\(\int (a+b x)^{3/2} (a c-b c x)^{3/2} \, dx\) [1146]

Optimal result

Integrand size = 23, antiderivative size = 102 \[ \int (a+b x)^{3/2} (a c-b c x)^{3/2} \, dx=\frac {3}{8} a^2 c x \sqrt {a+b x} \sqrt {a c-b c x}+\frac {1}{4} x (a+b x)^{3/2} (a c-b c x)^{3/2}+\frac {3 a^4 c^{3/2} \arctan \left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {c (a-b x)}}\right )}{4 b} \]

[Out]

1/4*x*(b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)+3/4*a^4*c^(3/2)*arctan(c^(1/2)*(b*x+a)^(1/2)/(c*(-b*x+a))^(1/2))/b+3/8*
a^2*c*x*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {38, 65, 223, 209} \[ \int (a+b x)^{3/2} (a c-b c x)^{3/2} \, dx=\frac {3 a^4 c^{3/2} \arctan \left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {c (a-b x)}}\right )}{4 b}+\frac {3}{8} a^2 c x \sqrt {a+b x} \sqrt {a c-b c x}+\frac {1}{4} x (a+b x)^{3/2} (a c-b c x)^{3/2} \]

[In]

Int[(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2),x]

[Out]

(3*a^2*c*x*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/8 + (x*(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2))/4 + (3*a^4*c^(3/2)*Arc
Tan[(Sqrt[c]*Sqrt[a + b*x])/Sqrt[c*(a - b*x)]])/(4*b)

Rule 38

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[x*(a + b*x)^m*((c + d*x)^m/(2*m + 1))
, x] + Dist[2*a*c*(m/(2*m + 1)), Int[(a + b*x)^(m - 1)*(c + d*x)^(m - 1), x], x] /; FreeQ[{a, b, c, d}, x] &&
EqQ[b*c + a*d, 0] && IGtQ[m + 1/2, 0]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x (a+b x)^{3/2} (a c-b c x)^{3/2}+\frac {1}{4} \left (3 a^2 c\right ) \int \sqrt {a+b x} \sqrt {a c-b c x} \, dx \\ & = \frac {3}{8} a^2 c x \sqrt {a+b x} \sqrt {a c-b c x}+\frac {1}{4} x (a+b x)^{3/2} (a c-b c x)^{3/2}+\frac {1}{8} \left (3 a^4 c^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx \\ & = \frac {3}{8} a^2 c x \sqrt {a+b x} \sqrt {a c-b c x}+\frac {1}{4} x (a+b x)^{3/2} (a c-b c x)^{3/2}+\frac {\left (3 a^4 c^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2 a c-c x^2}} \, dx,x,\sqrt {a+b x}\right )}{4 b} \\ & = \frac {3}{8} a^2 c x \sqrt {a+b x} \sqrt {a c-b c x}+\frac {1}{4} x (a+b x)^{3/2} (a c-b c x)^{3/2}+\frac {\left (3 a^4 c^2\right ) \text {Subst}\left (\int \frac {1}{1+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c (a-b x)}}\right )}{4 b} \\ & = \frac {3}{8} a^2 c x \sqrt {a+b x} \sqrt {a c-b c x}+\frac {1}{4} x (a+b x)^{3/2} (a c-b c x)^{3/2}+\frac {3 a^4 c^{3/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {c (a-b x)}}\right )}{4 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.90 \[ \int (a+b x)^{3/2} (a c-b c x)^{3/2} \, dx=\frac {(c (a-b x))^{3/2} \left (b x \sqrt {a-b x} \sqrt {a+b x} \left (5 a^2-2 b^2 x^2\right )+6 a^4 \arctan \left (\frac {\sqrt {a+b x}}{\sqrt {a-b x}}\right )\right )}{8 b (a-b x)^{3/2}} \]

[In]

Integrate[(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2),x]

[Out]

((c*(a - b*x))^(3/2)*(b*x*Sqrt[a - b*x]*Sqrt[a + b*x]*(5*a^2 - 2*b^2*x^2) + 6*a^4*ArcTan[Sqrt[a + b*x]/Sqrt[a
- b*x]]))/(8*b*(a - b*x)^(3/2))

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.23

method	result	size
risch	\(\frac {x \left (-2 b^{2} x^{2}+5 a^{2}\right ) \left (-b x +a \right ) \sqrt {b x +a}\, c^{2}}{8 \sqrt {-c \left (b x -a \right )}}+\frac {3 a^{4} \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {-b^{2} c \,x^{2}+a^{2} c}}\right ) \sqrt {-\left (b x +a \right ) c \left (b x -a \right )}\, c^{2}}{8 \sqrt {b^{2} c}\, \sqrt {b x +a}\, \sqrt {-c \left (b x -a \right )}}\)	\(125\)
default	\(-\frac {\left (b x +a \right )^{\frac {3}{2}} \left (-b c x +a c \right )^{\frac {5}{2}}}{4 b c}+\frac {3 a \left (-\frac {\sqrt {b x +a}\, \left (-b c x +a c \right )^{\frac {5}{2}}}{3 b c}+\frac {a \left (\frac {\left (-b c x +a c \right )^{\frac {3}{2}} \sqrt {b x +a}}{2 b}+\frac {3 a c \left (\frac {\sqrt {-b c x +a c}\, \sqrt {b x +a}}{b}+\frac {a c \sqrt {\left (b x +a \right ) \left (-b c x +a c \right )}\, \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {-b^{2} c \,x^{2}+a^{2} c}}\right )}{\sqrt {-b c x +a c}\, \sqrt {b x +a}\, \sqrt {b^{2} c}}\right )}{2}\right )}{3}\right )}{4}\)	\(184\)

[In]

int((b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/8*x*(-2*b^2*x^2+5*a^2)*(-b*x+a)*(b*x+a)^(1/2)/(-c*(b*x-a))^(1/2)*c^2+3/8*a^4/(b^2*c)^(1/2)*arctan((b^2*c)^(1
/2)*x/(-b^2*c*x^2+a^2*c)^(1/2))*(-(b*x+a)*c*(b*x-a))^(1/2)/(b*x+a)^(1/2)/(-c*(b*x-a))^(1/2)*c^2

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.89 \[ \int (a+b x)^{3/2} (a c-b c x)^{3/2} \, dx=\left [\frac {3 \, a^{4} \sqrt {-c} c \log \left (2 \, b^{2} c x^{2} + 2 \, \sqrt {-b c x + a c} \sqrt {b x + a} b \sqrt {-c} x - a^{2} c\right ) - 2 \, {\left (2 \, b^{3} c x^{3} - 5 \, a^{2} b c x\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{16 \, b}, -\frac {3 \, a^{4} c^{\frac {3}{2}} \arctan \left (\frac {\sqrt {-b c x + a c} \sqrt {b x + a} b \sqrt {c} x}{b^{2} c x^{2} - a^{2} c}\right ) + {\left (2 \, b^{3} c x^{3} - 5 \, a^{2} b c x\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{8 \, b}\right ] \]

[In]

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

[Out]

[1/16*(3*a^4*sqrt(-c)*c*log(2*b^2*c*x^2 + 2*sqrt(-b*c*x + a*c)*sqrt(b*x + a)*b*sqrt(-c)*x - a^2*c) - 2*(2*b^3*
c*x^3 - 5*a^2*b*c*x)*sqrt(-b*c*x + a*c)*sqrt(b*x + a))/b, -1/8*(3*a^4*c^(3/2)*arctan(sqrt(-b*c*x + a*c)*sqrt(b
*x + a)*b*sqrt(c)*x/(b^2*c*x^2 - a^2*c)) + (2*b^3*c*x^3 - 5*a^2*b*c*x)*sqrt(-b*c*x + a*c)*sqrt(b*x + a))/b]

Sympy [F]

\[ \int (a+b x)^{3/2} (a c-b c x)^{3/2} \, dx=\int \left (- c \left (- a + b x\right )\right )^{\frac {3}{2}} \left (a + b x\right )^{\frac {3}{2}}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.62 \[ \int (a+b x)^{3/2} (a c-b c x)^{3/2} \, dx=\frac {3 \, a^{4} c^{\frac {3}{2}} \arcsin \left (\frac {b x}{a}\right )}{8 \, b} + \frac {3}{8} \, \sqrt {-b^{2} c x^{2} + a^{2} c} a^{2} c x + \frac {1}{4} \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} x \]

[In]

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

[Out]

3/8*a^4*c^(3/2)*arcsin(b*x/a)/b + 3/8*sqrt(-b^2*c*x^2 + a^2*c)*a^2*c*x + 1/4*(-b^2*c*x^2 + a^2*c)^(3/2)*x

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 354 vs. \(2 (82) = 164\).

Time = 0.41 (sec) , antiderivative size = 354, normalized size of antiderivative = 3.47 \[ \int (a+b x)^{3/2} (a c-b c x)^{3/2} \, dx=-\frac {24 \, {\left (\frac {2 \, a c \log \left ({\left | -\sqrt {b x + a} \sqrt {-c} + \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \right |}\right )}{\sqrt {-c}} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \sqrt {b x + a}\right )} a^{3} c - 12 \, {\left (\frac {2 \, a^{2} c \log \left ({\left | -\sqrt {b x + a} \sqrt {-c} + \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \right |}\right )}{\sqrt {-c}} + \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \sqrt {b x + a} {\left (b x - 2 \, a\right )}\right )} a^{2} c - 4 \, {\left (\frac {6 \, a^{3} c \log \left ({\left | -\sqrt {b x + a} \sqrt {-c} + \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \right |}\right )}{\sqrt {-c}} - {\left ({\left (2 \, b x - 5 \, a\right )} {\left (b x + a\right )} + 9 \, a^{2}\right )} \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \sqrt {b x + a}\right )} a c + {\left (\frac {18 \, a^{4} c \log \left ({\left | -\sqrt {b x + a} \sqrt {-c} + \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \right |}\right )}{\sqrt {-c}} - {\left (39 \, a^{3} - {\left (2 \, {\left (3 \, b x - 10 \, a\right )} {\left (b x + a\right )} + 43 \, a^{2}\right )} {\left (b x + a\right )}\right )} \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \sqrt {b x + a}\right )} c}{24 \, b} \]

[In]

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

[Out]

-1/24*(24*(2*a*c*log(abs(-sqrt(b*x + a)*sqrt(-c) + sqrt(-(b*x + a)*c + 2*a*c)))/sqrt(-c) - sqrt(-(b*x + a)*c +
2*a*c)*sqrt(b*x + a))*a^3*c - 12*(2*a^2*c*log(abs(-sqrt(b*x + a)*sqrt(-c) + sqrt(-(b*x + a)*c + 2*a*c)))/sqrt
(-c) + sqrt(-(b*x + a)*c + 2*a*c)*sqrt(b*x + a)*(b*x - 2*a))*a^2*c - 4*(6*a^3*c*log(abs(-sqrt(b*x + a)*sqrt(-c
) + sqrt(-(b*x + a)*c + 2*a*c)))/sqrt(-c) - ((2*b*x - 5*a)*(b*x + a) + 9*a^2)*sqrt(-(b*x + a)*c + 2*a*c)*sqrt(
b*x + a))*a*c + (18*a^4*c*log(abs(-sqrt(b*x + a)*sqrt(-c) + sqrt(-(b*x + a)*c + 2*a*c)))/sqrt(-c) - (39*a^3 -
(2*(3*b*x - 10*a)*(b*x + a) + 43*a^2)*(b*x + a))*sqrt(-(b*x + a)*c + 2*a*c)*sqrt(b*x + a))*c)/b

Mupad [F(-1)]

Timed out. \[ \int (a+b x)^{3/2} (a c-b c x)^{3/2} \, dx=\int {\left (a\,c-b\,c\,x\right )}^{3/2}\,{\left (a+b\,x\right )}^{3/2} \,d x \]

[In]

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

[Out]

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