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

Optimal. Leaf size=126 \[ \frac {5}{16} a^2 c^2 x \sqrt {a+a x} \sqrt {c-c x}+\frac {5}{24} a c x (a+a x)^{3/2} (c-c x)^{3/2}+\frac {1}{6} x (a+a x)^{5/2} (c-c x)^{5/2}+\frac {5}{8} a^{5/2} c^{5/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a+a x}}{\sqrt {a} \sqrt {c-c x}}\right ) \]

[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)

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Rubi [A]
time = 0.04, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 6, 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 {5}{8} a^{5/2} c^{5/2} \tan ^{-1}\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} \end {gather*}

Antiderivative was successfully verified.

[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*} \int (a+a x)^{5/2} (c-c x)^{5/2} \, dx &=\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*}

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Mathematica [A]
time = 0.23, size = 86, normalized size = 0.68 \begin {gather*} \frac {c^2 (a (1+x))^{5/2} \left (x \sqrt {1+x} \sqrt {c-c x} \left (33-26 x^2+8 x^4\right )-30 \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {c-c x}}{\sqrt {c} \sqrt {1+x}}\right )\right )}{48 (1+x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Timed out

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(197\) vs. \(2(94)=188\).
time = 0.19, size = 198, normalized size = 1.57

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	\(-\frac {\left (a x +a \right )^{\frac {5}{2}} \left (-c x +c \right )^{\frac {7}{2}}}{6 c}+\frac {5 a \left (-\frac {\left (a x +a \right )^{\frac {3}{2}} \left (-c x +c \right )^{\frac {7}{2}}}{5 c}+\frac {3 a \left (-\frac {\sqrt {a x +a}\, \left (-c x +c \right )^{\frac {7}{2}}}{4 c}+\frac {a \left (\frac {\left (-c x +c \right )^{\frac {5}{2}} \sqrt {a x +a}}{3 a}+\frac {5 c \left (\frac {\left (-c x +c \right )^{\frac {3}{2}} \sqrt {a x +a}}{2 a}+\frac {3 c \left (\frac {\sqrt {-c x +c}\, \sqrt {a x +a}}{a}+\frac {c \sqrt {\left (-c x +c \right ) \left (a x +a \right )}\, \arctan \left (\frac {\sqrt {a c}\, x}{\sqrt {-a c \,x^{2}+a c}}\right )}{\sqrt {-c x +c}\, \sqrt {a x +a}\, \sqrt {a c}}\right )}{2}\right )}{3}\right )}{4}\right )}{5}\right )}{6}\)	\(198\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6/c*(a*x+a)^(5/2)*(-c*x+c)^(7/2)+5/6*a*(-1/5/c*(a*x+a)^(3/2)*(-c*x+c)^(7/2)+3/5*a*(-1/4/c*(a*x+a)^(1/2)*(-c
*x+c)^(7/2)+1/4*a*(1/3/a*(-c*x+c)^(5/2)*(a*x+a)^(1/2)+5/3*c*(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 = 0.36, size = 72, normalized size = 0.57 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]
time = 0.30, size = 201, normalized size = 1.60 \begin {gather*} \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 ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)]

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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 {5}{2}} \left (- c \left (x - 1\right )\right )^{\frac {5}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 679 vs. \(2 (94) = 188\).
time = 0.11, size = 1165, normalized size = 9.25 \begin {gather*} \frac {2 a^{2} c^{2} \left |a\right | \left (2 \left (\left (\left (\left (\left (\frac {\frac {1}{348364800}\cdot 14515200 a^{10} \sqrt {a x+a} \sqrt {a x+a}}{a^{15}}-\frac {\frac {1}{348364800}\cdot 89994240 a^{11}}{a^{15}}\right ) \sqrt {a x+a} \sqrt {a x+a}+\frac {\frac {1}{348364800}\cdot 232968960 a^{12}}{a^{15}}\right ) \sqrt {a x+a} \sqrt {a x+a}-\frac {\frac {1}{348364800}\cdot 327317760 a^{13}}{a^{15}}\right ) \sqrt {a x+a} \sqrt {a x+a}+\frac {\frac {1}{348364800}\cdot 270345600 a^{14}}{a^{15}}\right ) \sqrt {a x+a} \sqrt {a x+a}-\frac {\frac {1}{348364800}\cdot 146966400 a^{15}}{a^{15}}\right ) \sqrt {a x+a} \sqrt {2 a^{2} c-a c \left (a x+a\right )}+\frac {10 a^{2} c \ln \left |\sqrt {2 a^{2} c-a c \left (a x+a\right )}-\sqrt {-a c} \sqrt {a x+a}\right |}{32 \sqrt {-a c}}\right )}{a^{2}}+\frac {2 a^{2} c^{2} \left |a\right | \left (2 \left (\left (\left (\left (\frac {\frac {1}{806400}\cdot 40320 a^{6} \sqrt {a x+a} \sqrt {a x+a}}{a^{10}}-\frac {\frac {1}{806400}\cdot 211680 a^{7}}{a^{10}}\right ) \sqrt {a x+a} \sqrt {a x+a}+\frac {\frac {1}{806400}\cdot 446880 a^{8}}{a^{10}}\right ) \sqrt {a x+a} \sqrt {a x+a}-\frac {\frac {1}{806400}\cdot 495600 a^{9}}{a^{10}}\right ) \sqrt {a x+a} \sqrt {a x+a}+\frac {\frac {1}{806400}\cdot 327600 a^{10}}{a^{10}}\right ) \sqrt {a x+a} \sqrt {2 a^{2} c-a c \left (a x+a\right )}-\frac {6 a^{2} c \ln \left |\sqrt {2 a^{2} c-a c \left (a x+a\right )}-\sqrt {-a c} \sqrt {a x+a}\right |}{16 \sqrt {-a c}}\right )}{a^{2}}-\frac {4 a^{2} c^{2} \left |a\right | \left (2 \left (\left (\left (\frac {\frac {1}{92160}\cdot 5760 a^{3} \sqrt {a x+a} \sqrt {a x+a}}{a^{6}}-\frac {\frac {1}{92160}\cdot 24960 a^{4}}{a^{6}}\right ) \sqrt {a x+a} \sqrt {a x+a}+\frac {\frac {1}{92160}\cdot 41280 a^{5}}{a^{6}}\right ) \sqrt {a x+a} \sqrt {a x+a}-\frac {\frac {1}{92160}\cdot 37440 a^{6}}{a^{6}}\right ) \sqrt {a x+a} \sqrt {2 a^{2} c-a c \left (a x+a\right )}+\frac {6 a^{2} c \ln \left |\sqrt {2 a^{2} c-a c \left (a x+a\right )}-\sqrt {-a c} \sqrt {a x+a}\right |}{16 \sqrt {-a c}}\right )}{a^{2}}-\frac {4 a^{2} c^{2} \left |a\right | \left (2 \left (\left (\frac {\frac {1}{288}\cdot 24 a \sqrt {a x+a} \sqrt {a x+a}}{a^{3}}-\frac {\frac {1}{288}\cdot 84 a^{2}}{a^{3}}\right ) \sqrt {a x+a} \sqrt {a x+a}+\frac {\frac {1}{288}\cdot 108 a^{3}}{a^{3}}\right ) \sqrt {a x+a} \sqrt {2 a^{2} c-a c \left (a x+a\right )}-\frac {2 a^{2} c \ln \left |\sqrt {2 a^{2} c-a c \left (a x+a\right )}-\sqrt {-a c} \sqrt {a x+a}\right |}{4 \sqrt {-a c}}\right )}{a^{2}}+\frac {2 a^{2} c^{2} \left |a\right | \left (2 \left (\frac {1}{8} \sqrt {a x+a} \sqrt {a x+a}-\frac {12}{32} a\right ) \sqrt {a x+a} \sqrt {2 a^{2} c-a c \left (a x+a\right )}+\frac {2 a^{3} c \ln \left |\sqrt {2 a^{2} c-a c \left (a x+a\right )}-\sqrt {-a c} \sqrt {a x+a}\right |}{4 \sqrt {-a c}}\right )}{a^{2} a}+\frac {2 a^{2} c^{2} \left |a\right | \left (\frac {1}{2} \sqrt {a x+a} \sqrt {2 a^{2} c-a c \left (a x+a\right )}-\frac {2 a^{2} c \ln \left |\sqrt {2 a^{2} c-a c \left (a x+a\right )}-\sqrt {-a c} \sqrt {a x+a}\right |}{2 \sqrt {-a c}}\right )}{a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (a+a\,x\right )}^{5/2}\,{\left (c-c\,x\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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