3.12.51 \(\int \frac {1}{(a+b x)^{7/2} (a c-b c x)^{7/2}} \, dx\) [1151]

Optimal. Leaf size=100 \[ \frac {x}{5 a^2 c (a+b x)^{5/2} (a c-b c x)^{5/2}}+\frac {4 x}{15 a^4 c^2 (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac {8 x}{15 a^6 c^3 \sqrt {a+b x} \sqrt {a c-b c x}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 3, 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 {8 x}{15 a^6 c^3 \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {4 x}{15 a^4 c^2 (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac {x}{5 a^2 c (a+b x)^{5/2} (a c-b c x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x/(5*a^2*c*(a + b*x)^(5/2)*(a*c - b*c*x)^(5/2)) + (4*x)/(15*a^4*c^2*(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2)) + (8*
x)/(15*a^6*c^3*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)^{7/2} (a c-b c x)^{7/2}} \, dx &=\frac {x}{5 a^2 c (a+b x)^{5/2} (a c-b c x)^{5/2}}+\frac {4 \int \frac {1}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx}{5 a^2 c}\\ &=\frac {x}{5 a^2 c (a+b x)^{5/2} (a c-b c x)^{5/2}}+\frac {4 x}{15 a^4 c^2 (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac {8 \int \frac {1}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx}{15 a^4 c^2}\\ &=\frac {x}{5 a^2 c (a+b x)^{5/2} (a c-b c x)^{5/2}}+\frac {4 x}{15 a^4 c^2 (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac {8 x}{15 a^6 c^3 \sqrt {a+b x} \sqrt {a c-b c x}}\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 57, normalized size = 0.57 \begin {gather*} \frac {15 a^4 x-20 a^2 b^2 x^3+8 b^4 x^5}{15 a^6 c (c (a-b x))^{5/2} (a+b x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(201\) vs. \(2(82)=164\).
time = 0.17, size = 202, normalized size = 2.02

method	result	size
gosper	\(\frac {\left (-b x +a \right ) x \left (8 b^{4} x^{4}-20 a^{2} b^{2} x^{2}+15 a^{4}\right )}{15 \left (b x +a \right )^{\frac {5}{2}} a^{6} \left (-b c x +a c \right )^{\frac {7}{2}}}\)	\(56\)
default	\(-\frac {1}{5 a b c \left (b x +a \right )^{\frac {5}{2}} \left (-b c x +a c \right )^{\frac {5}{2}}}+\frac {-\frac {1}{3 a b c \left (b x +a \right )^{\frac {3}{2}} \left (-b c x +a c \right )^{\frac {5}{2}}}+\frac {-\frac {4}{3 a b c \sqrt {b x +a}\, \left (-b c x +a c \right )^{\frac {5}{2}}}+\frac {4 \left (\frac {3 \sqrt {b x +a}}{5 a b c \left (-b c x +a c \right )^{\frac {5}{2}}}+\frac {3 \left (\frac {2 \sqrt {b x +a}}{15 a b c \left (-b c x +a c \right )^{\frac {3}{2}}}+\frac {2 \sqrt {b x +a}}{15 b \,a^{2} c^{2} \sqrt {-b c x +a c}}\right )}{a c}\right )}{3 a}}{a}}{a}\)	\(202\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5/a/b/c/(b*x+a)^(5/2)/(-b*c*x+a*c)^(5/2)+1/a*(-1/3/a/b/c/(b*x+a)^(3/2)/(-b*c*x+a*c)^(5/2)+4/3/a*(-1/a/b/c/(
b*x+a)^(1/2)/(-b*c*x+a*c)^(5/2)+3/a*(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 = 79, normalized size = 0.79 \begin {gather*} \frac {x}{5 \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {5}{2}} a^{2} c} + \frac {4 \, x}{15 \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} a^{4} c^{2}} + \frac {8 \, x}{15 \, \sqrt {-b^{2} c x^{2} + a^{2} c} a^{6} c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 1.07, size = 98, normalized size = 0.98 \begin {gather*} -\frac {{\left (8 \, b^{4} x^{5} - 20 \, a^{2} b^{2} x^{3} + 15 \, a^{4} x\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{15 \, {\left (a^{6} b^{6} c^{4} x^{6} - 3 \, a^{8} b^{4} c^{4} x^{4} + 3 \, a^{10} b^{2} c^{4} x^{2} - a^{12} c^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(8*b^4*x^5 - 20*a^2*b^2*x^3 + 15*a^4*x)*sqrt(-b*c*x + a*c)*sqrt(b*x + a)/(a^6*b^6*c^4*x^6 - 3*a^8*b^4*c^
4*x^4 + 3*a^10*b^2*c^4*x^2 - a^12*c^4)

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Sympy [C] Result contains complex when optimal does not.
time = 39.18, size = 97, normalized size = 0.97 \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 {a^{2}}{b^{2} x^{2}}} \right )}}{15 \pi ^{\frac {3}{2}} a^{6} b 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 {a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{15 \pi ^{\frac {3}{2}} a^{6} b c^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)**(7/2)/(-b*c*x+a*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,)), a**2/(b**2*x**2))/(15*pi**(3/2)*a*
*6*b*c**(7/2)) + 2*meijerg(((-1/2, 0, 1/2, 5/4, 7/4, 1), ()), ((5/4, 7/4), (-1/2, 0, 3, 0)), a**2*exp_polar(-2
*I*pi)/(b**2*x**2))/(15*pi**(3/2)*a**6*b*c**(7/2))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 296 vs. \(2 (82) = 164\).
time = 1.79, size = 296, normalized size = 2.96 \begin {gather*} -\frac {\frac {\sqrt {-{\left (b x + a\right )} c + 2 \, a c} \sqrt {b x + a} {\left ({\left (b x + a\right )} {\left (\frac {64 \, {\left (b x + a\right )}}{a^{6} c} - \frac {275}{a^{5} c}\right )} + \frac {300}{a^{4} c}\right )}}{{\left ({\left (b x + a\right )} c - 2 \, a c\right )}^{3}} + \frac {4 \, {\left (45 \, {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{8} - 450 \, a {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{6} c + 1660 \, a^{2} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{4} c^{2} - 2200 \, a^{3} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2} c^{3} + 1024 \, a^{4} c^{4}\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 )}^{5} a^{5} \sqrt {-c} c^{2}}}{240 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.65, size = 111, normalized size = 1.11 \begin {gather*} \frac {15\,a^4\,x\,\sqrt {a\,c-b\,c\,x}+8\,b^4\,x^5\,\sqrt {a\,c-b\,c\,x}-20\,a^2\,b^2\,x^3\,\sqrt {a\,c-b\,c\,x}}{{\left (a\,c-b\,c\,x\right )}^3\,\left (60\,a^8\,c-\left (a\,c-b\,c\,x\right )\,\left (45\,a^7+15\,b\,x\,a^6\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)^(7/2)*(a + b*x)^(7/2)),x)

[Out]

(15*a^4*x*(a*c - b*c*x)^(1/2) + 8*b^4*x^5*(a*c - b*c*x)^(1/2) - 20*a^2*b^2*x^3*(a*c - b*c*x)^(1/2))/((a*c - b*
c*x)^3*(60*a^8*c - (a*c - b*c*x)*(45*a^7 + 15*a^6*b*x))*(a + b*x)^(1/2))

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