3.12.44 \(\int \frac {1}{(a+a x)^{9/2} (c-c x)^{9/2}} \, dx\) [1144]

Optimal. Leaf size=121 \[ \frac {x}{7 a c (a+a x)^{7/2} (c-c x)^{7/2}}+\frac {6 x}{35 a^2 c^2 (a+a x)^{5/2} (c-c x)^{5/2}}+\frac {8 x}{35 a^3 c^3 (a+a x)^{3/2} (c-c x)^{3/2}}+\frac {16 x}{35 a^4 c^4 \sqrt {a+a x} \sqrt {c-c x}} \]

[Out]

1/7*x/a/c/(a*x+a)^(7/2)/(-c*x+c)^(7/2)+6/35*x/a^2/c^2/(a*x+a)^(5/2)/(-c*x+c)^(5/2)+8/35*x/a^3/c^3/(a*x+a)^(3/2
)/(-c*x+c)^(3/2)+16/35*x/a^4/c^4/(a*x+a)^(1/2)/(-c*x+c)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {40, 39} \begin {gather*} \frac {16 x}{35 a^4 c^4 \sqrt {a x+a} \sqrt {c-c x}}+\frac {8 x}{35 a^3 c^3 (a x+a)^{3/2} (c-c x)^{3/2}}+\frac {6 x}{35 a^2 c^2 (a x+a)^{5/2} (c-c x)^{5/2}}+\frac {x}{7 a c (a x+a)^{7/2} (c-c x)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + a*x)^(9/2)*(c - c*x)^(9/2)),x]

[Out]

x/(7*a*c*(a + a*x)^(7/2)*(c - c*x)^(7/2)) + (6*x)/(35*a^2*c^2*(a + a*x)^(5/2)*(c - c*x)^(5/2)) + (8*x)/(35*a^3
*c^3*(a + a*x)^(3/2)*(c - c*x)^(3/2)) + (16*x)/(35*a^4*c^4*Sqrt[a + a*x]*Sqrt[c - c*x])

Rule 39

Int[1/(((a_) + (b_.)*(x_))^(3/2)*((c_) + (d_.)*(x_))^(3/2)), x_Symbol] :> Simp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d
*x]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0]

Rule 40

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-x)*(a + b*x)^(m + 1)*((c + d*x)^(m
+ 1)/(2*a*c*(m + 1))), x] + Dist[(2*m + 3)/(2*a*c*(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] && ILtQ[m + 3/2, 0]

Rubi steps

\begin {align*} \int \frac {1}{(a+a x)^{9/2} (c-c x)^{9/2}} \, dx &=\frac {x}{7 a c (a+a x)^{7/2} (c-c x)^{7/2}}+\frac {6 \int \frac {1}{(a+a x)^{7/2} (c-c x)^{7/2}} \, dx}{7 a c}\\ &=\frac {x}{7 a c (a+a x)^{7/2} (c-c x)^{7/2}}+\frac {6 x}{35 a^2 c^2 (a+a x)^{5/2} (c-c x)^{5/2}}+\frac {24 \int \frac {1}{(a+a x)^{5/2} (c-c x)^{5/2}} \, dx}{35 a^2 c^2}\\ &=\frac {x}{7 a c (a+a x)^{7/2} (c-c x)^{7/2}}+\frac {6 x}{35 a^2 c^2 (a+a x)^{5/2} (c-c x)^{5/2}}+\frac {8 x}{35 a^3 c^3 (a+a x)^{3/2} (c-c x)^{3/2}}+\frac {16 \int \frac {1}{(a+a x)^{3/2} (c-c x)^{3/2}} \, dx}{35 a^3 c^3}\\ &=\frac {x}{7 a c (a+a x)^{7/2} (c-c x)^{7/2}}+\frac {6 x}{35 a^2 c^2 (a+a x)^{5/2} (c-c x)^{5/2}}+\frac {8 x}{35 a^3 c^3 (a+a x)^{3/2} (c-c x)^{3/2}}+\frac {16 x}{35 a^4 c^4 \sqrt {a+a x} \sqrt {c-c x}}\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 54, normalized size = 0.45 \begin {gather*} \frac {x \left (-35+70 x^2-56 x^4+16 x^6\right )}{35 a^4 c^4 \sqrt {a (1+x)} \sqrt {c-c x} \left (-1+x^2\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + a*x)^(9/2)*(c - c*x)^(9/2)),x]

[Out]

(x*(-35 + 70*x^2 - 56*x^4 + 16*x^6))/(35*a^4*c^4*Sqrt[a*(1 + x)]*Sqrt[c - c*x]*(-1 + x^2)^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(220\) vs. \(2(97)=194\).
time = 0.14, size = 221, normalized size = 1.83

method	result	size
gosper	\(\frac {\left (1+x \right ) \left (-1+x \right ) x \left (16 x^{6}-56 x^{4}+70 x^{2}-35\right )}{35 \left (a x +a \right )^{\frac {9}{2}} \left (-c x +c \right )^{\frac {9}{2}}}\)	\(42\)
default	\(-\frac {1}{7 a c \left (a x +a \right )^{\frac {7}{2}} \left (-c x +c \right )^{\frac {7}{2}}}+\frac {-\frac {1}{5 a c \left (a x +a \right )^{\frac {5}{2}} \left (-c x +c \right )^{\frac {7}{2}}}+\frac {-\frac {2}{5 a c \left (a x +a \right )^{\frac {3}{2}} \left (-c x +c \right )^{\frac {7}{2}}}+\frac {6 \left (-\frac {5}{3 a c \sqrt {a x +a}\, \left (-c x +c \right )^{\frac {7}{2}}}+\frac {5 \left (\frac {4 \sqrt {a x +a}}{7 a c \left (-c x +c \right )^{\frac {7}{2}}}+\frac {4 \left (\frac {3 \sqrt {a x +a}}{35 a c \left (-c x +c \right )^{\frac {5}{2}}}+\frac {3 \left (\frac {2 \sqrt {a x +a}}{15 a c \left (-c x +c \right )^{\frac {3}{2}}}+\frac {2 \sqrt {a x +a}}{15 a \,c^{2} \sqrt {-c x +c}}\right )}{7 c}\right )}{c}\right )}{3 a}\right )}{5 a}}{a}}{a}\)	\(221\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x+a)^(9/2)/(-c*x+c)^(9/2),x,method=_RETURNVERBOSE)

[Out]

-1/7/a/c/(a*x+a)^(7/2)/(-c*x+c)^(7/2)+1/a*(-1/5/a/c/(a*x+a)^(5/2)/(-c*x+c)^(7/2)+6/5/a*(-1/3/a/c/(a*x+a)^(3/2)
/(-c*x+c)^(7/2)+5/3/a*(-1/a/c/(a*x+a)^(1/2)/(-c*x+c)^(7/2)+4/a*(1/7/a/c/(-c*x+c)^(7/2)*(a*x+a)^(1/2)+3/7/c*(1/
5/a/c/(-c*x+c)^(5/2)*(a*x+a)^(1/2)+2/5/c*(1/3/a/c/(-c*x+c)^(3/2)*(a*x+a)^(1/2)+1/3/a/c^2/(-c*x+c)^(1/2)*(a*x+a
)^(1/2)))))))

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Maxima [A]
time = 0.28, size = 89, normalized size = 0.74 \begin {gather*} \frac {x}{7 \, {\left (-a c x^{2} + a c\right )}^{\frac {7}{2}} a c} + \frac {6 \, x}{35 \, {\left (-a c x^{2} + a c\right )}^{\frac {5}{2}} a^{2} c^{2}} + \frac {8 \, x}{35 \, {\left (-a c x^{2} + a c\right )}^{\frac {3}{2}} a^{3} c^{3}} + \frac {16 \, x}{35 \, \sqrt {-a c x^{2} + a c} a^{4} c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+a)^(9/2)/(-c*x+c)^(9/2),x, algorithm="maxima")

[Out]

1/7*x/((-a*c*x^2 + a*c)^(7/2)*a*c) + 6/35*x/((-a*c*x^2 + a*c)^(5/2)*a^2*c^2) + 8/35*x/((-a*c*x^2 + a*c)^(3/2)*
a^3*c^3) + 16/35*x/(sqrt(-a*c*x^2 + a*c)*a^4*c^4)

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Fricas [A]
time = 1.01, size = 89, normalized size = 0.74 \begin {gather*} -\frac {{\left (16 \, x^{7} - 56 \, x^{5} + 70 \, x^{3} - 35 \, x\right )} \sqrt {a x + a} \sqrt {-c x + c}}{35 \, {\left (a^{5} c^{5} x^{8} - 4 \, a^{5} c^{5} x^{6} + 6 \, a^{5} c^{5} x^{4} - 4 \, a^{5} c^{5} x^{2} + a^{5} c^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+a)^(9/2)/(-c*x+c)^(9/2),x, algorithm="fricas")

[Out]

-1/35*(16*x^7 - 56*x^5 + 70*x^3 - 35*x)*sqrt(a*x + a)*sqrt(-c*x + c)/(a^5*c^5*x^8 - 4*a^5*c^5*x^6 + 6*a^5*c^5*
x^4 - 4*a^5*c^5*x^2 + a^5*c^5)

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Sympy [C] Result contains complex when optimal does not.
time = 180.07, size = 85, normalized size = 0.70 \begin {gather*} \frac {4 i {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {9}{4}, \frac {11}{4}, 1 & \frac {1}{2}, \frac {9}{2}, 5 \\\frac {9}{4}, \frac {11}{4}, 4, \frac {9}{2}, 5 & 0 \end {matrix} \middle | {\frac {1}{x^{2}}} \right )}}{105 \pi ^{\frac {3}{2}} a^{\frac {9}{2}} c^{\frac {9}{2}}} + \frac {4 {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, 0, \frac {1}{2}, \frac {7}{4}, \frac {9}{4}, 1 & \\\frac {7}{4}, \frac {9}{4} & - \frac {1}{2}, 0, 4, 0 \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{x^{2}}} \right )}}{105 \pi ^{\frac {3}{2}} a^{\frac {9}{2}} c^{\frac {9}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+a)**(9/2)/(-c*x+c)**(9/2),x)

[Out]

4*I*meijerg(((9/4, 11/4, 1), (1/2, 9/2, 5)), ((9/4, 11/4, 4, 9/2, 5), (0,)), x**(-2))/(105*pi**(3/2)*a**(9/2)*
c**(9/2)) + 4*meijerg(((-1/2, 0, 1/2, 7/4, 9/4, 1), ()), ((7/4, 9/4), (-1/2, 0, 4, 0)), exp_polar(-2*I*pi)/x**
2)/(105*pi**(3/2)*a**(9/2)*c**(9/2))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 437 vs. \(2 (97) = 194\).
time = 1.40, size = 437, normalized size = 3.61 \begin {gather*} -\frac {\sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} {\left ({\left (a x + a\right )} {\left ({\left (a x + a\right )} {\left (\frac {256 \, {\left (a x + a\right )} {\left | a \right |}}{a^{2} c} - \frac {1617 \, {\left | a \right |}}{a c}\right )} + \frac {3430 \, {\left | a \right |}}{c}\right )} - \frac {2450 \, a {\left | a \right |}}{c}\right )} \sqrt {a x + a}}{1120 \, {\left ({\left (a x + a\right )} a c - 2 \, a^{2} c\right )}^{4}} + \frac {16384 \, a^{12} c^{6} - 51744 \, {\left (\sqrt {-a c} \sqrt {a x + a} - \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{2} a^{10} c^{5} + 66416 \, {\left (\sqrt {-a c} \sqrt {a x + a} - \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{4} a^{8} c^{4} - 43120 \, {\left (\sqrt {-a c} \sqrt {a x + a} - \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{6} a^{6} c^{3} + 14280 \, {\left (\sqrt {-a c} \sqrt {a x + a} - \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{8} a^{4} c^{2} - 2450 \, {\left (\sqrt {-a c} \sqrt {a x + a} - \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{10} a^{2} c + 175 \, {\left (\sqrt {-a c} \sqrt {a x + a} - \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{12}}{280 \, {\left (2 \, a^{2} c - {\left (\sqrt {-a c} \sqrt {a x + a} - \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{2}\right )}^{7} \sqrt {-a c} a c^{3} {\left | a \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+a)^(9/2)/(-c*x+c)^(9/2),x, algorithm="giac")

[Out]

-1/1120*sqrt(-(a*x + a)*a*c + 2*a^2*c)*((a*x + a)*((a*x + a)*(256*(a*x + a)*abs(a)/(a^2*c) - 1617*abs(a)/(a*c)
) + 3430*abs(a)/c) - 2450*a*abs(a)/c)*sqrt(a*x + a)/((a*x + a)*a*c - 2*a^2*c)^4 + 1/280*(16384*a^12*c^6 - 5174
4*(sqrt(-a*c)*sqrt(a*x + a) - sqrt(-(a*x + a)*a*c + 2*a^2*c))^2*a^10*c^5 + 66416*(sqrt(-a*c)*sqrt(a*x + a) - s
qrt(-(a*x + a)*a*c + 2*a^2*c))^4*a^8*c^4 - 43120*(sqrt(-a*c)*sqrt(a*x + a) - sqrt(-(a*x + a)*a*c + 2*a^2*c))^6
*a^6*c^3 + 14280*(sqrt(-a*c)*sqrt(a*x + a) - sqrt(-(a*x + a)*a*c + 2*a^2*c))^8*a^4*c^2 - 2450*(sqrt(-a*c)*sqrt
(a*x + a) - sqrt(-(a*x + a)*a*c + 2*a^2*c))^10*a^2*c + 175*(sqrt(-a*c)*sqrt(a*x + a) - sqrt(-(a*x + a)*a*c + 2
*a^2*c))^12)/((2*a^2*c - (sqrt(-a*c)*sqrt(a*x + a) - sqrt(-(a*x + a)*a*c + 2*a^2*c))^2)^7*sqrt(-a*c)*a*c^3*abs
(a))

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Mupad [B]
time = 0.48, size = 66, normalized size = 0.55 \begin {gather*} -\frac {x\,\left (16\,x^6-56\,x^4+70\,x^2-35\right )}{35\,a^4\,\sqrt {a+a\,x}\,{\left (c-c\,x\right )}^{7/2}\,\left (c-x^2\,\left (c-c\,x\right )+7\,c\,x-4\,x\,\left (c-c\,x\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + a*x)^(9/2)*(c - c*x)^(9/2)),x)

[Out]

-(x*(70*x^2 - 56*x^4 + 16*x^6 - 35))/(35*a^4*(a + a*x)^(1/2)*(c - c*x)^(7/2)*(c - x^2*(c - c*x) + 7*c*x - 4*x*
(c - c*x)))

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