3.2802 \(\int (c (a+b x)^2)^{5/2} \, dx\)

Optimal. Leaf size=30 \[ \frac{c^2 (a+b x)^5 \sqrt{c (a+b x)^2}}{6 b} \]

[Out]

(c^2*(a + b*x)^5*Sqrt[c*(a + b*x)^2])/(6*b)

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Rubi [A]  time = 0.0111322, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {247, 15, 30} \[ \frac{c^2 (a+b x)^5 \sqrt{c (a+b x)^2}}{6 b} \]

Antiderivative was successfully verified.

[In]

Int[(c*(a + b*x)^2)^(5/2),x]

[Out]

(c^2*(a + b*x)^5*Sqrt[c*(a + b*x)^2])/(6*b)

Rule 247

Int[((a_.) + (b_.)*(v_)^(n_))^(p_), x_Symbol] :> Dist[1/Coefficient[v, x, 1], Subst[Int[(a + b*x^n)^p, x], x,
v], x] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && NeQ[v, x]

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \left (c (a+b x)^2\right )^{5/2} \, dx &=\frac{\operatorname{Subst}\left (\int \left (c x^2\right )^{5/2} \, dx,x,a+b x\right )}{b}\\ &=\frac{\left (c^2 \sqrt{c (a+b x)^2}\right ) \operatorname{Subst}\left (\int x^5 \, dx,x,a+b x\right )}{b (a+b x)}\\ &=\frac{c^2 (a+b x)^5 \sqrt{c (a+b x)^2}}{6 b}\\ \end{align*}

Mathematica [A]  time = 0.0201105, size = 25, normalized size = 0.83 \[ \frac{(a+b x) \left (c (a+b x)^2\right )^{5/2}}{6 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*(a + b*x)^2)^(5/2),x]

[Out]

((a + b*x)*(c*(a + b*x)^2)^(5/2))/(6*b)

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Maple [B]  time = 0.004, size = 73, normalized size = 2.4 \begin{align*}{\frac{x \left ({b}^{5}{x}^{5}+6\,a{b}^{4}{x}^{4}+15\,{a}^{2}{b}^{3}{x}^{3}+20\,{a}^{3}{b}^{2}{x}^{2}+15\,{a}^{4}bx+6\,{a}^{5} \right ) }{6\, \left ( bx+a \right ) ^{5}} \left ( c \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*(b*x+a)^2)^(5/2),x)

[Out]

1/6*x*(b^5*x^5+6*a*b^4*x^4+15*a^2*b^3*x^3+20*a^3*b^2*x^2+15*a^4*b*x+6*a^5)*(c*(b*x+a)^2)^(5/2)/(b*x+a)^5

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.2146, size = 213, normalized size = 7.1 \begin{align*} \frac{{\left (b^{5} c^{2} x^{6} + 6 \, a b^{4} c^{2} x^{5} + 15 \, a^{2} b^{3} c^{2} x^{4} + 20 \, a^{3} b^{2} c^{2} x^{3} + 15 \, a^{4} b c^{2} x^{2} + 6 \, a^{5} c^{2} x\right )} \sqrt{b^{2} c x^{2} + 2 \, a b c x + a^{2} c}}{6 \,{\left (b x + a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(b^5*c^2*x^6 + 6*a*b^4*c^2*x^5 + 15*a^2*b^3*c^2*x^4 + 20*a^3*b^2*c^2*x^3 + 15*a^4*b*c^2*x^2 + 6*a^5*c^2*x)
*sqrt(b^2*c*x^2 + 2*a*b*c*x + a^2*c)/(b*x + a)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*(b*x+a)**2)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 1.12215, size = 174, normalized size = 5.8 \begin{align*} \frac{1}{6} \,{\left (b^{5} c^{2} x^{6} \mathrm{sgn}\left (b x + a\right ) + 6 \, a b^{4} c^{2} x^{5} \mathrm{sgn}\left (b x + a\right ) + 15 \, a^{2} b^{3} c^{2} x^{4} \mathrm{sgn}\left (b x + a\right ) + 20 \, a^{3} b^{2} c^{2} x^{3} \mathrm{sgn}\left (b x + a\right ) + 15 \, a^{4} b c^{2} x^{2} \mathrm{sgn}\left (b x + a\right ) + 6 \, a^{5} c^{2} x \mathrm{sgn}\left (b x + a\right ) + \frac{a^{6} c^{2} \mathrm{sgn}\left (b x + a\right )}{b}\right )} \sqrt{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(b^5*c^2*x^6*sgn(b*x + a) + 6*a*b^4*c^2*x^5*sgn(b*x + a) + 15*a^2*b^3*c^2*x^4*sgn(b*x + a) + 20*a^3*b^2*c^
2*x^3*sgn(b*x + a) + 15*a^4*b*c^2*x^2*sgn(b*x + a) + 6*a^5*c^2*x*sgn(b*x + a) + a^6*c^2*sgn(b*x + a)/b)*sqrt(c
)