e^3*(a + b*
x))

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Rubi [A]  time = 0.152057, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {770, 21, 43} \[ \frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^8}{8 e^3 (a+b x)}-\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^7 (b d-a e)}{7 e^3 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^6 (b d-a e)^2}{6 e^3 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(d + e*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

((b*d - a*e)^2*(d + e*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*e^3*(a + b*x)) - (2*b*(b*d - a*e)*(d + e*x)^7*Sqr
t[a^2 + 2*a*b*x + b^2*x^2])/(7*e^3*(a + b*x)) + (b^2*(d + e*x)^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*e^3*(a + b*
x))

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
 n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
 d*x, a + b*x])

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) (d+e x)^5 \sqrt{a^2+2 a b x+b^2 x^2} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int (a+b x) \left (a b+b^2 x\right ) (d+e x)^5 \, dx}{a b+b^2 x}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int (a+b x)^2 (d+e x)^5 \, dx}{a b+b^2 x}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac{(-b d+a e)^2 (d+e x)^5}{e^2}-\frac{2 b (b d-a e) (d+e x)^6}{e^2}+\frac{b^2 (d+e x)^7}{e^2}\right ) \, dx}{a b+b^2 x}\\ &=\frac{(b d-a e)^2 (d+e x)^6 \sqrt{a^2+2 a b x+b^2 x^2}}{6 e^3 (a+b x)}-\frac{2 b (b d-a e) (d+e x)^7 \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^3 (a+b x)}+\frac{b^2 (d+e x)^8 \sqrt{a^2+2 a b x+b^2 x^2}}{8 e^3 (a+b x)}\\ \end{align*}

Mathematica [A]  time = 0.0680088, size = 196, normalized size = 1.34 \[ \frac{x \sqrt{(a+b x)^2} \left (28 a^2 \left (20 d^3 e^2 x^2+15 d^2 e^3 x^3+15 d^4 e x+6 d^5+6 d e^4 x^4+e^5 x^5\right )+8 a b x \left (105 d^3 e^2 x^2+84 d^2 e^3 x^3+70 d^4 e x+21 d^5+35 d e^4 x^4+6 e^5 x^5\right )+b^2 x^2 \left (336 d^3 e^2 x^2+280 d^2 e^3 x^3+210 d^4 e x+56 d^5+120 d e^4 x^4+21 e^5 x^5\right )\right )}{168 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(d + e*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

(x*Sqrt[(a + b*x)^2]*(28*a^2*(6*d^5 + 15*d^4*e*x + 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 + 6*d*e^4*x^4 + e^5*x^5) +
8*a*b*x*(21*d^5 + 70*d^4*e*x + 105*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 35*d*e^4*x^4 + 6*e^5*x^5) + b^2*x^2*(56*d^5
+ 210*d^4*e*x + 336*d^3*e^2*x^2 + 280*d^2*e^3*x^3 + 120*d*e^4*x^4 + 21*e^5*x^5)))/(168*(a + b*x))

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Maple [B]  time = 0.007, size = 230, normalized size = 1.6 \begin{align*}{\frac{x \left ( 21\,{b}^{2}{e}^{5}{x}^{7}+48\,{x}^{6}ab{e}^{5}+120\,{x}^{6}{b}^{2}d{e}^{4}+28\,{x}^{5}{a}^{2}{e}^{5}+280\,{x}^{5}abd{e}^{4}+280\,{x}^{5}{b}^{2}{d}^{2}{e}^{3}+168\,{a}^{2}d{e}^{4}{x}^{4}+672\,ab{d}^{2}{e}^{3}{x}^{4}+336\,{b}^{2}{d}^{3}{e}^{2}{x}^{4}+420\,{x}^{3}{a}^{2}{d}^{2}{e}^{3}+840\,{x}^{3}ab{d}^{3}{e}^{2}+210\,{x}^{3}{b}^{2}{d}^{4}e+560\,{x}^{2}{a}^{2}{d}^{3}{e}^{2}+560\,{x}^{2}ab{d}^{4}e+56\,{x}^{2}{b}^{2}{d}^{5}+420\,x{a}^{2}{d}^{4}e+168\,xab{d}^{5}+168\,{a}^{2}{d}^{5} \right ) }{168\,bx+168\,a}\sqrt{ \left ( bx+a \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(e*x+d)^5*((b*x+a)^2)^(1/2),x)

[Out]

1/168*x*(21*b^2*e^5*x^7+48*a*b*e^5*x^6+120*b^2*d*e^4*x^6+28*a^2*e^5*x^5+280*a*b*d*e^4*x^5+280*b^2*d^2*e^3*x^5+
168*a^2*d*e^4*x^4+672*a*b*d^2*e^3*x^4+336*b^2*d^3*e^2*x^4+420*a^2*d^2*e^3*x^3+840*a*b*d^3*e^2*x^3+210*b^2*d^4*
e*x^3+560*a^2*d^3*e^2*x^2+560*a*b*d^4*e*x^2+56*b^2*d^5*x^2+420*a^2*d^4*e*x+168*a*b*d^5*x+168*a^2*d^5)*((b*x+a)
^2)^(1/2)/(b*x+a)

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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)*(e*x+d)^5*((b*x+a)^2)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.50166, size = 417, normalized size = 2.86 \begin{align*} \frac{1}{8} \, b^{2} e^{5} x^{8} + a^{2} d^{5} x + \frac{1}{7} \,{\left (5 \, b^{2} d e^{4} + 2 \, a b e^{5}\right )} x^{7} + \frac{1}{6} \,{\left (10 \, b^{2} d^{2} e^{3} + 10 \, a b d e^{4} + a^{2} e^{5}\right )} x^{6} +{\left (2 \, b^{2} d^{3} e^{2} + 4 \, a b d^{2} e^{3} + a^{2} d e^{4}\right )} x^{5} + \frac{5}{4} \,{\left (b^{2} d^{4} e + 4 \, a b d^{3} e^{2} + 2 \, a^{2} d^{2} e^{3}\right )} x^{4} + \frac{1}{3} \,{\left (b^{2} d^{5} + 10 \, a b d^{4} e + 10 \, a^{2} d^{3} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (2 \, a b d^{5} + 5 \, a^{2} d^{4} e\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^5*((b*x+a)^2)^(1/2),x, algorithm="fricas")

[Out]

1/8*b^2*e^5*x^8 + a^2*d^5*x + 1/7*(5*b^2*d*e^4 + 2*a*b*e^5)*x^7 + 1/6*(10*b^2*d^2*e^3 + 10*a*b*d*e^4 + a^2*e^5
)*x^6 + (2*b^2*d^3*e^2 + 4*a*b*d^2*e^3 + a^2*d*e^4)*x^5 + 5/4*(b^2*d^4*e + 4*a*b*d^3*e^2 + 2*a^2*d^2*e^3)*x^4
+ 1/3*(b^2*d^5 + 10*a*b*d^4*e + 10*a^2*d^3*e^2)*x^3 + 1/2*(2*a*b*d^5 + 5*a^2*d^4*e)*x^2

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Sympy [B]  time = 0.143769, size = 218, normalized size = 1.49 \begin{align*} a^{2} d^{5} x + \frac{b^{2} e^{5} x^{8}}{8} + x^{7} \left (\frac{2 a b e^{5}}{7} + \frac{5 b^{2} d e^{4}}{7}\right ) + x^{6} \left (\frac{a^{2} e^{5}}{6} + \frac{5 a b d e^{4}}{3} + \frac{5 b^{2} d^{2} e^{3}}{3}\right ) + x^{5} \left (a^{2} d e^{4} + 4 a b d^{2} e^{3} + 2 b^{2} d^{3} e^{2}\right ) + x^{4} \left (\frac{5 a^{2} d^{2} e^{3}}{2} + 5 a b d^{3} e^{2} + \frac{5 b^{2} d^{4} e}{4}\right ) + x^{3} \left (\frac{10 a^{2} d^{3} e^{2}}{3} + \frac{10 a b d^{4} e}{3} + \frac{b^{2} d^{5}}{3}\right ) + x^{2} \left (\frac{5 a^{2} d^{4} e}{2} + a b d^{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)**5*((b*x+a)**2)**(1/2),x)

[Out]

a**2*d**5*x + b**2*e**5*x**8/8 + x**7*(2*a*b*e**5/7 + 5*b**2*d*e**4/7) + x**6*(a**2*e**5/6 + 5*a*b*d*e**4/3 +
5*b**2*d**2*e**3/3) + x**5*(a**2*d*e**4 + 4*a*b*d**2*e**3 + 2*b**2*d**3*e**2) + x**4*(5*a**2*d**2*e**3/2 + 5*a
*b*d**3*e**2 + 5*b**2*d**4*e/4) + x**3*(10*a**2*d**3*e**2/3 + 10*a*b*d**4*e/3 + b**2*d**5/3) + x**2*(5*a**2*d*
*4*e/2 + a*b*d**5)

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Giac [B]  time = 1.15787, size = 420, normalized size = 2.88 \begin{align*} \frac{1}{8} \, b^{2} x^{8} e^{5} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{7} \, b^{2} d x^{7} e^{4} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{3} \, b^{2} d^{2} x^{6} e^{3} \mathrm{sgn}\left (b x + a\right ) + 2 \, b^{2} d^{3} x^{5} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{4} \, b^{2} d^{4} x^{4} e \mathrm{sgn}\left (b x + a\right ) + \frac{1}{3} \, b^{2} d^{5} x^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{2}{7} \, a b x^{7} e^{5} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{3} \, a b d x^{6} e^{4} \mathrm{sgn}\left (b x + a\right ) + 4 \, a b d^{2} x^{5} e^{3} \mathrm{sgn}\left (b x + a\right ) + 5 \, a b d^{3} x^{4} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{10}{3} \, a b d^{4} x^{3} e \mathrm{sgn}\left (b x + a\right ) + a b d^{5} x^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{6} \, a^{2} x^{6} e^{5} \mathrm{sgn}\left (b x + a\right ) + a^{2} d x^{5} e^{4} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{2} \, a^{2} d^{2} x^{4} e^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{10}{3} \, a^{2} d^{3} x^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{2} \, a^{2} d^{4} x^{2} e \mathrm{sgn}\left (b x + a\right ) + a^{2} d^{5} x \mathrm{sgn}\left (b x + a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^5*((b*x+a)^2)^(1/2),x, algorithm="giac")

[Out]

1/8*b^2*x^8*e^5*sgn(b*x + a) + 5/7*b^2*d*x^7*e^4*sgn(b*x + a) + 5/3*b^2*d^2*x^6*e^3*sgn(b*x + a) + 2*b^2*d^3*x
^5*e^2*sgn(b*x + a) + 5/4*b^2*d^4*x^4*e*sgn(b*x + a) + 1/3*b^2*d^5*x^3*sgn(b*x + a) + 2/7*a*b*x^7*e^5*sgn(b*x
+ a) + 5/3*a*b*d*x^6*e^4*sgn(b*x + a) + 4*a*b*d^2*x^5*e^3*sgn(b*x + a) + 5*a*b*d^3*x^4*e^2*sgn(b*x + a) + 10/3
*a*b*d^4*x^3*e*sgn(b*x + a) + a*b*d^5*x^2*sgn(b*x + a) + 1/6*a^2*x^6*e^5*sgn(b*x + a) + a^2*d*x^5*e^4*sgn(b*x
+ a) + 5/2*a^2*d^2*x^4*e^3*sgn(b*x + a) + 10/3*a^2*d^3*x^3*e^2*sgn(b*x + a) + 5/2*a^2*d^4*x^2*e*sgn(b*x + a) +
 a^2*d^5*x*sgn(b*x + a)