3.2813 \(\int (c (a+b x)^3)^{5/2} \, dx\)

Optimal. Leaf size=30 \[ \frac{2 c^2 (a+b x)^7 \sqrt{c (a+b x)^3}}{17 b} \]

[Out]

(2*c^2*(a + b*x)^7*Sqrt[c*(a + b*x)^3])/(17*b)

________________________________________________________________________________________

Rubi [A]  time = 0.0152229, 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} \[ \frac{2 c^2 (a+b x)^7 \sqrt{c (a+b x)^3}}{17 b} \]

Antiderivative was successfully verified.

[In]

Int[(c*(a + b*x)^3)^(5/2),x]

[Out]

(2*c^2*(a + b*x)^7*Sqrt[c*(a + b*x)^3])/(17*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)^3\right )^{5/2} \, dx &=\frac{\operatorname{Subst}\left (\int \left (c x^3\right )^{5/2} \, dx,x,a+b x\right )}{b}\\ &=\frac{\left (c^2 \sqrt{c (a+b x)^3}\right ) \operatorname{Subst}\left (\int x^{15/2} \, dx,x,a+b x\right )}{b (a+b x)^{3/2}}\\ &=\frac{2 c^2 (a+b x)^7 \sqrt{c (a+b x)^3}}{17 b}\\ \end{align*}

Mathematica [A]  time = 0.0303587, size = 25, normalized size = 0.83 \[ \frac{2 (a+b x) \left (c (a+b x)^3\right )^{5/2}}{17 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*(a + b*x)^3)^(5/2),x]

[Out]

(2*(a + b*x)*(c*(a + b*x)^3)^(5/2))/(17*b)

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 22, normalized size = 0.7 \begin{align*}{\frac{2\,bx+2\,a}{17\,b} \left ( c \left ( bx+a \right ) ^{3} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*(b*x+a)^3)^(5/2),x)

[Out]

2/17*(b*x+a)*(c*(b*x+a)^3)^(5/2)/b

________________________________________________________________________________________

Maxima [B]  time = 1.01318, size = 146, normalized size = 4.87 \begin{align*} \frac{2 \,{\left (b^{7} c^{\frac{5}{2}} x^{7} + 7 \, a b^{6} c^{\frac{5}{2}} x^{6} + 21 \, a^{2} b^{5} c^{\frac{5}{2}} x^{5} + 35 \, a^{3} b^{4} c^{\frac{5}{2}} x^{4} + 35 \, a^{4} b^{3} c^{\frac{5}{2}} x^{3} + 21 \, a^{5} b^{2} c^{\frac{5}{2}} x^{2} + 7 \, a^{6} b c^{\frac{5}{2}} x + a^{7} c^{\frac{5}{2}}\right )}{\left (b x + a\right )}^{\frac{3}{2}}}{17 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*(b*x+a)^3)^(5/2),x, algorithm="maxima")

[Out]

2/17*(b^7*c^(5/2)*x^7 + 7*a*b^6*c^(5/2)*x^6 + 21*a^2*b^5*c^(5/2)*x^5 + 35*a^3*b^4*c^(5/2)*x^4 + 35*a^4*b^3*c^(
5/2)*x^3 + 21*a^5*b^2*c^(5/2)*x^2 + 7*a^6*b*c^(5/2)*x + a^7*c^(5/2))*(b*x + a)^(3/2)/b

________________________________________________________________________________________

Fricas [B]  time = 1.27736, size = 275, normalized size = 9.17 \begin{align*} \frac{2 \,{\left (b^{7} c^{2} x^{7} + 7 \, a b^{6} c^{2} x^{6} + 21 \, a^{2} b^{5} c^{2} x^{5} + 35 \, a^{3} b^{4} c^{2} x^{4} + 35 \, a^{4} b^{3} c^{2} x^{3} + 21 \, a^{5} b^{2} c^{2} x^{2} + 7 \, a^{6} b c^{2} x + a^{7} c^{2}\right )} \sqrt{b^{3} c x^{3} + 3 \, a b^{2} c x^{2} + 3 \, a^{2} b c x + a^{3} c}}{17 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*(b*x+a)^3)^(5/2),x, algorithm="fricas")

[Out]

2/17*(b^7*c^2*x^7 + 7*a*b^6*c^2*x^6 + 21*a^2*b^5*c^2*x^5 + 35*a^3*b^4*c^2*x^4 + 35*a^4*b^3*c^2*x^3 + 21*a^5*b^
2*c^2*x^2 + 7*a^6*b*c^2*x + a^7*c^2)*sqrt(b^3*c*x^3 + 3*a*b^2*c*x^2 + 3*a^2*b*c*x + a^3*c)/b

________________________________________________________________________________________

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)**3)**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.31209, size = 925, normalized size = 30.83 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*(b*x+a)^3)^(5/2),x, algorithm="giac")

[Out]

2/109395*(36465*(b*c*x + a*c)^(3/2)*a^7*c*sgn(b*x + a) - 51051*(5*(b*c*x + a*c)^(3/2)*a*c - 3*(b*c*x + a*c)^(5
/2))*a^6*sgn(b*x + a) + 21879*(35*(b*c*x + a*c)^(3/2)*a^2*c^2 - 42*(b*c*x + a*c)^(5/2)*a*c + 15*(b*c*x + a*c)^
(7/2))*a^5*sgn(b*x + a)/c - 12155*(105*(b*c*x + a*c)^(3/2)*a^3*c^3 - 189*(b*c*x + a*c)^(5/2)*a^2*c^2 + 135*(b*
c*x + a*c)^(7/2)*a*c - 35*(b*c*x + a*c)^(9/2))*a^4*sgn(b*x + a)/c^2 + 1105*(1155*(b*c*x + a*c)^(3/2)*a^4*c^4 -
2772*(b*c*x + a*c)^(5/2)*a^3*c^3 + 2970*(b*c*x + a*c)^(7/2)*a^2*c^2 - 1540*(b*c*x + a*c)^(9/2)*a*c + 315*(b*c
*x + a*c)^(11/2))*a^3*sgn(b*x + a)/c^3 - 255*(3003*(b*c*x + a*c)^(3/2)*a^5*c^5 - 9009*(b*c*x + a*c)^(5/2)*a^4*
c^4 + 12870*(b*c*x + a*c)^(7/2)*a^3*c^3 - 10010*(b*c*x + a*c)^(9/2)*a^2*c^2 + 4095*(b*c*x + a*c)^(11/2)*a*c -
693*(b*c*x + a*c)^(13/2))*a^2*sgn(b*x + a)/c^4 + 17*(15015*(b*c*x + a*c)^(3/2)*a^6*c^6 - 54054*(b*c*x + a*c)^(
5/2)*a^5*c^5 + 96525*(b*c*x + a*c)^(7/2)*a^4*c^4 - 100100*(b*c*x + a*c)^(9/2)*a^3*c^3 + 61425*(b*c*x + a*c)^(1
1/2)*a^2*c^2 - 20790*(b*c*x + a*c)^(13/2)*a*c + 3003*(b*c*x + a*c)^(15/2))*a*sgn(b*x + a)/c^5 - (36465*(b*c*x
+ a*c)^(3/2)*a^7*c^7 - 153153*(b*c*x + a*c)^(5/2)*a^6*c^6 + 328185*(b*c*x + a*c)^(7/2)*a^5*c^5 - 425425*(b*c*x
+ a*c)^(9/2)*a^4*c^4 + 348075*(b*c*x + a*c)^(11/2)*a^3*c^3 - 176715*(b*c*x + a*c)^(13/2)*a^2*c^2 + 51051*(b*c
*x + a*c)^(15/2)*a*c - 6435*(b*c*x + a*c)^(17/2))*sgn(b*x + a)/c^6)/b