3.12.38 \(\int (a+a x)^{3/2} (c-c x)^{3/2} \, dx\) [1138]

Optimal. Leaf size=96 \[ \frac {3}{8} a c x \sqrt {a+a x} \sqrt {c-c x}+\frac {1}{4} x (a+a x)^{3/2} (c-c x)^{3/2}+\frac {3}{4} a^{3/2} c^{3/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a+a x}}{\sqrt {a} \sqrt {c-c x}}\right ) \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {38, 65, 223, 209} \begin {gather*} \frac {3}{4} a^{3/2} c^{3/2} \text {ArcTan}\left (\frac {\sqrt {c} \sqrt {a x+a}}{\sqrt {a} \sqrt {c-c x}}\right )+\frac {3}{8} a c x \sqrt {a x+a} \sqrt {c-c x}+\frac {1}{4} x (a x+a)^{3/2} (c-c x)^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*a*c*x*Sqrt[a + a*x]*Sqrt[c - c*x])/8 + (x*(a + a*x)^(3/2)*(c - c*x)^(3/2))/4 + (3*a^(3/2)*c^(3/2)*ArcTan[(S
qrt[c]*Sqrt[a + a*x])/(Sqrt[a]*Sqrt[c - c*x])])/4

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

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Mathematica [A]
time = 0.17, size = 79, normalized size = 0.82 \begin {gather*} -\frac {c (a (1+x))^{3/2} \left (x \sqrt {1+x} \sqrt {c-c x} \left (-5+2 x^2\right )+6 \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {c-c x}}{\sqrt {c} \sqrt {1+x}}\right )\right )}{8 (1+x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/8*(c*(a*(1 + x))^(3/2)*(x*Sqrt[1 + x]*Sqrt[c - c*x]*(-5 + 2*x^2) + 6*Sqrt[c]*ArcTan[Sqrt[c - c*x]/(Sqrt[c]*
Sqrt[1 + x])]))/(1 + x)^(3/2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(149\) vs. \(2(70)=140\).
time = 0.18, size = 150, normalized size = 1.56

method	result	size
risch	\(\frac {x \left (2 x^{2}-5\right ) \left (1+x \right ) \left (-1+x \right ) a^{2} c^{2}}{8 \sqrt {a \left (1+x \right )}\, \sqrt {-c \left (-1+x \right )}}+\frac {3 \arctan \left (\frac {\sqrt {a c}\, x}{\sqrt {-a c \,x^{2}+a c}}\right ) a^{2} c^{2} \sqrt {-a \left (1+x \right ) c \left (-1+x \right )}}{8 \sqrt {a c}\, \sqrt {a \left (1+x \right )}\, \sqrt {-c \left (-1+x \right )}}\)	\(100\)
default	\(-\frac {\left (a x +a \right )^{\frac {3}{2}} \left (-c x +c \right )^{\frac {5}{2}}}{4 c}+\frac {3 a \left (-\frac {\sqrt {a x +a}\, \left (-c x +c \right )^{\frac {5}{2}}}{3 c}+\frac {a \left (\frac {\left (-c x +c \right )^{\frac {3}{2}} \sqrt {a x +a}}{2 a}+\frac {3 c \left (\frac {\sqrt {-c x +c}\, \sqrt {a x +a}}{a}+\frac {c \sqrt {\left (-c x +c \right ) \left (a x +a \right )}\, \arctan \left (\frac {\sqrt {a c}\, x}{\sqrt {-a c \,x^{2}+a c}}\right )}{\sqrt {-c x +c}\, \sqrt {a x +a}\, \sqrt {a c}}\right )}{2}\right )}{3}\right )}{4}\)	\(150\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.52, size = 50, normalized size = 0.52 \begin {gather*} \frac {3 \, a^{2} c^{2} \arcsin \left (x\right )}{8 \, \sqrt {a c}} + \frac {3}{8} \, \sqrt {-a c x^{2} + a c} a c x + \frac {1}{4} \, {\left (-a c x^{2} + a c\right )}^{\frac {3}{2}} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.70, size = 155, normalized size = 1.61 \begin {gather*} \left [\frac {3}{16} \, \sqrt {-a c} a c \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}{8} \, {\left (2 \, a c x^{3} - 5 \, a c x\right )} \sqrt {a x + a} \sqrt {-c x + c}, -\frac {3}{8} \, \sqrt {a c} a c \arctan \left (\frac {\sqrt {a c} \sqrt {a x + a} \sqrt {-c x + c} x}{a c x^{2} - a c}\right ) - \frac {1}{8} \, {\left (2 \, a c x^{3} - 5 \, a c x\right )} \sqrt {a x + a} \sqrt {-c x + c}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a \left (x + 1\right )\right )^{\frac {3}{2}} \left (- c \left (x - 1\right )\right )^{\frac {3}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a*(x + 1))**(3/2)*(-c*(x - 1))**(3/2), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 403 vs. \(2 (70) = 140\).
time = 1.72, size = 403, normalized size = 4.20 \begin {gather*} -\frac {{\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 {\left | a \right |}}{24 \, a} + \frac {{\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 {\left | a \right |}}{6 \, a} - \frac {{\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 {\left | a \right |}}{a} + \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 {\left | a \right |}}{2 \, a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(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*abs(a)/a
+ 1/6*(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*abs(a)/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*abs(a)/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*abs(a)/a^2

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (a+a\,x\right )}^{3/2}\,{\left (c-c\,x\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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