3.12.52 \(\int \frac {1}{(a+b x)^{9/2} (a c-b c x)^{9/2}} \, dx\) [1152]

Optimal. Leaf size=133 \[ \frac {x}{7 a^2 c (a+b x)^{7/2} (a c-b c x)^{7/2}}+\frac {6 x}{35 a^4 c^2 (a+b x)^{5/2} (a c-b c x)^{5/2}}+\frac {8 x}{35 a^6 c^3 (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac {16 x}{35 a^8 c^4 \sqrt {a+b x} \sqrt {a c-b c x}} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {40, 39} \begin {gather*} \frac {16 x}{35 a^8 c^4 \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {8 x}{35 a^6 c^3 (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac {6 x}{35 a^4 c^2 (a+b x)^{5/2} (a c-b c x)^{5/2}}+\frac {x}{7 a^2 c (a+b x)^{7/2} (a c-b c x)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]
time = 0.10, size = 76, normalized size = 0.57 \begin {gather*} \frac {\sqrt {c (a-b x)} \left (35 a^6 x-70 a^4 b^2 x^3+56 a^2 b^4 x^5-16 b^6 x^7\right )}{35 a^8 c^5 (a-b x)^4 (a+b x)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c*(a - b*x)]*(35*a^6*x - 70*a^4*b^2*x^3 + 56*a^2*b^4*x^5 - 16*b^6*x^7))/(35*a^8*c^5*(a - b*x)^4*(a + b*x
)^(7/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(274\) vs. \(2(109)=218\).
time = 0.15, size = 275, normalized size = 2.07

method	result	size
gosper	\(\frac {\left (-b x +a \right ) x \left (-16 x^{6} b^{6}+56 a^{2} x^{4} b^{4}-70 a^{4} x^{2} b^{2}+35 a^{6}\right )}{35 \left (b x +a \right )^{\frac {7}{2}} a^{8} \left (-b c x +a c \right )^{\frac {9}{2}}}\)	\(67\)
default	\(-\frac {1}{7 a b c \left (b x +a \right )^{\frac {7}{2}} \left (-b c x +a c \right )^{\frac {7}{2}}}+\frac {-\frac {1}{5 a b c \left (b x +a \right )^{\frac {5}{2}} \left (-b c x +a c \right )^{\frac {7}{2}}}+\frac {-\frac {2}{5 a b c \left (b x +a \right )^{\frac {3}{2}} \left (-b c x +a c \right )^{\frac {7}{2}}}+\frac {6 \left (-\frac {5}{3 a b c \sqrt {b x +a}\, \left (-b c x +a c \right )^{\frac {7}{2}}}+\frac {5 \left (\frac {4 \sqrt {b x +a}}{7 a b c \left (-b c x +a c \right )^{\frac {7}{2}}}+\frac {4 \left (\frac {3 \sqrt {b x +a}}{35 a b c \left (-b c x +a c \right )^{\frac {5}{2}}}+\frac {3 \left (\frac {2 \sqrt {b x +a}}{15 a b c \left (-b c x +a c \right )^{\frac {3}{2}}}+\frac {2 \sqrt {b x +a}}{15 b \,a^{2} c^{2} \sqrt {-b c x +a c}}\right )}{7 a c}\right )}{a c}\right )}{3 a}\right )}{5 a}}{a}}{a}\)	\(275\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 105, normalized size = 0.79 \begin {gather*} \frac {x}{7 \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {7}{2}} a^{2} c} + \frac {6 \, x}{35 \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {5}{2}} a^{4} c^{2}} + \frac {8 \, x}{35 \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} a^{6} c^{3}} + \frac {16 \, x}{35 \, \sqrt {-b^{2} c x^{2} + a^{2} c} a^{8} c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 1.28, size = 122, normalized size = 0.92 \begin {gather*} -\frac {{\left (16 \, b^{6} x^{7} - 56 \, a^{2} b^{4} x^{5} + 70 \, a^{4} b^{2} x^{3} - 35 \, a^{6} x\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{35 \, {\left (a^{8} b^{8} c^{5} x^{8} - 4 \, a^{10} b^{6} c^{5} x^{6} + 6 \, a^{12} b^{4} c^{5} x^{4} - 4 \, a^{14} b^{2} c^{5} x^{2} + a^{16} c^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/35*(16*b^6*x^7 - 56*a^2*b^4*x^5 + 70*a^4*b^2*x^3 - 35*a^6*x)*sqrt(-b*c*x + a*c)*sqrt(b*x + a)/(a^8*b^8*c^5*
x^8 - 4*a^10*b^6*c^5*x^6 + 6*a^12*b^4*c^5*x^4 - 4*a^14*b^2*c^5*x^2 + a^16*c^5)

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Sympy [C] Result contains complex when optimal does not.
time = 185.11, size = 97, normalized size = 0.73 \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 {a^{2}}{b^{2} x^{2}}} \right )}}{105 \pi ^{\frac {3}{2}} a^{8} b 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 {a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{105 \pi ^{\frac {3}{2}} a^{8} b c^{\frac {9}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)**(9/2)/(-b*c*x+a*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,)), a**2/(b**2*x**2))/(105*pi**(3/2)*
a**8*b*c**(9/2)) + 4*meijerg(((-1/2, 0, 1/2, 7/4, 9/4, 1), ()), ((7/4, 9/4), (-1/2, 0, 4, 0)), a**2*exp_polar(
-2*I*pi)/(b**2*x**2))/(105*pi**(3/2)*a**8*b*c**(9/2))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 393 vs. \(2 (109) = 218\).
time = 2.52, size = 393, normalized size = 2.95 \begin {gather*} -\frac {\frac {{\left ({\left (b x + a\right )} {\left ({\left (b x + a\right )} {\left (\frac {256 \, {\left (b x + a\right )}}{a^{8} c} - \frac {1617}{a^{7} c}\right )} + \frac {3430}{a^{6} c}\right )} - \frac {2450}{a^{5} c}\right )} \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \sqrt {b x + a}}{{\left ({\left (b x + a\right )} c - 2 \, a c\right )}^{4}} + \frac {4 \, {\left (175 \, {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{12} - 2450 \, a {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{10} c + 14280 \, a^{2} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{8} c^{2} - 43120 \, a^{3} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{6} c^{3} + 66416 \, a^{4} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{4} c^{4} - 51744 \, a^{5} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2} c^{5} + 16384 \, a^{6} c^{6}\right )}}{{\left ({\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2} - 2 \, a c\right )}^{7} a^{7} \sqrt {-c} c^{3}}}{1120 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1120*(((b*x + a)*((b*x + a)*(256*(b*x + a)/(a^8*c) - 1617/(a^7*c)) + 3430/(a^6*c)) - 2450/(a^5*c))*sqrt(-(b
*x + a)*c + 2*a*c)*sqrt(b*x + a)/((b*x + a)*c - 2*a*c)^4 + 4*(175*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c
+ 2*a*c))^12 - 2450*a*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^10*c + 14280*a^2*(sqrt(b*x + a)*sq
rt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^8*c^2 - 43120*a^3*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^6
*c^3 + 66416*a^4*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^4*c^4 - 51744*a^5*(sqrt(b*x + a)*sqrt(-
c) - sqrt(-(b*x + a)*c + 2*a*c))^2*c^5 + 16384*a^6*c^6)/(((sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c)
)^2 - 2*a*c)^7*a^7*sqrt(-c)*c^3))/b

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Mupad [B]
time = 0.71, size = 170, normalized size = 1.28 \begin {gather*} -\frac {35\,a^6\,x\,\sqrt {a\,c-b\,c\,x}-16\,b^6\,x^7\,\sqrt {a\,c-b\,c\,x}-70\,a^4\,b^2\,x^3\,\sqrt {a\,c-b\,c\,x}+56\,a^2\,b^4\,x^5\,\sqrt {a\,c-b\,c\,x}}{\left (\left (70\,a^9\,{\left (a\,c-b\,c\,x\right )}^5+35\,a^8\,{\left (a\,c-b\,c\,x\right )}^5\,\left (a+b\,x\right )\right )\,\left (a+b\,x\right )+{\left (a\,c-b\,c\,x\right )}^4\,\left (140\,a^{10}\,\left (a\,c-b\,c\,x\right )-280\,a^{11}\,c\right )\right )\,\sqrt {a+b\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(35*a^6*x*(a*c - b*c*x)^(1/2) - 16*b^6*x^7*(a*c - b*c*x)^(1/2) - 70*a^4*b^2*x^3*(a*c - b*c*x)^(1/2) + 56*a^2*
b^4*x^5*(a*c - b*c*x)^(1/2))/(((70*a^9*(a*c - b*c*x)^5 + 35*a^8*(a*c - b*c*x)^5*(a + b*x))*(a + b*x) + (a*c -
b*c*x)^4*(140*a^10*(a*c - b*c*x) - 280*a^11*c))*(a + b*x)^(1/2))

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