\(\int (a+b x)^{5/2} (a c-b c x)^{5/2} \, dx\) [1145]

Optimal result

Integrand size = 23, antiderivative size = 135 \[ \int (a+b x)^{5/2} (a c-b c x)^{5/2} \, dx=\frac {5}{16} a^4 c^2 x \sqrt {a+b x} \sqrt {a c-b c x}+\frac {5}{24} a^2 c x (a+b x)^{3/2} (a c-b c x)^{3/2}+\frac {1}{6} x (a+b x)^{5/2} (a c-b c x)^{5/2}+\frac {5 a^6 c^{5/2} \arctan \left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {c (a-b x)}}\right )}{8 b} \]

[Out]

5/24*a^2*c*x*(b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)+1/6*x*(b*x+a)^(5/2)*(-b*c*x+a*c)^(5/2)+5/8*a^6*c^(5/2)*arctan(c^
(1/2)*(b*x+a)^(1/2)/(c*(-b*x+a))^(1/2))/b+5/16*a^4*c^2*x*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {38, 65, 223, 209} \[ \int (a+b x)^{5/2} (a c-b c x)^{5/2} \, dx=\frac {5 a^6 c^{5/2} \arctan \left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {c (a-b x)}}\right )}{8 b}+\frac {5}{16} a^4 c^2 x \sqrt {a+b x} \sqrt {a c-b c x}+\frac {5}{24} a^2 c x (a+b x)^{3/2} (a c-b c x)^{3/2}+\frac {1}{6} x (a+b x)^{5/2} (a c-b c x)^{5/2} \]

[In]

Int[(a + b*x)^(5/2)*(a*c - b*c*x)^(5/2),x]

[Out]

(5*a^4*c^2*x*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/16 + (5*a^2*c*x*(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2))/24 + (x*(a
+ b*x)^(5/2)*(a*c - b*c*x)^(5/2))/6 + (5*a^6*c^(5/2)*ArcTan[(Sqrt[c]*Sqrt[a + b*x])/Sqrt[c*(a - b*x)]])/(8*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}{6} x (a+b x)^{5/2} (a c-b c x)^{5/2}+\frac {1}{6} \left (5 a^2 c\right ) \int (a+b x)^{3/2} (a c-b c x)^{3/2} \, dx \\ & = \frac {5}{24} a^2 c x (a+b x)^{3/2} (a c-b c x)^{3/2}+\frac {1}{6} x (a+b x)^{5/2} (a c-b c x)^{5/2}+\frac {1}{8} \left (5 a^4 c^2\right ) \int \sqrt {a+b x} \sqrt {a c-b c x} \, dx \\ & = \frac {5}{16} a^4 c^2 x \sqrt {a+b x} \sqrt {a c-b c x}+\frac {5}{24} a^2 c x (a+b x)^{3/2} (a c-b c x)^{3/2}+\frac {1}{6} x (a+b x)^{5/2} (a c-b c x)^{5/2}+\frac {1}{16} \left (5 a^6 c^3\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx \\ & = \frac {5}{16} a^4 c^2 x \sqrt {a+b x} \sqrt {a c-b c x}+\frac {5}{24} a^2 c x (a+b x)^{3/2} (a c-b c x)^{3/2}+\frac {1}{6} x (a+b x)^{5/2} (a c-b c x)^{5/2}+\frac {\left (5 a^6 c^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2 a c-c x^2}} \, dx,x,\sqrt {a+b x}\right )}{8 b} \\ & = \frac {5}{16} a^4 c^2 x \sqrt {a+b x} \sqrt {a c-b c x}+\frac {5}{24} a^2 c x (a+b x)^{3/2} (a c-b c x)^{3/2}+\frac {1}{6} x (a+b x)^{5/2} (a c-b c x)^{5/2}+\frac {\left (5 a^6 c^3\right ) \text {Subst}\left (\int \frac {1}{1+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c (a-b x)}}\right )}{8 b} \\ & = \frac {5}{16} a^4 c^2 x \sqrt {a+b x} \sqrt {a c-b c x}+\frac {5}{24} a^2 c x (a+b x)^{3/2} (a c-b c x)^{3/2}+\frac {1}{6} x (a+b x)^{5/2} (a c-b c x)^{5/2}+\frac {5 a^6 c^{5/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {c (a-b x)}}\right )}{8 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.76 \[ \int (a+b x)^{5/2} (a c-b c x)^{5/2} \, dx=\frac {(c (a-b x))^{5/2} \left (b x \sqrt {a-b x} \sqrt {a+b x} \left (33 a^4-26 a^2 b^2 x^2+8 b^4 x^4\right )+30 a^6 \arctan \left (\frac {\sqrt {a+b x}}{\sqrt {a-b x}}\right )\right )}{48 b (a-b x)^{5/2}} \]

[In]

Integrate[(a + b*x)^(5/2)*(a*c - b*c*x)^(5/2),x]

[Out]

((c*(a - b*x))^(5/2)*(b*x*Sqrt[a - b*x]*Sqrt[a + b*x]*(33*a^4 - 26*a^2*b^2*x^2 + 8*b^4*x^4) + 30*a^6*ArcTan[Sq
rt[a + b*x]/Sqrt[a - b*x]]))/(48*b*(a - b*x)^(5/2))

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.01

method	result	size
risch	\(\frac {x \left (8 b^{4} x^{4}-26 a^{2} b^{2} x^{2}+33 a^{4}\right ) \left (-b x +a \right ) \sqrt {b x +a}\, c^{3}}{48 \sqrt {-c \left (b x -a \right )}}+\frac {5 a^{6} \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {-b^{2} c \,x^{2}+a^{2} c}}\right ) \sqrt {-\left (b x +a \right ) c \left (b x -a \right )}\, c^{3}}{16 \sqrt {b^{2} c}\, \sqrt {b x +a}\, \sqrt {-c \left (b x -a \right )}}\)	\(136\)
default	\(-\frac {\left (b x +a \right )^{\frac {5}{2}} \left (-b c x +a c \right )^{\frac {7}{2}}}{6 b c}+\frac {5 a \left (-\frac {\left (b x +a \right )^{\frac {3}{2}} \left (-b c x +a c \right )^{\frac {7}{2}}}{5 b c}+\frac {3 a \left (-\frac {\sqrt {b x +a}\, \left (-b c x +a c \right )^{\frac {7}{2}}}{4 b c}+\frac {a \left (\frac {\left (-b c x +a c \right )^{\frac {5}{2}} \sqrt {b x +a}}{3 b}+\frac {5 a c \left (\frac {\left (-b c x +a c \right )^{\frac {3}{2}} \sqrt {b x +a}}{2 b}+\frac {3 a c \left (\frac {\sqrt {-b c x +a c}\, \sqrt {b x +a}}{b}+\frac {a c \sqrt {\left (b x +a \right ) \left (-b c x +a c \right )}\, \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {-b^{2} c \,x^{2}+a^{2} c}}\right )}{\sqrt {-b c x +a c}\, \sqrt {b x +a}\, \sqrt {b^{2} c}}\right )}{2}\right )}{3}\right )}{4}\right )}{5}\right )}{6}\)	\(242\)

[In]

int((b*x+a)^(5/2)*(-b*c*x+a*c)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/48*x*(8*b^4*x^4-26*a^2*b^2*x^2+33*a^4)*(-b*x+a)*(b*x+a)^(1/2)/(-c*(b*x-a))^(1/2)*c^3+5/16*a^6/(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
^3

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.72 \[ \int (a+b x)^{5/2} (a c-b c x)^{5/2} \, dx=\left [\frac {15 \, a^{6} \sqrt {-c} c^{2} \log \left (2 \, b^{2} c x^{2} + 2 \, \sqrt {-b c x + a c} \sqrt {b x + a} b \sqrt {-c} x - a^{2} c\right ) + 2 \, {\left (8 \, b^{5} c^{2} x^{5} - 26 \, a^{2} b^{3} c^{2} x^{3} + 33 \, a^{4} b c^{2} x\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{96 \, b}, -\frac {15 \, a^{6} c^{\frac {5}{2}} \arctan \left (\frac {\sqrt {-b c x + a c} \sqrt {b x + a} b \sqrt {c} x}{b^{2} c x^{2} - a^{2} c}\right ) - {\left (8 \, b^{5} c^{2} x^{5} - 26 \, a^{2} b^{3} c^{2} x^{3} + 33 \, a^{4} b c^{2} x\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{48 \, b}\right ] \]

[In]

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

[Out]

[1/96*(15*a^6*sqrt(-c)*c^2*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*(8*b
^5*c^2*x^5 - 26*a^2*b^3*c^2*x^3 + 33*a^4*b*c^2*x)*sqrt(-b*c*x + a*c)*sqrt(b*x + a))/b, -1/48*(15*a^6*c^(5/2)*a
rctan(sqrt(-b*c*x + a*c)*sqrt(b*x + a)*b*sqrt(c)*x/(b^2*c*x^2 - a^2*c)) - (8*b^5*c^2*x^5 - 26*a^2*b^3*c^2*x^3
+ 33*a^4*b*c^2*x)*sqrt(-b*c*x + a*c)*sqrt(b*x + a))/b]

Sympy [F]

\[ \int (a+b x)^{5/2} (a c-b c x)^{5/2} \, dx=\int \left (- c \left (- a + b x\right )\right )^{\frac {5}{2}} \left (a + b x\right )^{\frac {5}{2}}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.66 \[ \int (a+b x)^{5/2} (a c-b c x)^{5/2} \, dx=\frac {5 \, a^{6} c^{\frac {5}{2}} \arcsin \left (\frac {b x}{a}\right )}{16 \, b} + \frac {5}{16} \, \sqrt {-b^{2} c x^{2} + a^{2} c} a^{4} c^{2} x + \frac {5}{24} \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} a^{2} c x + \frac {1}{6} \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {5}{2}} x \]

[In]

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

[Out]

5/16*a^6*c^(5/2)*arcsin(b*x/a)/b + 5/16*sqrt(-b^2*c*x^2 + a^2*c)*a^4*c^2*x + 5/24*(-b^2*c*x^2 + a^2*c)^(3/2)*a
^2*c*x + 1/6*(-b^2*c*x^2 + a^2*c)^(5/2)*x

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 622 vs. \(2 (109) = 218\).

Time = 0.48 (sec) , antiderivative size = 622, normalized size of antiderivative = 4.61 \[ \int (a+b x)^{5/2} (a c-b c x)^{5/2} \, dx=-\frac {240 \, {\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^{5} c^{2} - 120 \, {\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^{4} c^{2} - 80 \, {\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^{3} c^{2} + 20 \, {\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 )} a^{2} c^{2} + 2 \, {\left (\frac {90 \, a^{5} 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 (195 \, a^{4} - {\left (295 \, a^{3} - 2 \, {\left (3 \, {\left (4 \, b x - 17 \, a\right )} {\left (b x + a\right )} + 133 \, a^{2}\right )} {\left (b x + a\right )}\right )} {\left (b x + a\right )}\right )} \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \sqrt {b x + a}\right )} a c^{2} - {\left (\frac {150 \, a^{6} 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 (405 \, a^{5} - {\left (745 \, a^{4} - 2 \, {\left (451 \, a^{3} - {\left (4 \, {\left (5 \, b x - 26 \, a\right )} {\left (b x + a\right )} + 321 \, a^{2}\right )} {\left (b x + a\right )}\right )} {\left (b x + a\right )}\right )} {\left (b x + a\right )}\right )} \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \sqrt {b x + a}\right )} c^{2}}{240 \, b} \]

[In]

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

[Out]

-1/240*(240*(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^5*c^2 - 120*(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^4*c^2 - 80*(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^3*c^2 + 20*(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))*a^2*
c^2 + 2*(90*a^5*c*log(abs(-sqrt(b*x + a)*sqrt(-c) + sqrt(-(b*x + a)*c + 2*a*c)))/sqrt(-c) - (195*a^4 - (295*a^
3 - 2*(3*(4*b*x - 17*a)*(b*x + a) + 133*a^2)*(b*x + a))*(b*x + a))*sqrt(-(b*x + a)*c + 2*a*c)*sqrt(b*x + a))*a
*c^2 - (150*a^6*c*log(abs(-sqrt(b*x + a)*sqrt(-c) + sqrt(-(b*x + a)*c + 2*a*c)))/sqrt(-c) - (405*a^5 - (745*a^
4 - 2*(451*a^3 - (4*(5*b*x - 26*a)*(b*x + a) + 321*a^2)*(b*x + a))*(b*x + a))*(b*x + a))*sqrt(-(b*x + a)*c + 2
*a*c)*sqrt(b*x + a))*c^2)/b

Mupad [F(-1)]

Timed out. \[ \int (a+b x)^{5/2} (a c-b c x)^{5/2} \, dx=\int {\left (a\,c-b\,c\,x\right )}^{5/2}\,{\left (a+b\,x\right )}^{5/2} \,d x \]

[In]

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

[Out]

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