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

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

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Rubi [A]  time = 0.03, 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] /; F
reeQ[{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 = 0.04, size = 76, normalized size = 0.57 \begin {gather*} \frac {x \left (35 a^6-70 a^4 b^2 x^2+56 a^2 b^4 x^4-16 b^6 x^6\right ) \sqrt {c (a-b x)}}{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]

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

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IntegrateAlgebraic [A]  time = 0.15, size = 195, normalized size = 1.47 \begin {gather*} \frac {(a+b x)^{7/2} \left (\frac {49 c^6 (a c-b c x)}{a+b x}+\frac {245 c^5 (a c-b c x)^2}{(a+b x)^2}+\frac {1225 c^4 (a c-b c x)^3}{(a+b x)^3}-\frac {1225 c^3 (a c-b c x)^4}{(a+b x)^4}-\frac {245 c^2 (a c-b c x)^5}{(a+b x)^5}-\frac {49 c (a c-b c x)^6}{(a+b x)^6}-\frac {5 (a c-b c x)^7}{(a+b x)^7}+5 c^7\right )}{4480 a^8 b c^8 (a c-b c x)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^(7/2)*(5*c^7 + (49*c^6*(a*c - b*c*x))/(a + b*x) + (245*c^5*(a*c - b*c*x)^2)/(a + b*x)^2 + (1225*c^4
*(a*c - b*c*x)^3)/(a + b*x)^3 - (1225*c^3*(a*c - b*c*x)^4)/(a + b*x)^4 - (245*c^2*(a*c - b*c*x)^5)/(a + b*x)^5
- (49*c*(a*c - b*c*x)^6)/(a + b*x)^6 - (5*(a*c - b*c*x)^7)/(a + b*x)^7))/(4480*a^8*b*c^8*(a*c - b*c*x)^(7/2))

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fricas [A]  time = 1.60, 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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giac [B]  time = 3.30, size = 487, normalized size = 3.66 \begin {gather*} -\frac {\sqrt {-b c x + a c} {\left ({\left (b c x - a c\right )} {\left ({\left (b c x - a c\right )} {\left (\frac {1617 \, {\left | c \right |}}{a^{7} b c} + \frac {256 \, {\left (b c x - a c\right )} {\left | c \right |}}{a^{8} b c^{2}}\right )} + \frac {3430 \, {\left | c \right |}}{a^{6} b}\right )} + \frac {2450 \, c {\left | c \right |}}{a^{5} b}\right )}}{1120 \, {\left (2 \, a c^{2} + {\left (b c x - a c\right )} c\right )}^{\frac {7}{2}}} - \frac {16384 \, a^{6} c^{12} - 51744 \, a^{5} {\left (\sqrt {-b c x + a c} \sqrt {-c} - \sqrt {2 \, a c^{2} + {\left (b c x - a c\right )} c}\right )}^{2} c^{10} + 66416 \, a^{4} {\left (\sqrt {-b c x + a c} \sqrt {-c} - \sqrt {2 \, a c^{2} + {\left (b c x - a c\right )} c}\right )}^{4} c^{8} - 43120 \, a^{3} {\left (\sqrt {-b c x + a c} \sqrt {-c} - \sqrt {2 \, a c^{2} + {\left (b c x - a c\right )} c}\right )}^{6} c^{6} + 14280 \, a^{2} {\left (\sqrt {-b c x + a c} \sqrt {-c} - \sqrt {2 \, a c^{2} + {\left (b c x - a c\right )} c}\right )}^{8} c^{4} - 2450 \, a {\left (\sqrt {-b c x + a c} \sqrt {-c} - \sqrt {2 \, a c^{2} + {\left (b c x - a c\right )} c}\right )}^{10} c^{2} + 175 \, {\left (\sqrt {-b c x + a c} \sqrt {-c} - \sqrt {2 \, a c^{2} + {\left (b c x - a c\right )} c}\right )}^{12}}{280 \, {\left (2 \, a c^{2} - {\left (\sqrt {-b c x + a c} \sqrt {-c} - \sqrt {2 \, a c^{2} + {\left (b c x - a c\right )} c}\right )}^{2}\right )}^{7} a^{7} b \sqrt {-c} c {\left | c \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="giac")

[Out]

-1/1120*sqrt(-b*c*x + a*c)*((b*c*x - a*c)*((b*c*x - a*c)*(1617*abs(c)/(a^7*b*c) + 256*(b*c*x - a*c)*abs(c)/(a^
8*b*c^2)) + 3430*abs(c)/(a^6*b)) + 2450*c*abs(c)/(a^5*b))/(2*a*c^2 + (b*c*x - a*c)*c)^(7/2) - 1/280*(16384*a^6
*c^12 - 51744*a^5*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x - a*c)*c))^2*c^10 + 66416*a^4*(sqrt(-b*
c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x - a*c)*c))^4*c^8 - 43120*a^3*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2
*a*c^2 + (b*c*x - a*c)*c))^6*c^6 + 14280*a^2*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x - a*c)*c))^8
*c^4 - 2450*a*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x - a*c)*c))^10*c^2 + 175*(sqrt(-b*c*x + a*c)
*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x - a*c)*c))^12)/((2*a*c^2 - (sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*
c*x - a*c)*c))^2)^7*a^7*b*sqrt(-c)*c*abs(c))

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maple [A]  time = 0.00, size = 67, normalized size = 0.50 \begin {gather*} \frac {\left (-b x +a \right ) \left (-16 b^{6} x^{6}+56 b^{4} x^{4} a^{2}-70 b^{2} x^{2} a^{4}+35 a^{6}\right ) x}{35 \left (b x +a \right )^{\frac {7}{2}} \left (-b c x +a c \right )^{\frac {9}{2}} a^{8}} \end {gather*}

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)

[Out]

1/35*(-b*x+a)*x*(-16*b^6*x^6+56*a^2*b^4*x^4-70*a^4*b^2*x^2+35*a^6)/(b*x+a)^(7/2)/a^8/(-b*c*x+a*c)^(9/2)

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maxima [A]  time = 1.29, 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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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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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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]

Timed out

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