3.12.43 \(\int \frac {1}{(a+a x)^{7/2} (c-c x)^{7/2}} \, dx\) [1143]

Optimal. Leaf size=91 \[ \frac {x}{5 a c (a+a x)^{5/2} (c-c x)^{5/2}}+\frac {4 x}{15 a^2 c^2 (a+a x)^{3/2} (c-c x)^{3/2}}+\frac {8 x}{15 a^3 c^3 \sqrt {a+a x} \sqrt {c-c x}} \]

[Out]

1/5*x/a/c/(a*x+a)^(5/2)/(-c*x+c)^(5/2)+4/15*x/a^2/c^2/(a*x+a)^(3/2)/(-c*x+c)^(3/2)+8/15*x/a^3/c^3/(a*x+a)^(1/2
)/(-c*x+c)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 3, 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 {8 x}{15 a^3 c^3 \sqrt {a x+a} \sqrt {c-c x}}+\frac {4 x}{15 a^2 c^2 (a x+a)^{3/2} (c-c x)^{3/2}}+\frac {x}{5 a c (a x+a)^{5/2} (c-c x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + a*x)^(7/2)*(c - c*x)^(7/2)),x]

[Out]

x/(5*a*c*(a + a*x)^(5/2)*(c - c*x)^(5/2)) + (4*x)/(15*a^2*c^2*(a + a*x)^(3/2)*(c - c*x)^(3/2)) + (8*x)/(15*a^3
*c^3*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)^{7/2} (c-c x)^{7/2}} \, dx &=\frac {x}{5 a c (a+a x)^{5/2} (c-c x)^{5/2}}+\frac {4 \int \frac {1}{(a+a x)^{5/2} (c-c x)^{5/2}} \, dx}{5 a c}\\ &=\frac {x}{5 a c (a+a x)^{5/2} (c-c x)^{5/2}}+\frac {4 x}{15 a^2 c^2 (a+a x)^{3/2} (c-c x)^{3/2}}+\frac {8 \int \frac {1}{(a+a x)^{3/2} (c-c x)^{3/2}} \, dx}{15 a^2 c^2}\\ &=\frac {x}{5 a c (a+a x)^{5/2} (c-c x)^{5/2}}+\frac {4 x}{15 a^2 c^2 (a+a x)^{3/2} (c-c x)^{3/2}}+\frac {8 x}{15 a^3 c^3 \sqrt {a+a x} \sqrt {c-c x}}\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 49, normalized size = 0.54 \begin {gather*} \frac {x \left (15-20 x^2+8 x^4\right )}{15 a^3 c^3 \sqrt {a (1+x)} \sqrt {c-c x} \left (-1+x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + a*x)^(7/2)*(c - c*x)^(7/2)),x]

[Out]

(x*(15 - 20*x^2 + 8*x^4))/(15*a^3*c^3*Sqrt[a*(1 + x)]*Sqrt[c - c*x]*(-1 + x^2)^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(162\) vs. \(2(73)=146\).
time = 0.16, size = 163, normalized size = 1.79

method	result	size
gosper	\(-\frac {\left (1+x \right ) \left (-1+x \right ) x \left (8 x^{4}-20 x^{2}+15\right )}{15 \left (a x +a \right )^{\frac {7}{2}} \left (-c x +c \right )^{\frac {7}{2}}}\)	\(37\)
default	\(-\frac {1}{5 a c \left (a x +a \right )^{\frac {5}{2}} \left (-c x +c \right )^{\frac {5}{2}}}+\frac {-\frac {1}{3 a c \left (a x +a \right )^{\frac {3}{2}} \left (-c x +c \right )^{\frac {5}{2}}}+\frac {-\frac {4}{3 a c \sqrt {a x +a}\, \left (-c x +c \right )^{\frac {5}{2}}}+\frac {4 \left (\frac {3 \sqrt {a x +a}}{5 a c \left (-c x +c \right )^{\frac {5}{2}}}+\frac {3 \left (\frac {2 \sqrt {a x +a}}{15 a c \left (-c x +c \right )^{\frac {3}{2}}}+\frac {2 \sqrt {a x +a}}{15 a \,c^{2} \sqrt {-c x +c}}\right )}{c}\right )}{3 a}}{a}}{a}\)	\(163\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x+a)^(7/2)/(-c*x+c)^(7/2),x,method=_RETURNVERBOSE)

[Out]

-1/5/a/c/(a*x+a)^(5/2)/(-c*x+c)^(5/2)+1/a*(-1/3/a/c/(a*x+a)^(3/2)/(-c*x+c)^(5/2)+4/3/a*(-1/a/c/(a*x+a)^(1/2)/(
-c*x+c)^(5/2)+3/a*(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 = 0.31, size = 67, normalized size = 0.74 \begin {gather*} \frac {x}{5 \, {\left (-a c x^{2} + a c\right )}^{\frac {5}{2}} a c} + \frac {4 \, x}{15 \, {\left (-a c x^{2} + a c\right )}^{\frac {3}{2}} a^{2} c^{2}} + \frac {8 \, x}{15 \, \sqrt {-a c x^{2} + a c} a^{3} c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+a)^(7/2)/(-c*x+c)^(7/2),x, algorithm="maxima")

[Out]

1/5*x/((-a*c*x^2 + a*c)^(5/2)*a*c) + 4/15*x/((-a*c*x^2 + a*c)^(3/2)*a^2*c^2) + 8/15*x/(sqrt(-a*c*x^2 + a*c)*a^
3*c^3)

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Fricas [A]
time = 1.21, size = 74, normalized size = 0.81 \begin {gather*} -\frac {{\left (8 \, x^{5} - 20 \, x^{3} + 15 \, x\right )} \sqrt {a x + a} \sqrt {-c x + c}}{15 \, {\left (a^{4} c^{4} x^{6} - 3 \, a^{4} c^{4} x^{4} + 3 \, a^{4} c^{4} x^{2} - a^{4} c^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+a)^(7/2)/(-c*x+c)^(7/2),x, algorithm="fricas")

[Out]

-1/15*(8*x^5 - 20*x^3 + 15*x)*sqrt(a*x + a)*sqrt(-c*x + c)/(a^4*c^4*x^6 - 3*a^4*c^4*x^4 + 3*a^4*c^4*x^2 - a^4*
c^4)

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Sympy [C] Result contains complex when optimal does not.
time = 35.64, size = 85, normalized size = 0.93 \begin {gather*} - \frac {2 i {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {7}{4}, \frac {9}{4}, 1 & \frac {1}{2}, \frac {7}{2}, 4 \\\frac {7}{4}, \frac {9}{4}, 3, \frac {7}{2}, 4 & 0 \end {matrix} \middle | {\frac {1}{x^{2}}} \right )}}{15 \pi ^{\frac {3}{2}} a^{\frac {7}{2}} c^{\frac {7}{2}}} + \frac {2 {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, 0, \frac {1}{2}, \frac {5}{4}, \frac {7}{4}, 1 & \\\frac {5}{4}, \frac {7}{4} & - \frac {1}{2}, 0, 3, 0 \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{x^{2}}} \right )}}{15 \pi ^{\frac {3}{2}} a^{\frac {7}{2}} c^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+a)**(7/2)/(-c*x+c)**(7/2),x)

[Out]

-2*I*meijerg(((7/4, 9/4, 1), (1/2, 7/2, 4)), ((7/4, 9/4, 3, 7/2, 4), (0,)), x**(-2))/(15*pi**(3/2)*a**(7/2)*c*
*(7/2)) + 2*meijerg(((-1/2, 0, 1/2, 5/4, 7/4, 1), ()), ((5/4, 7/4), (-1/2, 0, 3, 0)), exp_polar(-2*I*pi)/x**2)
/(15*pi**(3/2)*a**(7/2)*c**(7/2))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (73) = 146\).
time = 1.54, size = 333, normalized size = 3.66 \begin {gather*} -\frac {\sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} \sqrt {a x + a} {\left ({\left (a x + a\right )} {\left (\frac {64 \, {\left (a x + a\right )}}{c {\left | a \right |}} - \frac {275 \, a}{c {\left | a \right |}}\right )} + \frac {300 \, a^{2}}{c {\left | a \right |}}\right )}}{240 \, {\left ({\left (a x + a\right )} a c - 2 \, a^{2} c\right )}^{3}} + \frac {1024 \, a^{8} c^{4} - 2200 \, {\left (\sqrt {-a c} \sqrt {a x + a} - \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{2} a^{6} c^{3} + 1660 \, {\left (\sqrt {-a c} \sqrt {a x + a} - \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{4} a^{4} c^{2} - 450 \, {\left (\sqrt {-a c} \sqrt {a x + a} - \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{6} a^{2} c + 45 \, {\left (\sqrt {-a c} \sqrt {a x + a} - \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{8}}{60 \, {\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 )}^{5} \sqrt {-a c} c^{2} {\left | a \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+a)^(7/2)/(-c*x+c)^(7/2),x, algorithm="giac")

[Out]

-1/240*sqrt(-(a*x + a)*a*c + 2*a^2*c)*sqrt(a*x + a)*((a*x + a)*(64*(a*x + a)/(c*abs(a)) - 275*a/(c*abs(a))) +
300*a^2/(c*abs(a)))/((a*x + a)*a*c - 2*a^2*c)^3 + 1/60*(1024*a^8*c^4 - 2200*(sqrt(-a*c)*sqrt(a*x + a) - sqrt(-
(a*x + a)*a*c + 2*a^2*c))^2*a^6*c^3 + 1660*(sqrt(-a*c)*sqrt(a*x + a) - sqrt(-(a*x + a)*a*c + 2*a^2*c))^4*a^4*c
^2 - 450*(sqrt(-a*c)*sqrt(a*x + a) - sqrt(-(a*x + a)*a*c + 2*a^2*c))^6*a^2*c + 45*(sqrt(-a*c)*sqrt(a*x + a) -
sqrt(-(a*x + a)*a*c + 2*a^2*c))^8)/((2*a^2*c - (sqrt(-a*c)*sqrt(a*x + a) - sqrt(-(a*x + a)*a*c + 2*a^2*c))^2)^
5*sqrt(-a*c)*c^2*abs(a))

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Mupad [B]
time = 0.44, size = 50, normalized size = 0.55 \begin {gather*} \frac {x\,\left (8\,x^4-20\,x^2+15\right )}{15\,a^3\,\sqrt {a+a\,x}\,{\left (c-c\,x\right )}^{5/2}\,\left (c+3\,c\,x-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)^(7/2)*(c - c*x)^(7/2)),x)

[Out]

(x*(8*x^4 - 20*x^2 + 15))/(15*a^3*(a + a*x)^(1/2)*(c - c*x)^(5/2)*(c + 3*c*x - x*(c - c*x)))

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