3.1708 \(\int (A+B x) (d+e x)^4 \sqrt{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=158 \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^6 (-a B e-A b e+2 b B d)}{6 e^3 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^5 (b d-a e) (B d-A e)}{5 e^3 (a+b x)}+\frac{b B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^7}{7 e^3 (a+b x)} \]

[Out]

((b*d - a*e)*(B*d - A*e)*(d + e*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^3*(a + b*x)) - ((2*b*B*d - A*b*e - a*
B*e)*(d + e*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*e^3*(a + b*x)) + (b*B*(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 = 0.185132, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061, Rules used = {770, 77} \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^6 (-a B e-A b e+2 b B d)}{6 e^3 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^5 (b d-a e) (B d-A e)}{5 e^3 (a+b x)}+\frac{b B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^7}{7 e^3 (a+b x)} \]

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)*(B*d - A*e)*(d + e*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^3*(a + b*x)) - ((2*b*B*d - A*b*e - a*
B*e)*(d + e*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*e^3*(a + b*x)) + (b*B*(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 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

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 \left (a b+b^2 x\right ) (A+B x) (d+e x)^4 \, dx}{a b+b^2 x}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (-\frac{b (b d-a e) (-B d+A e) (d+e x)^4}{e^2}+\frac{b (-2 b B d+A b e+a B e) (d+e x)^5}{e^2}+\frac{b^2 B (d+e x)^6}{e^2}\right ) \, dx}{a b+b^2 x}\\ &=\frac{(b d-a e) (B d-A e) (d+e x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^3 (a+b x)}-\frac{(2 b B d-A b e-a B e) (d+e x)^6 \sqrt{a^2+2 a b x+b^2 x^2}}{6 e^3 (a+b x)}+\frac{b B (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 = 0.0872226, size = 208, normalized size = 1.32 \[ \frac{x \sqrt{(a+b x)^2} \left (7 a \left (6 A \left (10 d^2 e^2 x^2+10 d^3 e x+5 d^4+5 d e^3 x^3+e^4 x^4\right )+B x \left (45 d^2 e^2 x^2+40 d^3 e x+15 d^4+24 d e^3 x^3+5 e^4 x^4\right )\right )+b x \left (7 A \left (45 d^2 e^2 x^2+40 d^3 e x+15 d^4+24 d e^3 x^3+5 e^4 x^4\right )+2 B x \left (126 d^2 e^2 x^2+105 d^3 e x+35 d^4+70 d e^3 x^3+15 e^4 x^4\right )\right )\right )}{210 (a+b x)} \]

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]*(7*a*(6*A*(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) + B*x*(15*d^4 + 4
0*d^3*e*x + 45*d^2*e^2*x^2 + 24*d*e^3*x^3 + 5*e^4*x^4)) + b*x*(7*A*(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) + 2*B*x*(35*d^4 + 105*d^3*e*x + 126*d^2*e^2*x^2 + 70*d*e^3*x^3 + 15*e^4*x^4))))/(210*(a
+ b*x))

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Maple [A]  time = 0.006, size = 232, normalized size = 1.5 \begin{align*}{\frac{x \left ( 30\,bB{e}^{4}{x}^{6}+35\,{x}^{5}Ab{e}^{4}+35\,{x}^{5}B{e}^{4}a+140\,{x}^{5}bBd{e}^{3}+42\,{x}^{4}aA{e}^{4}+168\,{x}^{4}Abd{e}^{3}+168\,{x}^{4}Bad{e}^{3}+252\,{x}^{4}bB{d}^{2}{e}^{2}+210\,{x}^{3}aAd{e}^{3}+315\,{x}^{3}Ab{d}^{2}{e}^{2}+315\,{x}^{3}Ba{d}^{2}{e}^{2}+210\,{x}^{3}bB{d}^{3}e+420\,{x}^{2}aA{d}^{2}{e}^{2}+280\,{x}^{2}Ab{d}^{3}e+280\,{x}^{2}Ba{d}^{3}e+70\,{x}^{2}bB{d}^{4}+420\,xaA{d}^{3}e+105\,xAb{d}^{4}+105\,xBa{d}^{4}+210\,aA{d}^{4} \right ) }{210\,bx+210\,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]

1/210*x*(30*B*b*e^4*x^6+35*A*b*e^4*x^5+35*B*a*e^4*x^5+140*B*b*d*e^3*x^5+42*A*a*e^4*x^4+168*A*b*d*e^3*x^4+168*B
*a*d*e^3*x^4+252*B*b*d^2*e^2*x^4+210*A*a*d*e^3*x^3+315*A*b*d^2*e^2*x^3+315*B*a*d^2*e^2*x^3+210*B*b*d^3*e*x^3+4
20*A*a*d^2*e^2*x^2+280*A*b*d^3*e*x^2+280*B*a*d^3*e*x^2+70*B*b*d^4*x^2+420*A*a*d^3*e*x+105*A*b*d^4*x+105*B*a*d^
4*x+210*A*a*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 = 1.61146, size = 393, normalized size = 2.49 \begin{align*} \frac{1}{7} \, B b e^{4} x^{7} + A a d^{4} x + \frac{1}{6} \,{\left (4 \, B b d e^{3} +{\left (B a + A b\right )} e^{4}\right )} x^{6} + \frac{1}{5} \,{\left (6 \, B b d^{2} e^{2} + A a e^{4} + 4 \,{\left (B a + A b\right )} d e^{3}\right )} x^{5} + \frac{1}{2} \,{\left (2 \, B b d^{3} e + 2 \, A a d e^{3} + 3 \,{\left (B a + A b\right )} d^{2} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (B b d^{4} + 6 \, A a d^{2} e^{2} + 4 \,{\left (B a + A b\right )} d^{3} e\right )} x^{3} + \frac{1}{2} \,{\left (4 \, A a d^{3} e +{\left (B a + A b\right )} d^{4}\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*b*e^4*x^7 + A*a*d^4*x + 1/6*(4*B*b*d*e^3 + (B*a + A*b)*e^4)*x^6 + 1/5*(6*B*b*d^2*e^2 + A*a*e^4 + 4*(B*a
+ A*b)*d*e^3)*x^5 + 1/2*(2*B*b*d^3*e + 2*A*a*d*e^3 + 3*(B*a + A*b)*d^2*e^2)*x^4 + 1/3*(B*b*d^4 + 6*A*a*d^2*e^2
+ 4*(B*a + A*b)*d^3*e)*x^3 + 1/2*(4*A*a*d^3*e + (B*a + A*b)*d^4)*x^2

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Sympy [A]  time = 0.150686, size = 226, normalized size = 1.43 \begin{align*} A a d^{4} x + \frac{B b e^{4} x^{7}}{7} + x^{6} \left (\frac{A b e^{4}}{6} + \frac{B a e^{4}}{6} + \frac{2 B b d e^{3}}{3}\right ) + x^{5} \left (\frac{A a e^{4}}{5} + \frac{4 A b d e^{3}}{5} + \frac{4 B a d e^{3}}{5} + \frac{6 B b d^{2} e^{2}}{5}\right ) + x^{4} \left (A a d e^{3} + \frac{3 A b d^{2} e^{2}}{2} + \frac{3 B a d^{2} e^{2}}{2} + B b d^{3} e\right ) + x^{3} \left (2 A a d^{2} e^{2} + \frac{4 A b d^{3} e}{3} + \frac{4 B a d^{3} e}{3} + \frac{B b d^{4}}{3}\right ) + x^{2} \left (2 A a d^{3} e + \frac{A b d^{4}}{2} + \frac{B a d^{4}}{2}\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*a*d**4*x + B*b*e**4*x**7/7 + x**6*(A*b*e**4/6 + B*a*e**4/6 + 2*B*b*d*e**3/3) + x**5*(A*a*e**4/5 + 4*A*b*d*e*
*3/5 + 4*B*a*d*e**3/5 + 6*B*b*d**2*e**2/5) + x**4*(A*a*d*e**3 + 3*A*b*d**2*e**2/2 + 3*B*a*d**2*e**2/2 + B*b*d*
*3*e) + x**3*(2*A*a*d**2*e**2 + 4*A*b*d**3*e/3 + 4*B*a*d**3*e/3 + B*b*d**4/3) + x**2*(2*A*a*d**3*e + A*b*d**4/
2 + B*a*d**4/2)

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Giac [B]  time = 1.16898, size = 443, normalized size = 2.8 \begin{align*} \frac{1}{7} \, B b x^{7} e^{4} \mathrm{sgn}\left (b x + a\right ) + \frac{2}{3} \, B b d x^{6} e^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{6}{5} \, B b d^{2} x^{5} e^{2} \mathrm{sgn}\left (b x + a\right ) + B b d^{3} x^{4} e \mathrm{sgn}\left (b x + a\right ) + \frac{1}{3} \, B b d^{4} x^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{6} \, B a x^{6} e^{4} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{6} \, A b x^{6} e^{4} \mathrm{sgn}\left (b x + a\right ) + \frac{4}{5} \, B a d x^{5} e^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{4}{5} \, A b d x^{5} e^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{2} \, B a d^{2} x^{4} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{2} \, A b d^{2} x^{4} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{4}{3} \, B a d^{3} x^{3} e \mathrm{sgn}\left (b x + a\right ) + \frac{4}{3} \, A b d^{3} x^{3} e \mathrm{sgn}\left (b x + a\right ) + \frac{1}{2} \, B a d^{4} x^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{2} \, A b d^{4} x^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{5} \, A a x^{5} e^{4} \mathrm{sgn}\left (b x + a\right ) + A a d x^{4} e^{3} \mathrm{sgn}\left (b x + a\right ) + 2 \, A a d^{2} x^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) + 2 \, A a d^{3} x^{2} e \mathrm{sgn}\left (b x + a\right ) + A a 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*b*x^7*e^4*sgn(b*x + a) + 2/3*B*b*d*x^6*e^3*sgn(b*x + a) + 6/5*B*b*d^2*x^5*e^2*sgn(b*x + a) + B*b*d^3*x^4
*e*sgn(b*x + a) + 1/3*B*b*d^4*x^3*sgn(b*x + a) + 1/6*B*a*x^6*e^4*sgn(b*x + a) + 1/6*A*b*x^6*e^4*sgn(b*x + a) +
4/5*B*a*d*x^5*e^3*sgn(b*x + a) + 4/5*A*b*d*x^5*e^3*sgn(b*x + a) + 3/2*B*a*d^2*x^4*e^2*sgn(b*x + a) + 3/2*A*b*
d^2*x^4*e^2*sgn(b*x + a) + 4/3*B*a*d^3*x^3*e*sgn(b*x + a) + 4/3*A*b*d^3*x^3*e*sgn(b*x + a) + 1/2*B*a*d^4*x^2*s
gn(b*x + a) + 1/2*A*b*d^4*x^2*sgn(b*x + a) + 1/5*A*a*x^5*e^4*sgn(b*x + a) + A*a*d*x^4*e^3*sgn(b*x + a) + 2*A*a
*d^2*x^3*e^2*sgn(b*x + a) + 2*A*a*d^3*x^2*e*sgn(b*x + a) + A*a*d^4*x*sgn(b*x + a)