### 3.138 $$\int x^3 \sqrt{a^2+2 a b x+b^2 x^2} \, dx$$

Optimal. Leaf size=71 $\frac{b x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{a x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)}$

[Out]

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

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Rubi [A]  time = 0.0235502, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.083, Rules used = {646, 43} $\frac{b x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{a x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)}$

Antiderivative was successfully veriﬁed.

[In]

Int[x^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

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

Mathematica [A]  time = 0.0087176, size = 33, normalized size = 0.46 $\frac{x^4 \sqrt{(a+b x)^2} (5 a+4 b x)}{20 (a+b x)}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

(x^4*Sqrt[(a + b*x)^2]*(5*a + 4*b*x))/(20*(a + b*x))

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Maple [A]  time = 0.049, size = 30, normalized size = 0.4 \begin{align*}{\frac{{x}^{4} \left ( 4\,bx+5\,a \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(x^3*((b*x+a)^2)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.86497, size = 31, normalized size = 0.44 \begin{align*} \frac{1}{5} \, b x^{5} + \frac{1}{4} \, a x^{4} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b*x^5 + 1/4*a*x^4

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Sympy [A]  time = 0.208986, size = 12, normalized size = 0.17 \begin{align*} \frac{a x^{4}}{4} + \frac{b x^{5}}{5} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x**4/4 + b*x**5/5

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Giac [A]  time = 1.25429, size = 53, normalized size = 0.75 \begin{align*} \frac{1}{5} \, b x^{5} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{4} \, a x^{4} \mathrm{sgn}\left (b x + a\right ) - \frac{a^{5} \mathrm{sgn}\left (b x + a\right )}{20 \, b^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b*x^5*sgn(b*x + a) + 1/4*a*x^4*sgn(b*x + a) - 1/20*a^5*sgn(b*x + a)/b^4