3.2814 \(\int (c (a+b x)^3)^{3/2} \, dx\)

Optimal. Leaf size=28 \[ \frac {2 c (a+b x)^4 \sqrt {c (a+b x)^3}}{11 b} \]

[Out]

2/11*c*(b*x+a)^4*(c*(b*x+a)^3)^(1/2)/b

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, 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} \[ \frac {2 c (a+b x)^4 \sqrt {c (a+b x)^3}}{11 b} \]

Antiderivative was successfully verified.

[In]

Int[(c*(a + b*x)^3)^(3/2),x]

[Out]

(2*c*(a + b*x)^4*Sqrt[c*(a + b*x)^3])/(11*b)

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]

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]

Rubi steps

\begin {align*} \int \left (c (a+b x)^3\right )^{3/2} \, dx &=\frac {\operatorname {Subst}\left (\int \left (c x^3\right )^{3/2} \, dx,x,a+b x\right )}{b}\\ &=\frac {\left (c \sqrt {c (a+b x)^3}\right ) \operatorname {Subst}\left (\int x^{9/2} \, dx,x,a+b x\right )}{b (a+b x)^{3/2}}\\ &=\frac {2 c (a+b x)^4 \sqrt {c (a+b x)^3}}{11 b}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 25, normalized size = 0.89 \[ \frac {2 (a+b x) \left (c (a+b x)^3\right )^{3/2}}{11 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*(a + b*x)^3)^(3/2),x]

[Out]

(2*(a + b*x)*(c*(a + b*x)^3)^(3/2))/(11*b)

________________________________________________________________________________________

fricas [B]  time = 0.60, size = 83, normalized size = 2.96 \[ \frac {2 \, {\left (b^{4} c x^{4} + 4 \, a b^{3} c x^{3} + 6 \, a^{2} b^{2} c x^{2} + 4 \, a^{3} b c x + a^{4} c\right )} \sqrt {b^{3} c x^{3} + 3 \, a b^{2} c x^{2} + 3 \, a^{2} b c x + a^{3} c}}{11 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/11*(b^4*c*x^4 + 4*a*b^3*c*x^3 + 6*a^2*b^2*c*x^2 + 4*a^3*b*c*x + a^4*c)*sqrt(b^3*c*x^3 + 3*a*b^2*c*x^2 + 3*a^
2*b*c*x + a^3*c)/b

________________________________________________________________________________________

giac [B]  time = 0.20, size = 408, normalized size = 14.57 \[ \frac {2 \, {\left (693 \, \sqrt {b c x + a c} a^{5} \mathrm {sgn}\left (b x + a\right ) - \frac {1155 \, {\left (3 \, \sqrt {b c x + a c} a c - {\left (b c x + a c\right )}^{\frac {3}{2}}\right )} a^{4} \mathrm {sgn}\left (b x + a\right )}{c} + \frac {462 \, {\left (15 \, \sqrt {b c x + a c} a^{2} c^{2} - 10 \, {\left (b c x + a c\right )}^{\frac {3}{2}} a c + 3 \, {\left (b c x + a c\right )}^{\frac {5}{2}}\right )} a^{3} \mathrm {sgn}\left (b x + a\right )}{c^{2}} - \frac {198 \, {\left (35 \, \sqrt {b c x + a c} a^{3} c^{3} - 35 \, {\left (b c x + a c\right )}^{\frac {3}{2}} a^{2} c^{2} + 21 \, {\left (b c x + a c\right )}^{\frac {5}{2}} a c - 5 \, {\left (b c x + a c\right )}^{\frac {7}{2}}\right )} a^{2} \mathrm {sgn}\left (b x + a\right )}{c^{3}} + \frac {11 \, {\left (315 \, \sqrt {b c x + a c} a^{4} c^{4} - 420 \, {\left (b c x + a c\right )}^{\frac {3}{2}} a^{3} c^{3} + 378 \, {\left (b c x + a c\right )}^{\frac {5}{2}} a^{2} c^{2} - 180 \, {\left (b c x + a c\right )}^{\frac {7}{2}} a c + 35 \, {\left (b c x + a c\right )}^{\frac {9}{2}}\right )} a \mathrm {sgn}\left (b x + a\right )}{c^{4}} - \frac {{\left (693 \, \sqrt {b c x + a c} a^{5} c^{5} - 1155 \, {\left (b c x + a c\right )}^{\frac {3}{2}} a^{4} c^{4} + 1386 \, {\left (b c x + a c\right )}^{\frac {5}{2}} a^{3} c^{3} - 990 \, {\left (b c x + a c\right )}^{\frac {7}{2}} a^{2} c^{2} + 385 \, {\left (b c x + a c\right )}^{\frac {9}{2}} a c - 63 \, {\left (b c x + a c\right )}^{\frac {11}{2}}\right )} \mathrm {sgn}\left (b x + a\right )}{c^{5}}\right )} c}{693 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/693*(693*sqrt(b*c*x + a*c)*a^5*sgn(b*x + a) - 1155*(3*sqrt(b*c*x + a*c)*a*c - (b*c*x + a*c)^(3/2))*a^4*sgn(b
*x + a)/c + 462*(15*sqrt(b*c*x + a*c)*a^2*c^2 - 10*(b*c*x + a*c)^(3/2)*a*c + 3*(b*c*x + a*c)^(5/2))*a^3*sgn(b*
x + a)/c^2 - 198*(35*sqrt(b*c*x + a*c)*a^3*c^3 - 35*(b*c*x + a*c)^(3/2)*a^2*c^2 + 21*(b*c*x + a*c)^(5/2)*a*c -
5*(b*c*x + a*c)^(7/2))*a^2*sgn(b*x + a)/c^3 + 11*(315*sqrt(b*c*x + a*c)*a^4*c^4 - 420*(b*c*x + a*c)^(3/2)*a^3
*c^3 + 378*(b*c*x + a*c)^(5/2)*a^2*c^2 - 180*(b*c*x + a*c)^(7/2)*a*c + 35*(b*c*x + a*c)^(9/2))*a*sgn(b*x + a)/
c^4 - (693*sqrt(b*c*x + a*c)*a^5*c^5 - 1155*(b*c*x + a*c)^(3/2)*a^4*c^4 + 1386*(b*c*x + a*c)^(5/2)*a^3*c^3 - 9
90*(b*c*x + a*c)^(7/2)*a^2*c^2 + 385*(b*c*x + a*c)^(9/2)*a*c - 63*(b*c*x + a*c)^(11/2))*sgn(b*x + a)/c^5)*c/b

________________________________________________________________________________________

maple [A]  time = 0.00, size = 22, normalized size = 0.79 \[ \frac {2 \left (b x +a \right ) \left (\left (b x +a \right )^{3} c \right )^{\frac {3}{2}}}{11 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/11*(b*x+a)*((b*x+a)^3*c)^(3/2)/b

________________________________________________________________________________________

maxima [B]  time = 0.69, size = 66, normalized size = 2.36 \[ \frac {2 \, {\left (b^{4} c^{\frac {3}{2}} x^{4} + 4 \, a b^{3} c^{\frac {3}{2}} x^{3} + 6 \, a^{2} b^{2} c^{\frac {3}{2}} x^{2} + 4 \, a^{3} b c^{\frac {3}{2}} x + a^{4} c^{\frac {3}{2}}\right )} {\left (b x + a\right )}^{\frac {3}{2}}}{11 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/11*(b^4*c^(3/2)*x^4 + 4*a*b^3*c^(3/2)*x^3 + 6*a^2*b^2*c^(3/2)*x^2 + 4*a^3*b*c^(3/2)*x + a^4*c^(3/2))*(b*x +
a)^(3/2)/b

________________________________________________________________________________________

mupad [B]  time = 1.25, size = 58, normalized size = 2.07 \[ \sqrt {c\,{\left (a+b\,x\right )}^3}\,\left (\frac {2\,a^4\,c}{11\,b}+\frac {2\,b^3\,c\,x^4}{11}+\frac {8\,a^3\,c\,x}{11}+\frac {12\,a^2\,b\,c\,x^2}{11}+\frac {8\,a\,b^2\,c\,x^3}{11}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*(a + b*x)^3)^(3/2),x)

[Out]

(c*(a + b*x)^3)^(1/2)*((2*a^4*c)/(11*b) + (2*b^3*c*x^4)/11 + (8*a^3*c*x)/11 + (12*a^2*b*c*x^2)/11 + (8*a*b^2*c
*x^3)/11)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c \left (a + b x\right )^{3}\right )^{\frac {3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((c*(a + b*x)**3)**(3/2), x)

________________________________________________________________________________________