3.1955 \(\int (a+b x) (d+e x)^4 \sqrt{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=146 \[ \frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^7}{7 e^3 (a+b x)}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^6 (b d-a e)}{3 e^3 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^5 (b d-a e)^2}{5 e^3 (a+b x)} \]

[Out]

((b*d - a*e)^2*(d + e*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^3*(a + b*x)) - (b*(b*d - a*e)*(d + e*x)^6*Sqrt[
a^2 + 2*a*b*x + b^2*x^2])/(3*e^3*(a + b*x)) + (b^2*(d + e*x)^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^3*(a + b*x)
)

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Rubi [A]  time = 0.132888, 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} \[ \frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^7}{7 e^3 (a+b x)}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^6 (b d-a e)}{3 e^3 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^5 (b d-a e)^2}{5 e^3 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(d + e*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

((b*d - a*e)^2*(d + e*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^3*(a + b*x)) - (b*(b*d - a*e)*(d + e*x)^6*Sqrt[
a^2 + 2*a*b*x + b^2*x^2])/(3*e^3*(a + b*x)) + (b^2*(d + e*x)^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*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)^4 \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)^4 \, 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)^4 \, 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)^4}{e^2}-\frac{2 b (b d-a e) (d+e x)^5}{e^2}+\frac{b^2 (d+e x)^6}{e^2}\right ) \, dx}{a b+b^2 x}\\ &=\frac{(b d-a e)^2 (d+e x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^3 (a+b x)}-\frac{b (b d-a e) (d+e x)^6 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^3 (a+b x)}+\frac{b^2 (d+e x)^7 \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^3 (a+b x)}\\ \end{align*}

Mathematica [A]  time = 0.0569503, size = 163, normalized size = 1.12 \[ \frac{x \sqrt{(a+b x)^2} \left (21 a^2 \left (10 d^2 e^2 x^2+10 d^3 e x+5 d^4+5 d e^3 x^3+e^4 x^4\right )+7 a b x \left (45 d^2 e^2 x^2+40 d^3 e x+15 d^4+24 d e^3 x^3+5 e^4 x^4\right )+b^2 x^2 \left (126 d^2 e^2 x^2+105 d^3 e x+35 d^4+70 d e^3 x^3+15 e^4 x^4\right )\right )}{105 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(d + e*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

(x*Sqrt[(a + b*x)^2]*(21*a^2*(5*d^4 + 10*d^3*e*x + 10*d^2*e^2*x^2 + 5*d*e^3*x^3 + e^4*x^4) + 7*a*b*x*(15*d^4 +
40*d^3*e*x + 45*d^2*e^2*x^2 + 24*d*e^3*x^3 + 5*e^4*x^4) + b^2*x^2*(35*d^4 + 105*d^3*e*x + 126*d^2*e^2*x^2 + 7
0*d*e^3*x^3 + 15*e^4*x^4)))/(105*(a + b*x))

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Maple [A]  time = 0.005, size = 189, normalized size = 1.3 \begin{align*}{\frac{x \left ( 15\,{b}^{2}{e}^{4}{x}^{6}+35\,{x}^{5}ab{e}^{4}+70\,{x}^{5}{b}^{2}d{e}^{3}+21\,{x}^{4}{a}^{2}{e}^{4}+168\,{x}^{4}abd{e}^{3}+126\,{x}^{4}{b}^{2}{d}^{2}{e}^{2}+105\,{a}^{2}d{e}^{3}{x}^{3}+315\,ab{d}^{2}{e}^{2}{x}^{3}+105\,{b}^{2}{d}^{3}e{x}^{3}+210\,{x}^{2}{a}^{2}{d}^{2}{e}^{2}+280\,{x}^{2}ab{d}^{3}e+35\,{x}^{2}{b}^{2}{d}^{4}+210\,{a}^{2}{d}^{3}ex+105\,b{d}^{4}ax+105\,{a}^{2}{d}^{4} \right ) }{105\,bx+105\,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)^4*((b*x+a)^2)^(1/2),x)

[Out]

1/105*x*(15*b^2*e^4*x^6+35*a*b*e^4*x^5+70*b^2*d*e^3*x^5+21*a^2*e^4*x^4+168*a*b*d*e^3*x^4+126*b^2*d^2*e^2*x^4+1
05*a^2*d*e^3*x^3+315*a*b*d^2*e^2*x^3+105*b^2*d^3*e*x^3+210*a^2*d^2*e^2*x^2+280*a*b*d^3*e*x^2+35*b^2*d^4*x^2+21
0*a^2*d^3*e*x+105*a*b*d^4*x+105*a^2*d^4)*((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)^4*((b*x+a)^2)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.49689, size = 323, normalized size = 2.21 \begin{align*} \frac{1}{7} \, b^{2} e^{4} x^{7} + a^{2} d^{4} x + \frac{1}{3} \,{\left (2 \, b^{2} d e^{3} + a b e^{4}\right )} x^{6} + \frac{1}{5} \,{\left (6 \, b^{2} d^{2} e^{2} + 8 \, a b d e^{3} + a^{2} e^{4}\right )} x^{5} +{\left (b^{2} d^{3} e + 3 \, a b d^{2} e^{2} + a^{2} d e^{3}\right )} x^{4} + \frac{1}{3} \,{\left (b^{2} d^{4} + 8 \, a b d^{3} e + 6 \, a^{2} d^{2} e^{2}\right )} x^{3} +{\left (a b d^{4} + 2 \, a^{2} d^{3} e\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^4*((b*x+a)^2)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.131263, size = 168, normalized size = 1.15 \begin{align*} a^{2} d^{4} x + \frac{b^{2} e^{4} x^{7}}{7} + x^{6} \left (\frac{a b e^{4}}{3} + \frac{2 b^{2} d e^{3}}{3}\right ) + x^{5} \left (\frac{a^{2} e^{4}}{5} + \frac{8 a b d e^{3}}{5} + \frac{6 b^{2} d^{2} e^{2}}{5}\right ) + x^{4} \left (a^{2} d e^{3} + 3 a b d^{2} e^{2} + b^{2} d^{3} e\right ) + x^{3} \left (2 a^{2} d^{2} e^{2} + \frac{8 a b d^{3} e}{3} + \frac{b^{2} d^{4}}{3}\right ) + x^{2} \left (2 a^{2} d^{3} e + a b d^{4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)**4*((b*x+a)**2)**(1/2),x)

[Out]

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

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Giac [B]  time = 1.13681, size = 343, normalized size = 2.35 \begin{align*} \frac{1}{7} \, b^{2} x^{7} e^{4} \mathrm{sgn}\left (b x + a\right ) + \frac{2}{3} \, b^{2} d x^{6} e^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{6}{5} \, b^{2} d^{2} x^{5} e^{2} \mathrm{sgn}\left (b x + a\right ) + b^{2} d^{3} x^{4} e \mathrm{sgn}\left (b x + a\right ) + \frac{1}{3} \, b^{2} d^{4} x^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{3} \, a b x^{6} e^{4} \mathrm{sgn}\left (b x + a\right ) + \frac{8}{5} \, a b d x^{5} e^{3} \mathrm{sgn}\left (b x + a\right ) + 3 \, a b d^{2} x^{4} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{8}{3} \, a b d^{3} x^{3} e \mathrm{sgn}\left (b x + a\right ) + a b d^{4} x^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{5} \, a^{2} x^{5} e^{4} \mathrm{sgn}\left (b x + a\right ) + a^{2} d x^{4} e^{3} \mathrm{sgn}\left (b x + a\right ) + 2 \, a^{2} d^{2} x^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) + 2 \, a^{2} d^{3} x^{2} e \mathrm{sgn}\left (b x + a\right ) + a^{2} d^{4} 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)^4*((b*x+a)^2)^(1/2),x, algorithm="giac")

[Out]

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