3.16.1 \(\int \frac {1}{(a+b x)^{11/2} \sqrt {c+d x}} \, dx\) [1501]

Optimal. Leaf size=171 \[ -\frac {2 \sqrt {c+d x}}{9 (b c-a d) (a+b x)^{9/2}}+\frac {16 d \sqrt {c+d x}}{63 (b c-a d)^2 (a+b x)^{7/2}}-\frac {32 d^2 \sqrt {c+d x}}{105 (b c-a d)^3 (a+b x)^{5/2}}+\frac {128 d^3 \sqrt {c+d x}}{315 (b c-a d)^4 (a+b x)^{3/2}}-\frac {256 d^4 \sqrt {c+d x}}{315 (b c-a d)^5 \sqrt {a+b x}} \]

[Out]

-2/9*(d*x+c)^(1/2)/(-a*d+b*c)/(b*x+a)^(9/2)+16/63*d*(d*x+c)^(1/2)/(-a*d+b*c)^2/(b*x+a)^(7/2)-32/105*d^2*(d*x+c
)^(1/2)/(-a*d+b*c)^3/(b*x+a)^(5/2)+128/315*d^3*(d*x+c)^(1/2)/(-a*d+b*c)^4/(b*x+a)^(3/2)-256/315*d^4*(d*x+c)^(1
/2)/(-a*d+b*c)^5/(b*x+a)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {47, 37} \begin {gather*} -\frac {256 d^4 \sqrt {c+d x}}{315 \sqrt {a+b x} (b c-a d)^5}+\frac {128 d^3 \sqrt {c+d x}}{315 (a+b x)^{3/2} (b c-a d)^4}-\frac {32 d^2 \sqrt {c+d x}}{105 (a+b x)^{5/2} (b c-a d)^3}+\frac {16 d \sqrt {c+d x}}{63 (a+b x)^{7/2} (b c-a d)^2}-\frac {2 \sqrt {c+d x}}{9 (a+b x)^{9/2} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)^(11/2)*Sqrt[c + d*x]),x]

[Out]

(-2*Sqrt[c + d*x])/(9*(b*c - a*d)*(a + b*x)^(9/2)) + (16*d*Sqrt[c + d*x])/(63*(b*c - a*d)^2*(a + b*x)^(7/2)) -
(32*d^2*Sqrt[c + d*x])/(105*(b*c - a*d)^3*(a + b*x)^(5/2)) + (128*d^3*Sqrt[c + d*x])/(315*(b*c - a*d)^4*(a +
b*x)^(3/2)) - (256*d^4*Sqrt[c + d*x])/(315*(b*c - a*d)^5*Sqrt[a + b*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 47

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])

Rubi steps

\begin {align*} \int \frac {1}{(a+b x)^{11/2} \sqrt {c+d x}} \, dx &=-\frac {2 \sqrt {c+d x}}{9 (b c-a d) (a+b x)^{9/2}}-\frac {(8 d) \int \frac {1}{(a+b x)^{9/2} \sqrt {c+d x}} \, dx}{9 (b c-a d)}\\ &=-\frac {2 \sqrt {c+d x}}{9 (b c-a d) (a+b x)^{9/2}}+\frac {16 d \sqrt {c+d x}}{63 (b c-a d)^2 (a+b x)^{7/2}}+\frac {\left (16 d^2\right ) \int \frac {1}{(a+b x)^{7/2} \sqrt {c+d x}} \, dx}{21 (b c-a d)^2}\\ &=-\frac {2 \sqrt {c+d x}}{9 (b c-a d) (a+b x)^{9/2}}+\frac {16 d \sqrt {c+d x}}{63 (b c-a d)^2 (a+b x)^{7/2}}-\frac {32 d^2 \sqrt {c+d x}}{105 (b c-a d)^3 (a+b x)^{5/2}}-\frac {\left (64 d^3\right ) \int \frac {1}{(a+b x)^{5/2} \sqrt {c+d x}} \, dx}{105 (b c-a d)^3}\\ &=-\frac {2 \sqrt {c+d x}}{9 (b c-a d) (a+b x)^{9/2}}+\frac {16 d \sqrt {c+d x}}{63 (b c-a d)^2 (a+b x)^{7/2}}-\frac {32 d^2 \sqrt {c+d x}}{105 (b c-a d)^3 (a+b x)^{5/2}}+\frac {128 d^3 \sqrt {c+d x}}{315 (b c-a d)^4 (a+b x)^{3/2}}+\frac {\left (128 d^4\right ) \int \frac {1}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx}{315 (b c-a d)^4}\\ &=-\frac {2 \sqrt {c+d x}}{9 (b c-a d) (a+b x)^{9/2}}+\frac {16 d \sqrt {c+d x}}{63 (b c-a d)^2 (a+b x)^{7/2}}-\frac {32 d^2 \sqrt {c+d x}}{105 (b c-a d)^3 (a+b x)^{5/2}}+\frac {128 d^3 \sqrt {c+d x}}{315 (b c-a d)^4 (a+b x)^{3/2}}-\frac {256 d^4 \sqrt {c+d x}}{315 (b c-a d)^5 \sqrt {a+b x}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.14, size = 115, normalized size = 0.67 \begin {gather*} -\frac {2 \sqrt {c+d x} \left (315 d^4 (a+b x)^4-420 b d^3 (a+b x)^3 (c+d x)+378 b^2 d^2 (a+b x)^2 (c+d x)^2-180 b^3 d (a+b x) (c+d x)^3+35 b^4 (c+d x)^4\right )}{315 (b c-a d)^5 (a+b x)^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x)^(11/2)*Sqrt[c + d*x]),x]

[Out]

(-2*Sqrt[c + d*x]*(315*d^4*(a + b*x)^4 - 420*b*d^3*(a + b*x)^3*(c + d*x) + 378*b^2*d^2*(a + b*x)^2*(c + d*x)^2
- 180*b^3*d*(a + b*x)*(c + d*x)^3 + 35*b^4*(c + d*x)^4))/(315*(b*c - a*d)^5*(a + b*x)^(9/2))

________________________________________________________________________________________

Maple [A]
time = 0.16, size = 175, normalized size = 1.02

method	result	size
default	\(-\frac {2 \sqrt {d x +c}}{9 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {9}{2}}}-\frac {8 d \left (-\frac {2 \sqrt {d x +c}}{7 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {7}{2}}}-\frac {6 d \left (-\frac {2 \sqrt {d x +c}}{5 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {5}{2}}}-\frac {4 d \left (-\frac {2 \sqrt {d x +c}}{3 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {3}{2}}}+\frac {4 d \sqrt {d x +c}}{3 \left (-a d +b c \right )^{2} \sqrt {b x +a}}\right )}{5 \left (-a d +b c \right )}\right )}{7 \left (-a d +b c \right )}\right )}{9 \left (-a d +b c \right )}\)	\(175\)
gosper	\(\frac {2 \sqrt {d x +c}\, \left (128 d^{4} x^{4} b^{4}+576 a \,b^{3} d^{4} x^{3}-64 b^{4} c \,d^{3} x^{3}+1008 a^{2} b^{2} d^{4} x^{2}-288 a \,b^{3} c \,d^{3} x^{2}+48 b^{4} c^{2} d^{2} x^{2}+840 a^{3} b \,d^{4} x -504 a^{2} b^{2} c \,d^{3} x +216 a \,b^{3} c^{2} d^{2} x -40 b^{4} c^{3} d x +315 a^{4} d^{4}-420 a^{3} b c \,d^{3}+378 a^{2} b^{2} c^{2} d^{2}-180 a \,b^{3} c^{3} d +35 b^{4} c^{4}\right )}{315 \left (b x +a \right )^{\frac {9}{2}} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}\)	\(256\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)^(11/2)/(d*x+c)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/9*(d*x+c)^(1/2)/(-a*d+b*c)/(b*x+a)^(9/2)-8/9*d/(-a*d+b*c)*(-2/7*(d*x+c)^(1/2)/(-a*d+b*c)/(b*x+a)^(7/2)-6/7*
d/(-a*d+b*c)*(-2/5*(d*x+c)^(1/2)/(-a*d+b*c)/(b*x+a)^(5/2)-4/5*d/(-a*d+b*c)*(-2/3*(d*x+c)^(1/2)/(-a*d+b*c)/(b*x
+a)^(3/2)+4/3*d*(d*x+c)^(1/2)/(-a*d+b*c)^2/(b*x+a)^(1/2))))

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for
more detail

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 638 vs. \(2 (141) = 282\).
time = 3.68, size = 638, normalized size = 3.73 \begin {gather*} -\frac {2 \, {\left (128 \, b^{4} d^{4} x^{4} + 35 \, b^{4} c^{4} - 180 \, a b^{3} c^{3} d + 378 \, a^{2} b^{2} c^{2} d^{2} - 420 \, a^{3} b c d^{3} + 315 \, a^{4} d^{4} - 64 \, {\left (b^{4} c d^{3} - 9 \, a b^{3} d^{4}\right )} x^{3} + 48 \, {\left (b^{4} c^{2} d^{2} - 6 \, a b^{3} c d^{3} + 21 \, a^{2} b^{2} d^{4}\right )} x^{2} - 8 \, {\left (5 \, b^{4} c^{3} d - 27 \, a b^{3} c^{2} d^{2} + 63 \, a^{2} b^{2} c d^{3} - 105 \, a^{3} b d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{315 \, {\left (a^{5} b^{5} c^{5} - 5 \, a^{6} b^{4} c^{4} d + 10 \, a^{7} b^{3} c^{3} d^{2} - 10 \, a^{8} b^{2} c^{2} d^{3} + 5 \, a^{9} b c d^{4} - a^{10} d^{5} + {\left (b^{10} c^{5} - 5 \, a b^{9} c^{4} d + 10 \, a^{2} b^{8} c^{3} d^{2} - 10 \, a^{3} b^{7} c^{2} d^{3} + 5 \, a^{4} b^{6} c d^{4} - a^{5} b^{5} d^{5}\right )} x^{5} + 5 \, {\left (a b^{9} c^{5} - 5 \, a^{2} b^{8} c^{4} d + 10 \, a^{3} b^{7} c^{3} d^{2} - 10 \, a^{4} b^{6} c^{2} d^{3} + 5 \, a^{5} b^{5} c d^{4} - a^{6} b^{4} d^{5}\right )} x^{4} + 10 \, {\left (a^{2} b^{8} c^{5} - 5 \, a^{3} b^{7} c^{4} d + 10 \, a^{4} b^{6} c^{3} d^{2} - 10 \, a^{5} b^{5} c^{2} d^{3} + 5 \, a^{6} b^{4} c d^{4} - a^{7} b^{3} d^{5}\right )} x^{3} + 10 \, {\left (a^{3} b^{7} c^{5} - 5 \, a^{4} b^{6} c^{4} d + 10 \, a^{5} b^{5} c^{3} d^{2} - 10 \, a^{6} b^{4} c^{2} d^{3} + 5 \, a^{7} b^{3} c d^{4} - a^{8} b^{2} d^{5}\right )} x^{2} + 5 \, {\left (a^{4} b^{6} c^{5} - 5 \, a^{5} b^{5} c^{4} d + 10 \, a^{6} b^{4} c^{3} d^{2} - 10 \, a^{7} b^{3} c^{2} d^{3} + 5 \, a^{8} b^{2} c d^{4} - a^{9} b d^{5}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/315*(128*b^4*d^4*x^4 + 35*b^4*c^4 - 180*a*b^3*c^3*d + 378*a^2*b^2*c^2*d^2 - 420*a^3*b*c*d^3 + 315*a^4*d^4 -
64*(b^4*c*d^3 - 9*a*b^3*d^4)*x^3 + 48*(b^4*c^2*d^2 - 6*a*b^3*c*d^3 + 21*a^2*b^2*d^4)*x^2 - 8*(5*b^4*c^3*d - 2
7*a*b^3*c^2*d^2 + 63*a^2*b^2*c*d^3 - 105*a^3*b*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c)/(a^5*b^5*c^5 - 5*a^6*b^4*c^
4*d + 10*a^7*b^3*c^3*d^2 - 10*a^8*b^2*c^2*d^3 + 5*a^9*b*c*d^4 - a^10*d^5 + (b^10*c^5 - 5*a*b^9*c^4*d + 10*a^2*
b^8*c^3*d^2 - 10*a^3*b^7*c^2*d^3 + 5*a^4*b^6*c*d^4 - a^5*b^5*d^5)*x^5 + 5*(a*b^9*c^5 - 5*a^2*b^8*c^4*d + 10*a^
3*b^7*c^3*d^2 - 10*a^4*b^6*c^2*d^3 + 5*a^5*b^5*c*d^4 - a^6*b^4*d^5)*x^4 + 10*(a^2*b^8*c^5 - 5*a^3*b^7*c^4*d +
10*a^4*b^6*c^3*d^2 - 10*a^5*b^5*c^2*d^3 + 5*a^6*b^4*c*d^4 - a^7*b^3*d^5)*x^3 + 10*(a^3*b^7*c^5 - 5*a^4*b^6*c^4
*d + 10*a^5*b^5*c^3*d^2 - 10*a^6*b^4*c^2*d^3 + 5*a^7*b^3*c*d^4 - a^8*b^2*d^5)*x^2 + 5*(a^4*b^6*c^5 - 5*a^5*b^5
*c^4*d + 10*a^6*b^4*c^3*d^2 - 10*a^7*b^3*c^2*d^3 + 5*a^8*b^2*c*d^4 - a^9*b*d^5)*x)

________________________________________________________________________________________

Sympy [F(-1)] Timed out
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)**(11/2)/(d*x+c)**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 596 vs. \(2 (141) = 282\).
time = 2.20, size = 596, normalized size = 3.49 \begin {gather*} -\frac {512 \, {\left (b^{8} c^{4} - 4 \, a b^{7} c^{3} d + 6 \, a^{2} b^{6} c^{2} d^{2} - 4 \, a^{3} b^{5} c d^{3} + a^{4} b^{4} d^{4} - 9 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{6} c^{3} + 27 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{5} c^{2} d - 27 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{4} c d^{2} + 9 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{3} b^{3} d^{3} + 36 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} b^{4} c^{2} - 72 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a b^{3} c d + 36 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{2} b^{2} d^{2} - 84 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} b^{2} c + 84 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} a b d + 126 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{8}\right )} \sqrt {b d} b^{5} d^{4}}{315 \, {\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{9} {\left | b \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-512/315*(b^8*c^4 - 4*a*b^7*c^3*d + 6*a^2*b^6*c^2*d^2 - 4*a^3*b^5*c*d^3 + a^4*b^4*d^4 - 9*(sqrt(b*d)*sqrt(b*x
+ a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^6*c^3 + 27*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b
*d - a*b*d))^2*a*b^5*c^2*d - 27*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2*b^4*c*d^
2 + 9*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^3*b^3*d^3 + 36*(sqrt(b*d)*sqrt(b*x +
a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^4*c^2 - 72*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*
d - a*b*d))^4*a*b^3*c*d + 36*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^2*b^2*d^2 - 8
4*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*b^2*c + 84*(sqrt(b*d)*sqrt(b*x + a) - sqrt
(b^2*c + (b*x + a)*b*d - a*b*d))^6*a*b*d + 126*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))
^8)*sqrt(b*d)*b^5*d^4/((b^2*c - a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)^9*a
bs(b))

________________________________________________________________________________________

Mupad [B]
time = 1.37, size = 303, normalized size = 1.77 \begin {gather*} \frac {\sqrt {c+d\,x}\,\left (\frac {256\,d^4\,x^4}{315\,{\left (a\,d-b\,c\right )}^5}+\frac {630\,a^4\,d^4-840\,a^3\,b\,c\,d^3+756\,a^2\,b^2\,c^2\,d^2-360\,a\,b^3\,c^3\,d+70\,b^4\,c^4}{315\,b^4\,{\left (a\,d-b\,c\right )}^5}+\frac {x\,\left (1680\,a^3\,b\,d^4-1008\,a^2\,b^2\,c\,d^3+432\,a\,b^3\,c^2\,d^2-80\,b^4\,c^3\,d\right )}{315\,b^4\,{\left (a\,d-b\,c\right )}^5}+\frac {128\,d^3\,x^3\,\left (9\,a\,d-b\,c\right )}{315\,b\,{\left (a\,d-b\,c\right )}^5}+\frac {32\,d^2\,x^2\,\left (21\,a^2\,d^2-6\,a\,b\,c\,d+b^2\,c^2\right )}{105\,b^2\,{\left (a\,d-b\,c\right )}^5}\right )}{x^4\,\sqrt {a+b\,x}+\frac {a^4\,\sqrt {a+b\,x}}{b^4}+\frac {6\,a^2\,x^2\,\sqrt {a+b\,x}}{b^2}+\frac {4\,a\,x^3\,\sqrt {a+b\,x}}{b}+\frac {4\,a^3\,x\,\sqrt {a+b\,x}}{b^3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*x)^(11/2)*(c + d*x)^(1/2)),x)

[Out]

((c + d*x)^(1/2)*((256*d^4*x^4)/(315*(a*d - b*c)^5) + (630*a^4*d^4 + 70*b^4*c^4 + 756*a^2*b^2*c^2*d^2 - 360*a*
b^3*c^3*d - 840*a^3*b*c*d^3)/(315*b^4*(a*d - b*c)^5) + (x*(1680*a^3*b*d^4 - 80*b^4*c^3*d + 432*a*b^3*c^2*d^2 -
1008*a^2*b^2*c*d^3))/(315*b^4*(a*d - b*c)^5) + (128*d^3*x^3*(9*a*d - b*c))/(315*b*(a*d - b*c)^5) + (32*d^2*x^
2*(21*a^2*d^2 + b^2*c^2 - 6*a*b*c*d))/(105*b^2*(a*d - b*c)^5)))/(x^4*(a + b*x)^(1/2) + (a^4*(a + b*x)^(1/2))/b
^4 + (6*a^2*x^2*(a + b*x)^(1/2))/b^2 + (4*a*x^3*(a + b*x)^(1/2))/b + (4*a^3*x*(a + b*x)^(1/2))/b^3)

________________________________________________________________________________________