3.5.98 \(\int \frac {A+B x}{x^{11/2} \sqrt {a+b x}} \, dx\)

Optimal. Leaf size=150 \[ -\frac {32 b^3 \sqrt {a+b x} (8 A b-9 a B)}{315 a^5 \sqrt {x}}+\frac {16 b^2 \sqrt {a+b x} (8 A b-9 a B)}{315 a^4 x^{3/2}}-\frac {4 b \sqrt {a+b x} (8 A b-9 a B)}{105 a^3 x^{5/2}}+\frac {2 \sqrt {a+b x} (8 A b-9 a B)}{63 a^2 x^{7/2}}-\frac {2 A \sqrt {a+b x}}{9 a x^{9/2}} \]

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Rubi [A]  time = 0.06, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {78, 45, 37} \begin {gather*} \frac {16 b^2 \sqrt {a+b x} (8 A b-9 a B)}{315 a^4 x^{3/2}}-\frac {32 b^3 \sqrt {a+b x} (8 A b-9 a B)}{315 a^5 \sqrt {x}}-\frac {4 b \sqrt {a+b x} (8 A b-9 a B)}{105 a^3 x^{5/2}}+\frac {2 \sqrt {a+b x} (8 A b-9 a B)}{63 a^2 x^{7/2}}-\frac {2 A \sqrt {a+b x}}{9 a x^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(x^(11/2)*Sqrt[a + b*x]),x]

[Out]

(-2*A*Sqrt[a + b*x])/(9*a*x^(9/2)) + (2*(8*A*b - 9*a*B)*Sqrt[a + b*x])/(63*a^2*x^(7/2)) - (4*b*(8*A*b - 9*a*B)
*Sqrt[a + b*x])/(105*a^3*x^(5/2)) + (16*b^2*(8*A*b - 9*a*B)*Sqrt[a + b*x])/(315*a^4*x^(3/2)) - (32*b^3*(8*A*b
- 9*a*B)*Sqrt[a + b*x])/(315*a^5*Sqrt[x])

Rule 37

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rubi steps

\begin {align*} \int \frac {A+B x}{x^{11/2} \sqrt {a+b x}} \, dx &=-\frac {2 A \sqrt {a+b x}}{9 a x^{9/2}}+\frac {\left (2 \left (-4 A b+\frac {9 a B}{2}\right )\right ) \int \frac {1}{x^{9/2} \sqrt {a+b x}} \, dx}{9 a}\\ &=-\frac {2 A \sqrt {a+b x}}{9 a x^{9/2}}+\frac {2 (8 A b-9 a B) \sqrt {a+b x}}{63 a^2 x^{7/2}}+\frac {(2 b (8 A b-9 a B)) \int \frac {1}{x^{7/2} \sqrt {a+b x}} \, dx}{21 a^2}\\ &=-\frac {2 A \sqrt {a+b x}}{9 a x^{9/2}}+\frac {2 (8 A b-9 a B) \sqrt {a+b x}}{63 a^2 x^{7/2}}-\frac {4 b (8 A b-9 a B) \sqrt {a+b x}}{105 a^3 x^{5/2}}-\frac {\left (8 b^2 (8 A b-9 a B)\right ) \int \frac {1}{x^{5/2} \sqrt {a+b x}} \, dx}{105 a^3}\\ &=-\frac {2 A \sqrt {a+b x}}{9 a x^{9/2}}+\frac {2 (8 A b-9 a B) \sqrt {a+b x}}{63 a^2 x^{7/2}}-\frac {4 b (8 A b-9 a B) \sqrt {a+b x}}{105 a^3 x^{5/2}}+\frac {16 b^2 (8 A b-9 a B) \sqrt {a+b x}}{315 a^4 x^{3/2}}+\frac {\left (16 b^3 (8 A b-9 a B)\right ) \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx}{315 a^4}\\ &=-\frac {2 A \sqrt {a+b x}}{9 a x^{9/2}}+\frac {2 (8 A b-9 a B) \sqrt {a+b x}}{63 a^2 x^{7/2}}-\frac {4 b (8 A b-9 a B) \sqrt {a+b x}}{105 a^3 x^{5/2}}+\frac {16 b^2 (8 A b-9 a B) \sqrt {a+b x}}{315 a^4 x^{3/2}}-\frac {32 b^3 (8 A b-9 a B) \sqrt {a+b x}}{315 a^5 \sqrt {x}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 95, normalized size = 0.63 \begin {gather*} -\frac {2 \sqrt {a+b x} \left (5 a^4 (7 A+9 B x)-2 a^3 b x (20 A+27 B x)+24 a^2 b^2 x^2 (2 A+3 B x)-16 a b^3 x^3 (4 A+9 B x)+128 A b^4 x^4\right )}{315 a^5 x^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(x^(11/2)*Sqrt[a + b*x]),x]

[Out]

(-2*Sqrt[a + b*x]*(128*A*b^4*x^4 + 24*a^2*b^2*x^2*(2*A + 3*B*x) - 16*a*b^3*x^3*(4*A + 9*B*x) + 5*a^4*(7*A + 9*
B*x) - 2*a^3*b*x*(20*A + 27*B*x)))/(315*a^5*x^(9/2))

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IntegrateAlgebraic [A]  time = 0.20, size = 106, normalized size = 0.71 \begin {gather*} \frac {2 \sqrt {a+b x} \left (-35 a^4 A-45 a^4 B x+40 a^3 A b x+54 a^3 b B x^2-48 a^2 A b^2 x^2-72 a^2 b^2 B x^3+64 a A b^3 x^3+144 a b^3 B x^4-128 A b^4 x^4\right )}{315 a^5 x^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(A + B*x)/(x^(11/2)*Sqrt[a + b*x]),x]

[Out]

(2*Sqrt[a + b*x]*(-35*a^4*A + 40*a^3*A*b*x - 45*a^4*B*x - 48*a^2*A*b^2*x^2 + 54*a^3*b*B*x^2 + 64*a*A*b^3*x^3 -
72*a^2*b^2*B*x^3 - 128*A*b^4*x^4 + 144*a*b^3*B*x^4))/(315*a^5*x^(9/2))

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fricas [A]  time = 1.62, size = 102, normalized size = 0.68 \begin {gather*} -\frac {2 \, {\left (35 \, A a^{4} - 16 \, {\left (9 \, B a b^{3} - 8 \, A b^{4}\right )} x^{4} + 8 \, {\left (9 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} x^{3} - 6 \, {\left (9 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} x^{2} + 5 \, {\left (9 \, B a^{4} - 8 \, A a^{3} b\right )} x\right )} \sqrt {b x + a}}{315 \, a^{5} x^{\frac {9}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(11/2)/(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

-2/315*(35*A*a^4 - 16*(9*B*a*b^3 - 8*A*b^4)*x^4 + 8*(9*B*a^2*b^2 - 8*A*a*b^3)*x^3 - 6*(9*B*a^3*b - 8*A*a^2*b^2
)*x^2 + 5*(9*B*a^4 - 8*A*a^3*b)*x)*sqrt(b*x + a)/(a^5*x^(9/2))

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giac [A]  time = 1.43, size = 169, normalized size = 1.13 \begin {gather*} \frac {2 \, {\left ({\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (9 \, B a b^{8} - 8 \, A b^{9}\right )} {\left (b x + a\right )}}{a^{5}} - \frac {9 \, {\left (9 \, B a^{2} b^{8} - 8 \, A a b^{9}\right )}}{a^{5}}\right )} + \frac {63 \, {\left (9 \, B a^{3} b^{8} - 8 \, A a^{2} b^{9}\right )}}{a^{5}}\right )} - \frac {105 \, {\left (9 \, B a^{4} b^{8} - 8 \, A a^{3} b^{9}\right )}}{a^{5}}\right )} {\left (b x + a\right )} + \frac {315 \, {\left (B a^{5} b^{8} - A a^{4} b^{9}\right )}}{a^{5}}\right )} \sqrt {b x + a} b}{315 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {9}{2}} {\left | b \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(11/2)/(b*x+a)^(1/2),x, algorithm="giac")

[Out]

2/315*((2*(b*x + a)*(4*(b*x + a)*(2*(9*B*a*b^8 - 8*A*b^9)*(b*x + a)/a^5 - 9*(9*B*a^2*b^8 - 8*A*a*b^9)/a^5) + 6
3*(9*B*a^3*b^8 - 8*A*a^2*b^9)/a^5) - 105*(9*B*a^4*b^8 - 8*A*a^3*b^9)/a^5)*(b*x + a) + 315*(B*a^5*b^8 - A*a^4*b
^9)/a^5)*sqrt(b*x + a)*b/(((b*x + a)*b - a*b)^(9/2)*abs(b))

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maple [A]  time = 0.01, size = 101, normalized size = 0.67 \begin {gather*} -\frac {2 \sqrt {b x +a}\, \left (128 A \,b^{4} x^{4}-144 B a \,b^{3} x^{4}-64 A a \,b^{3} x^{3}+72 B \,a^{2} b^{2} x^{3}+48 A \,a^{2} b^{2} x^{2}-54 B \,a^{3} b \,x^{2}-40 A \,a^{3} b x +45 B \,a^{4} x +35 A \,a^{4}\right )}{315 a^{5} x^{\frac {9}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/x^(11/2)/(b*x+a)^(1/2),x)

[Out]

-2/315*(b*x+a)^(1/2)*(128*A*b^4*x^4-144*B*a*b^3*x^4-64*A*a*b^3*x^3+72*B*a^2*b^2*x^3+48*A*a^2*b^2*x^2-54*B*a^3*
b*x^2-40*A*a^3*b*x+45*B*a^4*x+35*A*a^4)/x^(9/2)/a^5

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maxima [A]  time = 0.92, size = 198, normalized size = 1.32 \begin {gather*} \frac {32 \, \sqrt {b x^{2} + a x} B b^{3}}{35 \, a^{4} x} - \frac {256 \, \sqrt {b x^{2} + a x} A b^{4}}{315 \, a^{5} x} - \frac {16 \, \sqrt {b x^{2} + a x} B b^{2}}{35 \, a^{3} x^{2}} + \frac {128 \, \sqrt {b x^{2} + a x} A b^{3}}{315 \, a^{4} x^{2}} + \frac {12 \, \sqrt {b x^{2} + a x} B b}{35 \, a^{2} x^{3}} - \frac {32 \, \sqrt {b x^{2} + a x} A b^{2}}{105 \, a^{3} x^{3}} - \frac {2 \, \sqrt {b x^{2} + a x} B}{7 \, a x^{4}} + \frac {16 \, \sqrt {b x^{2} + a x} A b}{63 \, a^{2} x^{4}} - \frac {2 \, \sqrt {b x^{2} + a x} A}{9 \, a x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(11/2)/(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

32/35*sqrt(b*x^2 + a*x)*B*b^3/(a^4*x) - 256/315*sqrt(b*x^2 + a*x)*A*b^4/(a^5*x) - 16/35*sqrt(b*x^2 + a*x)*B*b^
2/(a^3*x^2) + 128/315*sqrt(b*x^2 + a*x)*A*b^3/(a^4*x^2) + 12/35*sqrt(b*x^2 + a*x)*B*b/(a^2*x^3) - 32/105*sqrt(
b*x^2 + a*x)*A*b^2/(a^3*x^3) - 2/7*sqrt(b*x^2 + a*x)*B/(a*x^4) + 16/63*sqrt(b*x^2 + a*x)*A*b/(a^2*x^4) - 2/9*s
qrt(b*x^2 + a*x)*A/(a*x^5)

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mupad [B]  time = 0.96, size = 99, normalized size = 0.66 \begin {gather*} -\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A}{9\,a}+\frac {x\,\left (90\,B\,a^4-80\,A\,a^3\,b\right )}{315\,a^5}+\frac {x^4\,\left (256\,A\,b^4-288\,B\,a\,b^3\right )}{315\,a^5}-\frac {16\,b^2\,x^3\,\left (8\,A\,b-9\,B\,a\right )}{315\,a^4}+\frac {4\,b\,x^2\,\left (8\,A\,b-9\,B\,a\right )}{105\,a^3}\right )}{x^{9/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)/(x^(11/2)*(a + b*x)^(1/2)),x)

[Out]

-((a + b*x)^(1/2)*((2*A)/(9*a) + (x*(90*B*a^4 - 80*A*a^3*b))/(315*a^5) + (x^4*(256*A*b^4 - 288*B*a*b^3))/(315*
a^5) - (16*b^2*x^3*(8*A*b - 9*B*a))/(315*a^4) + (4*b*x^2*(8*A*b - 9*B*a))/(105*a^3)))/x^(9/2)

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sympy [B]  time = 107.07, size = 1255, normalized size = 8.37 \begin {gather*} - \frac {70 A a^{8} b^{\frac {33}{2}} \sqrt {\frac {a}{b x} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{5} + 1890 a^{7} b^{18} x^{6} + 1260 a^{6} b^{19} x^{7} + 315 a^{5} b^{20} x^{8}} - \frac {200 A a^{7} b^{\frac {35}{2}} x \sqrt {\frac {a}{b x} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{5} + 1890 a^{7} b^{18} x^{6} + 1260 a^{6} b^{19} x^{7} + 315 a^{5} b^{20} x^{8}} - \frac {196 A a^{6} b^{\frac {37}{2}} x^{2} \sqrt {\frac {a}{b x} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{5} + 1890 a^{7} b^{18} x^{6} + 1260 a^{6} b^{19} x^{7} + 315 a^{5} b^{20} x^{8}} - \frac {56 A a^{5} b^{\frac {39}{2}} x^{3} \sqrt {\frac {a}{b x} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{5} + 1890 a^{7} b^{18} x^{6} + 1260 a^{6} b^{19} x^{7} + 315 a^{5} b^{20} x^{8}} - \frac {70 A a^{4} b^{\frac {41}{2}} x^{4} \sqrt {\frac {a}{b x} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{5} + 1890 a^{7} b^{18} x^{6} + 1260 a^{6} b^{19} x^{7} + 315 a^{5} b^{20} x^{8}} - \frac {560 A a^{3} b^{\frac {43}{2}} x^{5} \sqrt {\frac {a}{b x} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{5} + 1890 a^{7} b^{18} x^{6} + 1260 a^{6} b^{19} x^{7} + 315 a^{5} b^{20} x^{8}} - \frac {1120 A a^{2} b^{\frac {45}{2}} x^{6} \sqrt {\frac {a}{b x} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{5} + 1890 a^{7} b^{18} x^{6} + 1260 a^{6} b^{19} x^{7} + 315 a^{5} b^{20} x^{8}} - \frac {896 A a b^{\frac {47}{2}} x^{7} \sqrt {\frac {a}{b x} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{5} + 1890 a^{7} b^{18} x^{6} + 1260 a^{6} b^{19} x^{7} + 315 a^{5} b^{20} x^{8}} - \frac {256 A b^{\frac {49}{2}} x^{8} \sqrt {\frac {a}{b x} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{5} + 1890 a^{7} b^{18} x^{6} + 1260 a^{6} b^{19} x^{7} + 315 a^{5} b^{20} x^{8}} - \frac {10 B a^{6} b^{\frac {19}{2}} \sqrt {\frac {a}{b x} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{4} + 105 a^{5} b^{11} x^{5} + 35 a^{4} b^{12} x^{6}} - \frac {18 B a^{5} b^{\frac {21}{2}} x \sqrt {\frac {a}{b x} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{4} + 105 a^{5} b^{11} x^{5} + 35 a^{4} b^{12} x^{6}} - \frac {10 B a^{4} b^{\frac {23}{2}} x^{2} \sqrt {\frac {a}{b x} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{4} + 105 a^{5} b^{11} x^{5} + 35 a^{4} b^{12} x^{6}} + \frac {10 B a^{3} b^{\frac {25}{2}} x^{3} \sqrt {\frac {a}{b x} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{4} + 105 a^{5} b^{11} x^{5} + 35 a^{4} b^{12} x^{6}} + \frac {60 B a^{2} b^{\frac {27}{2}} x^{4} \sqrt {\frac {a}{b x} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{4} + 105 a^{5} b^{11} x^{5} + 35 a^{4} b^{12} x^{6}} + \frac {80 B a b^{\frac {29}{2}} x^{5} \sqrt {\frac {a}{b x} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{4} + 105 a^{5} b^{11} x^{5} + 35 a^{4} b^{12} x^{6}} + \frac {32 B b^{\frac {31}{2}} x^{6} \sqrt {\frac {a}{b x} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{4} + 105 a^{5} b^{11} x^{5} + 35 a^{4} b^{12} x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x**(11/2)/(b*x+a)**(1/2),x)

[Out]

-70*A*a**8*b**(33/2)*sqrt(a/(b*x) + 1)/(315*a**9*b**16*x**4 + 1260*a**8*b**17*x**5 + 1890*a**7*b**18*x**6 + 12
60*a**6*b**19*x**7 + 315*a**5*b**20*x**8) - 200*A*a**7*b**(35/2)*x*sqrt(a/(b*x) + 1)/(315*a**9*b**16*x**4 + 12
60*a**8*b**17*x**5 + 1890*a**7*b**18*x**6 + 1260*a**6*b**19*x**7 + 315*a**5*b**20*x**8) - 196*A*a**6*b**(37/2)
*x**2*sqrt(a/(b*x) + 1)/(315*a**9*b**16*x**4 + 1260*a**8*b**17*x**5 + 1890*a**7*b**18*x**6 + 1260*a**6*b**19*x
**7 + 315*a**5*b**20*x**8) - 56*A*a**5*b**(39/2)*x**3*sqrt(a/(b*x) + 1)/(315*a**9*b**16*x**4 + 1260*a**8*b**17
*x**5 + 1890*a**7*b**18*x**6 + 1260*a**6*b**19*x**7 + 315*a**5*b**20*x**8) - 70*A*a**4*b**(41/2)*x**4*sqrt(a/(
b*x) + 1)/(315*a**9*b**16*x**4 + 1260*a**8*b**17*x**5 + 1890*a**7*b**18*x**6 + 1260*a**6*b**19*x**7 + 315*a**5
*b**20*x**8) - 560*A*a**3*b**(43/2)*x**5*sqrt(a/(b*x) + 1)/(315*a**9*b**16*x**4 + 1260*a**8*b**17*x**5 + 1890*
a**7*b**18*x**6 + 1260*a**6*b**19*x**7 + 315*a**5*b**20*x**8) - 1120*A*a**2*b**(45/2)*x**6*sqrt(a/(b*x) + 1)/(
315*a**9*b**16*x**4 + 1260*a**8*b**17*x**5 + 1890*a**7*b**18*x**6 + 1260*a**6*b**19*x**7 + 315*a**5*b**20*x**8
) - 896*A*a*b**(47/2)*x**7*sqrt(a/(b*x) + 1)/(315*a**9*b**16*x**4 + 1260*a**8*b**17*x**5 + 1890*a**7*b**18*x**
6 + 1260*a**6*b**19*x**7 + 315*a**5*b**20*x**8) - 256*A*b**(49/2)*x**8*sqrt(a/(b*x) + 1)/(315*a**9*b**16*x**4
+ 1260*a**8*b**17*x**5 + 1890*a**7*b**18*x**6 + 1260*a**6*b**19*x**7 + 315*a**5*b**20*x**8) - 10*B*a**6*b**(19
/2)*sqrt(a/(b*x) + 1)/(35*a**7*b**9*x**3 + 105*a**6*b**10*x**4 + 105*a**5*b**11*x**5 + 35*a**4*b**12*x**6) - 1
8*B*a**5*b**(21/2)*x*sqrt(a/(b*x) + 1)/(35*a**7*b**9*x**3 + 105*a**6*b**10*x**4 + 105*a**5*b**11*x**5 + 35*a**
4*b**12*x**6) - 10*B*a**4*b**(23/2)*x**2*sqrt(a/(b*x) + 1)/(35*a**7*b**9*x**3 + 105*a**6*b**10*x**4 + 105*a**5
*b**11*x**5 + 35*a**4*b**12*x**6) + 10*B*a**3*b**(25/2)*x**3*sqrt(a/(b*x) + 1)/(35*a**7*b**9*x**3 + 105*a**6*b
**10*x**4 + 105*a**5*b**11*x**5 + 35*a**4*b**12*x**6) + 60*B*a**2*b**(27/2)*x**4*sqrt(a/(b*x) + 1)/(35*a**7*b*
*9*x**3 + 105*a**6*b**10*x**4 + 105*a**5*b**11*x**5 + 35*a**4*b**12*x**6) + 80*B*a*b**(29/2)*x**5*sqrt(a/(b*x)
+ 1)/(35*a**7*b**9*x**3 + 105*a**6*b**10*x**4 + 105*a**5*b**11*x**5 + 35*a**4*b**12*x**6) + 32*B*b**(31/2)*x*
*6*sqrt(a/(b*x) + 1)/(35*a**7*b**9*x**3 + 105*a**6*b**10*x**4 + 105*a**5*b**11*x**5 + 35*a**4*b**12*x**6)

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