\(\int (a+a x)^{5/2} (c-c x)^{5/2} \, dx\) [1137]

Optimal result

Integrand size = 20, antiderivative size = 126 \[ \int (a+a x)^{5/2} (c-c x)^{5/2} \, dx=\frac {5}{16} a^2 c^2 x \sqrt {a+a x} \sqrt {c-c x}+\frac {5}{24} a c x (a+a x)^{3/2} (c-c x)^{3/2}+\frac {1}{6} x (a+a x)^{5/2} (c-c x)^{5/2}+\frac {5}{8} a^{5/2} c^{5/2} \arctan \left (\frac {\sqrt {c} \sqrt {a+a x}}{\sqrt {a} \sqrt {c-c x}}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {38, 65, 223, 209} \[ \int (a+a x)^{5/2} (c-c x)^{5/2} \, dx=\frac {5}{8} a^{5/2} c^{5/2} \arctan \left (\frac {\sqrt {c} \sqrt {a x+a}}{\sqrt {a} \sqrt {c-c x}}\right )+\frac {5}{16} a^2 c^2 x \sqrt {a x+a} \sqrt {c-c x}+\frac {5}{24} a c x (a x+a)^{3/2} (c-c x)^{3/2}+\frac {1}{6} x (a x+a)^{5/2} (c-c x)^{5/2} \]

[In]

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

[Out]

(5*a^2*c^2*x*Sqrt[a + a*x]*Sqrt[c - c*x])/16 + (5*a*c*x*(a + a*x)^(3/2)*(c - c*x)^(3/2))/24 + (x*(a + a*x)^(5/
2)*(c - c*x)^(5/2))/6 + (5*a^(5/2)*c^(5/2)*ArcTan[(Sqrt[c]*Sqrt[a + a*x])/(Sqrt[a]*Sqrt[c - c*x])])/8

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

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.90 \[ \int (a+a x)^{5/2} (c-c x)^{5/2} \, dx=\frac {c^{3/2} (a (1+x))^{5/2} \sqrt {c-c x} \left (\sqrt {c} x \sqrt {1+x} \left (-33+33 x+26 x^2-26 x^3-8 x^4+8 x^5\right )+30 \sqrt {c-c x} \arcsin \left (\frac {\sqrt {c-c x}}{\sqrt {2} \sqrt {c}}\right )\right )}{48 (-1+x) (1+x)^{5/2}} \]

[In]

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

[Out]

(c^(3/2)*(a*(1 + x))^(5/2)*Sqrt[c - c*x]*(Sqrt[c]*x*Sqrt[1 + x]*(-33 + 33*x + 26*x^2 - 26*x^3 - 8*x^4 + 8*x^5)
+ 30*Sqrt[c - c*x]*ArcSin[Sqrt[c - c*x]/(Sqrt[2]*Sqrt[c])]))/(48*(-1 + x)*(1 + x)^(5/2))

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.83

method	result	size
risch	\(-\frac {x \left (8 x^{4}-26 x^{2}+33\right ) \left (-1+x \right ) \left (1+x \right ) a^{3} c^{3}}{48 \sqrt {a \left (1+x \right )}\, \sqrt {-c \left (-1+x \right )}}+\frac {5 \arctan \left (\frac {\sqrt {a c}\, x}{\sqrt {-a c \,x^{2}+a c}}\right ) a^{3} c^{3} \sqrt {-a \left (1+x \right ) c \left (-1+x \right )}}{16 \sqrt {a c}\, \sqrt {a \left (1+x \right )}\, \sqrt {-c \left (-1+x \right )}}\)	\(105\)
default	\(-\frac {\left (a x +a \right )^{\frac {5}{2}} \left (-c x +c \right )^{\frac {7}{2}}}{6 c}+\frac {5 a \left (-\frac {\left (a x +a \right )^{\frac {3}{2}} \left (-c x +c \right )^{\frac {7}{2}}}{5 c}+\frac {3 a \left (-\frac {\sqrt {a x +a}\, \left (-c x +c \right )^{\frac {7}{2}}}{4 c}+\frac {a \left (\frac {\left (-c x +c \right )^{\frac {5}{2}} \sqrt {a x +a}}{3 a}+\frac {5 c \left (\frac {\left (-c x +c \right )^{\frac {3}{2}} \sqrt {a x +a}}{2 a}+\frac {3 c \left (\frac {\sqrt {-c x +c}\, \sqrt {a x +a}}{a}+\frac {c \sqrt {\left (-c x +c \right ) \left (a x +a \right )}\, \arctan \left (\frac {\sqrt {a c}\, x}{\sqrt {-a c \,x^{2}+a c}}\right )}{\sqrt {-c x +c}\, \sqrt {a x +a}\, \sqrt {a c}}\right )}{2}\right )}{3}\right )}{4}\right )}{5}\right )}{6}\)	\(198\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.60 \[ \int (a+a x)^{5/2} (c-c x)^{5/2} \, dx=\left [\frac {5}{32} \, \sqrt {-a c} a^{2} c^{2} \log \left (2 \, a c x^{2} + 2 \, \sqrt {-a c} \sqrt {a x + a} \sqrt {-c x + c} x - a c\right ) + \frac {1}{48} \, {\left (8 \, a^{2} c^{2} x^{5} - 26 \, a^{2} c^{2} x^{3} + 33 \, a^{2} c^{2} x\right )} \sqrt {a x + a} \sqrt {-c x + c}, -\frac {5}{16} \, \sqrt {a c} a^{2} c^{2} \arctan \left (\frac {\sqrt {a c} \sqrt {a x + a} \sqrt {-c x + c} x}{a c x^{2} - a c}\right ) + \frac {1}{48} \, {\left (8 \, a^{2} c^{2} x^{5} - 26 \, a^{2} c^{2} x^{3} + 33 \, a^{2} c^{2} x\right )} \sqrt {a x + a} \sqrt {-c x + c}\right ] \]

[In]

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

[Out]

[5/32*sqrt(-a*c)*a^2*c^2*log(2*a*c*x^2 + 2*sqrt(-a*c)*sqrt(a*x + a)*sqrt(-c*x + c)*x - a*c) + 1/48*(8*a^2*c^2*
x^5 - 26*a^2*c^2*x^3 + 33*a^2*c^2*x)*sqrt(a*x + a)*sqrt(-c*x + c), -5/16*sqrt(a*c)*a^2*c^2*arctan(sqrt(a*c)*sq
rt(a*x + a)*sqrt(-c*x + c)*x/(a*c*x^2 - a*c)) + 1/48*(8*a^2*c^2*x^5 - 26*a^2*c^2*x^3 + 33*a^2*c^2*x)*sqrt(a*x
+ a)*sqrt(-c*x + c)]

Sympy [F]

\[ \int (a+a x)^{5/2} (c-c x)^{5/2} \, dx=\int \left (a \left (x + 1\right )\right )^{\frac {5}{2}} \left (- c \left (x - 1\right )\right )^{\frac {5}{2}}\, dx \]

[In]

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

[Out]

Integral((a*(x + 1))**(5/2)*(-c*(x - 1))**(5/2), x)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.57 \[ \int (a+a x)^{5/2} (c-c x)^{5/2} \, dx=\frac {5 \, a^{3} c^{3} \arcsin \left (x\right )}{16 \, \sqrt {a c}} + \frac {5}{16} \, \sqrt {-a c x^{2} + a c} a^{2} c^{2} x + \frac {5}{24} \, {\left (-a c x^{2} + a c\right )}^{\frac {3}{2}} a c x + \frac {1}{6} \, {\left (-a c x^{2} + a c\right )}^{\frac {5}{2}} x \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 679 vs. \(2 (94) = 188\).

Time = 0.55 (sec) , antiderivative size = 679, normalized size of antiderivative = 5.39 \[ \int (a+a x)^{5/2} (c-c x)^{5/2} \, dx=\frac {1}{240} \, {\left (\frac {150 \, a^{2} c \log \left ({\left | -\sqrt {-a c} \sqrt {a x + a} + \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} \right |}\right )}{\sqrt {-a c}} + \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} {\left ({\left (2 \, {\left ({\left (a x + a\right )} {\left (4 \, {\left (a x + a\right )} {\left (\frac {5 \, {\left (a x + a\right )}}{a^{5}} - \frac {31}{a^{4}}\right )} + \frac {321}{a^{3}}\right )} - \frac {451}{a^{2}}\right )} {\left (a x + a\right )} + \frac {745}{a}\right )} {\left (a x + a\right )} - 405\right )} \sqrt {a x + a}\right )} c^{2} {\left | a \right |} - \frac {1}{120} \, {\left (\frac {90 \, a^{2} c \log \left ({\left | -\sqrt {-a c} \sqrt {a x + a} + \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} \right |}\right )}{\sqrt {-a c}} - \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} {\left ({\left (2 \, {\left (a x + a\right )} {\left (3 \, {\left (a x + a\right )} {\left (\frac {4 \, {\left (a x + a\right )}}{a^{4}} - \frac {21}{a^{3}}\right )} + \frac {133}{a^{2}}\right )} - \frac {295}{a}\right )} {\left (a x + a\right )} + 195\right )} \sqrt {a x + a}\right )} c^{2} {\left | a \right |} - \frac {1}{12} \, {\left (\frac {18 \, a^{2} c \log \left ({\left | -\sqrt {-a c} \sqrt {a x + a} + \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} \right |}\right )}{\sqrt {-a c}} + \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} {\left ({\left (a x + a\right )} {\left (2 \, {\left (a x + a\right )} {\left (\frac {3 \, {\left (a x + a\right )}}{a^{3}} - \frac {13}{a^{2}}\right )} + \frac {43}{a}\right )} - 39\right )} \sqrt {a x + a}\right )} c^{2} {\left | a \right |} + \frac {1}{3} \, {\left (\frac {6 \, a^{2} c \log \left ({\left | -\sqrt {-a c} \sqrt {a x + a} + \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} \right |}\right )}{\sqrt {-a c}} - \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} \sqrt {a x + a} {\left ({\left (a x + a\right )} {\left (\frac {2 \, {\left (a x + a\right )}}{a^{2}} - \frac {7}{a}\right )} + 9\right )}\right )} c^{2} {\left | a \right |} - {\left (\frac {2 \, a^{2} c \log \left ({\left | -\sqrt {-a c} \sqrt {a x + a} + \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} \right |}\right )}{\sqrt {-a c}} - \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} \sqrt {a x + a}\right )} c^{2} {\left | a \right |} + \frac {{\left (\frac {2 \, a^{3} c \log \left ({\left | -\sqrt {-a c} \sqrt {a x + a} + \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} \right |}\right )}{\sqrt {-a c}} + \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} \sqrt {a x + a} {\left (a x - 2 \, a\right )}\right )} c^{2} {\left | a \right |}}{2 \, a} \]

[In]

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

[Out]

1/240*(150*a^2*c*log(abs(-sqrt(-a*c)*sqrt(a*x + a) + sqrt(-(a*x + a)*a*c + 2*a^2*c)))/sqrt(-a*c) + sqrt(-(a*x
+ a)*a*c + 2*a^2*c)*((2*((a*x + a)*(4*(a*x + a)*(5*(a*x + a)/a^5 - 31/a^4) + 321/a^3) - 451/a^2)*(a*x + a) + 7
45/a)*(a*x + a) - 405)*sqrt(a*x + a))*c^2*abs(a) - 1/120*(90*a^2*c*log(abs(-sqrt(-a*c)*sqrt(a*x + a) + sqrt(-(
a*x + a)*a*c + 2*a^2*c)))/sqrt(-a*c) - sqrt(-(a*x + a)*a*c + 2*a^2*c)*((2*(a*x + a)*(3*(a*x + a)*(4*(a*x + a)/
a^4 - 21/a^3) + 133/a^2) - 295/a)*(a*x + a) + 195)*sqrt(a*x + a))*c^2*abs(a) - 1/12*(18*a^2*c*log(abs(-sqrt(-a
*c)*sqrt(a*x + a) + sqrt(-(a*x + a)*a*c + 2*a^2*c)))/sqrt(-a*c) + sqrt(-(a*x + a)*a*c + 2*a^2*c)*((a*x + a)*(2
*(a*x + a)*(3*(a*x + a)/a^3 - 13/a^2) + 43/a) - 39)*sqrt(a*x + a))*c^2*abs(a) + 1/3*(6*a^2*c*log(abs(-sqrt(-a*
c)*sqrt(a*x + a) + sqrt(-(a*x + a)*a*c + 2*a^2*c)))/sqrt(-a*c) - sqrt(-(a*x + a)*a*c + 2*a^2*c)*sqrt(a*x + a)*
((a*x + a)*(2*(a*x + a)/a^2 - 7/a) + 9))*c^2*abs(a) - (2*a^2*c*log(abs(-sqrt(-a*c)*sqrt(a*x + a) + sqrt(-(a*x
+ a)*a*c + 2*a^2*c)))/sqrt(-a*c) - sqrt(-(a*x + a)*a*c + 2*a^2*c)*sqrt(a*x + a))*c^2*abs(a) + 1/2*(2*a^3*c*log
(abs(-sqrt(-a*c)*sqrt(a*x + a) + sqrt(-(a*x + a)*a*c + 2*a^2*c)))/sqrt(-a*c) + sqrt(-(a*x + a)*a*c + 2*a^2*c)*
sqrt(a*x + a)*(a*x - 2*a))*c^2*abs(a)/a

Mupad [F(-1)]

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

[In]

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

[Out]

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