### 3.1544 $$\int (d+e x)^3 \sqrt{a^2+2 a b x+b^2 x^2} \, dx$$

Optimal. Leaf size=92 $\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^5}{5 e^2 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^4 (b d-a e)}{4 e^2 (a+b x)}$

[Out]

-((b*d - a*e)*(d + e*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^2*(a + b*x)) + (b*(d + e*x)^5*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2])/(5*e^2*(a + b*x))

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Rubi [A]  time = 0.0358736, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.071, Rules used = {646, 43} $\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^5}{5 e^2 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^4 (b d-a e)}{4 e^2 (a+b x)}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

-((b*d - a*e)*(d + e*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^2*(a + b*x)) + (b*(d + e*x)^5*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2])/(5*e^2*(a + b*x))

Rule 646

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b,
c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && NeQ[2*c*d - b*e, 0]

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

Mathematica [A]  time = 0.0282263, size = 89, normalized size = 0.97 $\frac{x \sqrt{(a+b x)^2} \left (5 a \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right )+b x \left (20 d^2 e x+10 d^3+15 d e^2 x^2+4 e^3 x^3\right )\right )}{20 (a+b x)}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

(x*Sqrt[(a + b*x)^2]*(5*a*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + b*x*(10*d^3 + 20*d^2*e*x + 15*d*e^2*x^
2 + 4*e^3*x^3)))/(20*(a + b*x))

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Maple [A]  time = 0.042, size = 90, normalized size = 1. \begin{align*}{\frac{x \left ( 4\,b{e}^{3}{x}^{4}+5\,{x}^{3}a{e}^{3}+15\,{x}^{3}bd{e}^{2}+20\,ad{e}^{2}{x}^{2}+20\,b{d}^{2}e{x}^{2}+30\,xa{d}^{2}e+10\,xb{d}^{3}+20\,a{d}^{3} \right ) }{20\,bx+20\,a}\sqrt{ \left ( bx+a \right ) ^{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3*((b*x+a)^2)^(1/2),x)

[Out]

1/20*x*(4*b*e^3*x^4+5*a*e^3*x^3+15*b*d*e^2*x^3+20*a*d*e^2*x^2+20*b*d^2*e*x^2+30*a*d^2*e*x+10*b*d^3*x+20*a*d^3)
*((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((e*x+d)^3*((b*x+a)^2)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.85812, size = 150, normalized size = 1.63 \begin{align*} \frac{1}{5} \, b e^{3} x^{5} + a d^{3} x + \frac{1}{4} \,{\left (3 \, b d e^{2} + a e^{3}\right )} x^{4} +{\left (b d^{2} e + a d e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (b d^{3} + 3 \, a d^{2} e\right )} x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*((b*x+a)^2)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.131303, size = 73, normalized size = 0.79 \begin{align*} a d^{3} x + \frac{b e^{3} x^{5}}{5} + x^{4} \left (\frac{a e^{3}}{4} + \frac{3 b d e^{2}}{4}\right ) + x^{3} \left (a d e^{2} + b d^{2} e\right ) + x^{2} \left (\frac{3 a d^{2} e}{2} + \frac{b d^{3}}{2}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3*((b*x+a)**2)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.16381, size = 159, normalized size = 1.73 \begin{align*} \frac{1}{5} \, b x^{5} e^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{4} \, b d x^{4} e^{2} \mathrm{sgn}\left (b x + a\right ) + b d^{2} x^{3} e \mathrm{sgn}\left (b x + a\right ) + \frac{1}{2} \, b d^{3} x^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{4} \, a x^{4} e^{3} \mathrm{sgn}\left (b x + a\right ) + a d x^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{2} \, a d^{2} x^{2} e \mathrm{sgn}\left (b x + a\right ) + a d^{3} x \mathrm{sgn}\left (b x + a\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*((b*x+a)^2)^(1/2),x, algorithm="giac")

[Out]

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