3.15.80 \(\int \frac {(c+d x)^{3/2}}{(a+b x)^{13/2}} \, dx\) [1480]

Optimal. Leaf size=136 \[ -\frac {2 (c+d x)^{5/2}}{11 (b c-a d) (a+b x)^{11/2}}+\frac {4 d (c+d x)^{5/2}}{33 (b c-a d)^2 (a+b x)^{9/2}}-\frac {16 d^2 (c+d x)^{5/2}}{231 (b c-a d)^3 (a+b x)^{7/2}}+\frac {32 d^3 (c+d x)^{5/2}}{1155 (b c-a d)^4 (a+b x)^{5/2}} \]

[Out]

-2/11*(d*x+c)^(5/2)/(-a*d+b*c)/(b*x+a)^(11/2)+4/33*d*(d*x+c)^(5/2)/(-a*d+b*c)^2/(b*x+a)^(9/2)-16/231*d^2*(d*x+
c)^(5/2)/(-a*d+b*c)^3/(b*x+a)^(7/2)+32/1155*d^3*(d*x+c)^(5/2)/(-a*d+b*c)^4/(b*x+a)^(5/2)

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Rubi [A]
time = 0.02, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 4, 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 {32 d^3 (c+d x)^{5/2}}{1155 (a+b x)^{5/2} (b c-a d)^4}-\frac {16 d^2 (c+d x)^{5/2}}{231 (a+b x)^{7/2} (b c-a d)^3}+\frac {4 d (c+d x)^{5/2}}{33 (a+b x)^{9/2} (b c-a d)^2}-\frac {2 (c+d x)^{5/2}}{11 (a+b x)^{11/2} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^(3/2)/(a + b*x)^(13/2),x]

[Out]

(-2*(c + d*x)^(5/2))/(11*(b*c - a*d)*(a + b*x)^(11/2)) + (4*d*(c + d*x)^(5/2))/(33*(b*c - a*d)^2*(a + b*x)^(9/
2)) - (16*d^2*(c + d*x)^(5/2))/(231*(b*c - a*d)^3*(a + b*x)^(7/2)) + (32*d^3*(c + d*x)^(5/2))/(1155*(b*c - a*d
)^4*(a + b*x)^(5/2))

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 {(c+d x)^{3/2}}{(a+b x)^{13/2}} \, dx &=-\frac {2 (c+d x)^{5/2}}{11 (b c-a d) (a+b x)^{11/2}}-\frac {(6 d) \int \frac {(c+d x)^{3/2}}{(a+b x)^{11/2}} \, dx}{11 (b c-a d)}\\ &=-\frac {2 (c+d x)^{5/2}}{11 (b c-a d) (a+b x)^{11/2}}+\frac {4 d (c+d x)^{5/2}}{33 (b c-a d)^2 (a+b x)^{9/2}}+\frac {\left (8 d^2\right ) \int \frac {(c+d x)^{3/2}}{(a+b x)^{9/2}} \, dx}{33 (b c-a d)^2}\\ &=-\frac {2 (c+d x)^{5/2}}{11 (b c-a d) (a+b x)^{11/2}}+\frac {4 d (c+d x)^{5/2}}{33 (b c-a d)^2 (a+b x)^{9/2}}-\frac {16 d^2 (c+d x)^{5/2}}{231 (b c-a d)^3 (a+b x)^{7/2}}-\frac {\left (16 d^3\right ) \int \frac {(c+d x)^{3/2}}{(a+b x)^{7/2}} \, dx}{231 (b c-a d)^3}\\ &=-\frac {2 (c+d x)^{5/2}}{11 (b c-a d) (a+b x)^{11/2}}+\frac {4 d (c+d x)^{5/2}}{33 (b c-a d)^2 (a+b x)^{9/2}}-\frac {16 d^2 (c+d x)^{5/2}}{231 (b c-a d)^3 (a+b x)^{7/2}}+\frac {32 d^3 (c+d x)^{5/2}}{1155 (b c-a d)^4 (a+b x)^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 95, normalized size = 0.70 \begin {gather*} -\frac {2 (c+d x)^{11/2} \left (105 b^3-\frac {231 d^3 (a+b x)^3}{(c+d x)^3}+\frac {495 b d^2 (a+b x)^2}{(c+d x)^2}-\frac {385 b^2 d (a+b x)}{c+d x}\right )}{1155 (b c-a d)^4 (a+b x)^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^(3/2)/(a + b*x)^(13/2),x]

[Out]

(-2*(c + d*x)^(11/2)*(105*b^3 - (231*d^3*(a + b*x)^3)/(c + d*x)^3 + (495*b*d^2*(a + b*x)^2)/(c + d*x)^2 - (385
*b^2*d*(a + b*x))/(c + d*x)))/(1155*(b*c - a*d)^4*(a + b*x)^(11/2))

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Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[(c + d*x)^(3/2)/(a + b*x)^(13/2),x]')

[Out]

Timed out

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(280\) vs. \(2(112)=224\).
time = 0.17, size = 281, normalized size = 2.07

method	result	size
gosper	\(\frac {2 \left (d x +c \right )^{\frac {5}{2}} \left (16 b^{3} x^{3} d^{3}+88 d^{3} a \,x^{2} b^{2}-40 b^{3} c \,d^{2} x^{2}+198 a^{2} b \,d^{3} x -220 a \,b^{2} c \,d^{2} x +70 b^{3} c^{2} d x +231 a^{3} d^{3}-495 a^{2} b c \,d^{2}+385 a \,b^{2} c^{2} d -105 b^{3} c^{3}\right )}{1155 \left (b x +a \right )^{\frac {11}{2}} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\)	\(171\)
default	\(-\frac {\left (d x +c \right )^{\frac {3}{2}}}{4 b \left (b x +a \right )^{\frac {11}{2}}}+\frac {3 \left (a d -b c \right ) \left (-\frac {\sqrt {d x +c}}{5 b \left (b x +a \right )^{\frac {11}{2}}}+\frac {\left (a d -b c \right ) \left (-\frac {2 \sqrt {d x +c}}{11 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {11}{2}}}-\frac {10 d \left (-\frac {2 \sqrt {d x +c}}{9 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {9}{2}}}-\frac {8 d \left (-\frac {2 \sqrt {d x +c}}{7 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {7}{2}}}-\frac {6 d \left (-\frac {2 \sqrt {d x +c}}{5 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {5}{2}}}-\frac {4 d \left (-\frac {2 \sqrt {d x +c}}{3 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {3}{2}}}+\frac {4 d \sqrt {d x +c}}{3 \left (-a d +b c \right )^{2} \sqrt {b x +a}}\right )}{5 \left (-a d +b c \right )}\right )}{7 \left (-a d +b c \right )}\right )}{9 \left (-a d +b c \right )}\right )}{11 \left (-a d +b c \right )}\right )}{10 b}\right )}{8 b}\)	\(281\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(3/2)/(b*x+a)^(13/2),x,method=_RETURNVERBOSE)

[Out]

-1/4/b*(d*x+c)^(3/2)/(b*x+a)^(11/2)+3/8*(a*d-b*c)/b*(-1/5/b*(d*x+c)^(1/2)/(b*x+a)^(11/2)+1/10*(a*d-b*c)/b*(-2/
11/(-a*d+b*c)/(b*x+a)^(11/2)*(d*x+c)^(1/2)-10/11*d/(-a*d+b*c)*(-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)))))))

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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((d*x+c)^(3/2)/(b*x+a)^(13/2),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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 649 vs. \(2 (112) = 224\).
time = 5.50, size = 649, normalized size = 4.77 \begin {gather*} \frac {2 \, {\left (16 \, b^{3} d^{5} x^{5} - 105 \, b^{3} c^{5} + 385 \, a b^{2} c^{4} d - 495 \, a^{2} b c^{3} d^{2} + 231 \, a^{3} c^{2} d^{3} - 8 \, {\left (b^{3} c d^{4} - 11 \, a b^{2} d^{5}\right )} x^{4} + 2 \, {\left (3 \, b^{3} c^{2} d^{3} - 22 \, a b^{2} c d^{4} + 99 \, a^{2} b d^{5}\right )} x^{3} - {\left (5 \, b^{3} c^{3} d^{2} - 33 \, a b^{2} c^{2} d^{3} + 99 \, a^{2} b c d^{4} - 231 \, a^{3} d^{5}\right )} x^{2} - 2 \, {\left (70 \, b^{3} c^{4} d - 275 \, a b^{2} c^{3} d^{2} + 396 \, a^{2} b c^{2} d^{3} - 231 \, a^{3} c d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{1155 \, {\left (a^{6} b^{4} c^{4} - 4 \, a^{7} b^{3} c^{3} d + 6 \, a^{8} b^{2} c^{2} d^{2} - 4 \, a^{9} b c d^{3} + a^{10} d^{4} + {\left (b^{10} c^{4} - 4 \, a b^{9} c^{3} d + 6 \, a^{2} b^{8} c^{2} d^{2} - 4 \, a^{3} b^{7} c d^{3} + a^{4} b^{6} d^{4}\right )} x^{6} + 6 \, {\left (a b^{9} c^{4} - 4 \, a^{2} b^{8} c^{3} d + 6 \, a^{3} b^{7} c^{2} d^{2} - 4 \, a^{4} b^{6} c d^{3} + a^{5} b^{5} d^{4}\right )} x^{5} + 15 \, {\left (a^{2} b^{8} c^{4} - 4 \, a^{3} b^{7} c^{3} d + 6 \, a^{4} b^{6} c^{2} d^{2} - 4 \, a^{5} b^{5} c d^{3} + a^{6} b^{4} d^{4}\right )} x^{4} + 20 \, {\left (a^{3} b^{7} c^{4} - 4 \, a^{4} b^{6} c^{3} d + 6 \, a^{5} b^{5} c^{2} d^{2} - 4 \, a^{6} b^{4} c d^{3} + a^{7} b^{3} d^{4}\right )} x^{3} + 15 \, {\left (a^{4} b^{6} c^{4} - 4 \, a^{5} b^{5} c^{3} d + 6 \, a^{6} b^{4} c^{2} d^{2} - 4 \, a^{7} b^{3} c d^{3} + a^{8} b^{2} d^{4}\right )} x^{2} + 6 \, {\left (a^{5} b^{5} c^{4} - 4 \, a^{6} b^{4} c^{3} d + 6 \, a^{7} b^{3} c^{2} d^{2} - 4 \, a^{8} b^{2} c d^{3} + a^{9} b d^{4}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(3/2)/(b*x+a)^(13/2),x, algorithm="fricas")

[Out]

2/1155*(16*b^3*d^5*x^5 - 105*b^3*c^5 + 385*a*b^2*c^4*d - 495*a^2*b*c^3*d^2 + 231*a^3*c^2*d^3 - 8*(b^3*c*d^4 -
11*a*b^2*d^5)*x^4 + 2*(3*b^3*c^2*d^3 - 22*a*b^2*c*d^4 + 99*a^2*b*d^5)*x^3 - (5*b^3*c^3*d^2 - 33*a*b^2*c^2*d^3
+ 99*a^2*b*c*d^4 - 231*a^3*d^5)*x^2 - 2*(70*b^3*c^4*d - 275*a*b^2*c^3*d^2 + 396*a^2*b*c^2*d^3 - 231*a^3*c*d^4)
*x)*sqrt(b*x + a)*sqrt(d*x + c)/(a^6*b^4*c^4 - 4*a^7*b^3*c^3*d + 6*a^8*b^2*c^2*d^2 - 4*a^9*b*c*d^3 + a^10*d^4
+ (b^10*c^4 - 4*a*b^9*c^3*d + 6*a^2*b^8*c^2*d^2 - 4*a^3*b^7*c*d^3 + a^4*b^6*d^4)*x^6 + 6*(a*b^9*c^4 - 4*a^2*b^
8*c^3*d + 6*a^3*b^7*c^2*d^2 - 4*a^4*b^6*c*d^3 + a^5*b^5*d^4)*x^5 + 15*(a^2*b^8*c^4 - 4*a^3*b^7*c^3*d + 6*a^4*b
^6*c^2*d^2 - 4*a^5*b^5*c*d^3 + a^6*b^4*d^4)*x^4 + 20*(a^3*b^7*c^4 - 4*a^4*b^6*c^3*d + 6*a^5*b^5*c^2*d^2 - 4*a^
6*b^4*c*d^3 + a^7*b^3*d^4)*x^3 + 15*(a^4*b^6*c^4 - 4*a^5*b^5*c^3*d + 6*a^6*b^4*c^2*d^2 - 4*a^7*b^3*c*d^3 + a^8
*b^2*d^4)*x^2 + 6*(a^5*b^5*c^4 - 4*a^6*b^4*c^3*d + 6*a^7*b^3*c^2*d^2 - 4*a^8*b^2*c*d^3 + a^9*b*d^4)*x)

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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((d*x+c)**(3/2)/(b*x+a)**(13/2),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 546 vs. \(2 (112) = 224\).
time = 0.25, size = 671, normalized size = 4.93 \begin {gather*} \frac {2 \left (\left (\left (-\frac {\left (1496880 b^{9} d^{12} c-1496880 b^{8} d^{13} a\right ) \sqrt {c+d x} \sqrt {c+d x}}{-108056025 b^{10} c^{5} \left |d\right |[phone] b^{9} d a c^{4} \left |d\right |-1080560250 b^{8} d^{2} a^{2} c^{3} \left |d\right |[phone] b^{7} d^{3} a^{3} c^{2} \left |d\right |-540280125 b^{6} d^{4} a^{4} c \left |d\right |[phone] b^{5} d^{5} a^{5} \left |d\right |}-\frac {-8232840 b^{9} d^{12} c^{2}[phone] b^{8} d^{13} a c-8232840 b^{7} d^{14} a^{2}}{-108056025 b^{10} c^{5} \left |d\right |[phone] b^{9} d a c^{4} \left |d\right |-1080560250 b^{8} d^{2} a^{2} c^{3} \left |d\right |[phone] b^{7} d^{3} a^{3} c^{2} \left |d\right |-540280125 b^{6} d^{4} a^{4} c \left |d\right |[phone] b^{5} d^{5} a^{5} \left |d\right |}\right ) \sqrt {c+d x} \sqrt {c+d x}-\frac {18523890 b^{9} d^{12} c^{3}-55571670 b^{8} d^{13} a c^{2}[phone] b^{7} d^{14} a^{2} c-18523890 b^{6} d^{15} a^{3}}{-108056025 b^{10} c^{5} \left |d\right |[phone] b^{9} d a c^{4} \left |d\right |-1080560250 b^{8} d^{2} a^{2} c^{3} \left |d\right |[phone] b^{7} d^{3} a^{3} c^{2} \left |d\right |-540280125 b^{6} d^{4} a^{4} c \left |d\right |[phone] b^{5} d^{5} a^{5} \left |d\right |}\right ) \sqrt {c+d x} \sqrt {c+d x}-\frac {-21611205 b^{9} d^{12} c^{4}[phone] b^{8} d^{13} a c^{3}-129667230 b^{7} d^{14} a^{2} c^{2}[phone] b^{6} d^{15} a^{3} c-21611205 b^{5} d^{16} a^{4}}{-108056025 b^{10} c^{5} \left |d\right |[phone] b^{9} d a c^{4} \left |d\right |-1080560250 b^{8} d^{2} a^{2} c^{3} \left |d\right |[phone] b^{7} d^{3} a^{3} c^{2} \left |d\right |-540280125 b^{6} d^{4} a^{4} c \left |d\right |[phone] b^{5} d^{5} a^{5} \left |d\right |}\right ) \sqrt {c+d x} \sqrt {c+d x} \sqrt {c+d x} \sqrt {c+d x} \sqrt {c+d x} \sqrt {a d^{2}-b c d+b d \left (c+d x\right )}}{\left (a d^{2}-b c d+b d \left (c+d x\right )\right )^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(3/2)/(b*x+a)^(13/2),x)

[Out]

2/1155*(2*(4*(d*x + c)*(2*(b^9*c*d^12 - a*b^8*d^13)*(d*x + c)/(b^10*c^5*abs(d) - 5*a*b^9*c^4*d*abs(d) + 10*a^2
*b^8*c^3*d^2*abs(d) - 10*a^3*b^7*c^2*d^3*abs(d) + 5*a^4*b^6*c*d^4*abs(d) - a^5*b^5*d^5*abs(d)) - 11*(b^9*c^2*d
^12 - 2*a*b^8*c*d^13 + a^2*b^7*d^14)/(b^10*c^5*abs(d) - 5*a*b^9*c^4*d*abs(d) + 10*a^2*b^8*c^3*d^2*abs(d) - 10*
a^3*b^7*c^2*d^3*abs(d) + 5*a^4*b^6*c*d^4*abs(d) - a^5*b^5*d^5*abs(d))) + 99*(b^9*c^3*d^12 - 3*a*b^8*c^2*d^13 +
3*a^2*b^7*c*d^14 - a^3*b^6*d^15)/(b^10*c^5*abs(d) - 5*a*b^9*c^4*d*abs(d) + 10*a^2*b^8*c^3*d^2*abs(d) - 10*a^3
*b^7*c^2*d^3*abs(d) + 5*a^4*b^6*c*d^4*abs(d) - a^5*b^5*d^5*abs(d)))*(d*x + c) - 231*(b^9*c^4*d^12 - 4*a*b^8*c^
3*d^13 + 6*a^2*b^7*c^2*d^14 - 4*a^3*b^6*c*d^15 + a^4*b^5*d^16)/(b^10*c^5*abs(d) - 5*a*b^9*c^4*d*abs(d) + 10*a^
2*b^8*c^3*d^2*abs(d) - 10*a^3*b^7*c^2*d^3*abs(d) + 5*a^4*b^6*c*d^4*abs(d) - a^5*b^5*d^5*abs(d)))*(d*x + c)^(5/
2)/((d*x + c)*b*d - b*c*d + a*d^2)^(11/2)

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Mupad [B]
time = 1.33, size = 376, normalized size = 2.76 \begin {gather*} \frac {\sqrt {c+d\,x}\,\left (\frac {x^2\,\left (462\,a^3\,d^5-198\,a^2\,b\,c\,d^4+66\,a\,b^2\,c^2\,d^3-10\,b^3\,c^3\,d^2\right )}{1155\,b^5\,{\left (a\,d-b\,c\right )}^4}-\frac {-462\,a^3\,c^2\,d^3+990\,a^2\,b\,c^3\,d^2-770\,a\,b^2\,c^4\,d+210\,b^3\,c^5}{1155\,b^5\,{\left (a\,d-b\,c\right )}^4}+\frac {x\,\left (924\,a^3\,c\,d^4-1584\,a^2\,b\,c^2\,d^3+1100\,a\,b^2\,c^3\,d^2-280\,b^3\,c^4\,d\right )}{1155\,b^5\,{\left (a\,d-b\,c\right )}^4}+\frac {32\,d^5\,x^5}{1155\,b^2\,{\left (a\,d-b\,c\right )}^4}+\frac {16\,d^4\,x^4\,\left (11\,a\,d-b\,c\right )}{1155\,b^3\,{\left (a\,d-b\,c\right )}^4}+\frac {4\,d^3\,x^3\,\left (99\,a^2\,d^2-22\,a\,b\,c\,d+3\,b^2\,c^2\right )}{1155\,b^4\,{\left (a\,d-b\,c\right )}^4}\right )}{x^5\,\sqrt {a+b\,x}+\frac {a^5\,\sqrt {a+b\,x}}{b^5}+\frac {10\,a^2\,x^3\,\sqrt {a+b\,x}}{b^2}+\frac {10\,a^3\,x^2\,\sqrt {a+b\,x}}{b^3}+\frac {5\,a\,x^4\,\sqrt {a+b\,x}}{b}+\frac {5\,a^4\,x\,\sqrt {a+b\,x}}{b^4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^(3/2)/(a + b*x)^(13/2),x)

[Out]

((c + d*x)^(1/2)*((x^2*(462*a^3*d^5 - 10*b^3*c^3*d^2 + 66*a*b^2*c^2*d^3 - 198*a^2*b*c*d^4))/(1155*b^5*(a*d - b
*c)^4) - (210*b^3*c^5 - 462*a^3*c^2*d^3 + 990*a^2*b*c^3*d^2 - 770*a*b^2*c^4*d)/(1155*b^5*(a*d - b*c)^4) + (x*(
924*a^3*c*d^4 - 280*b^3*c^4*d + 1100*a*b^2*c^3*d^2 - 1584*a^2*b*c^2*d^3))/(1155*b^5*(a*d - b*c)^4) + (32*d^5*x
^5)/(1155*b^2*(a*d - b*c)^4) + (16*d^4*x^4*(11*a*d - b*c))/(1155*b^3*(a*d - b*c)^4) + (4*d^3*x^3*(99*a^2*d^2 +
3*b^2*c^2 - 22*a*b*c*d))/(1155*b^4*(a*d - b*c)^4)))/(x^5*(a + b*x)^(1/2) + (a^5*(a + b*x)^(1/2))/b^5 + (10*a^
2*x^3*(a + b*x)^(1/2))/b^2 + (10*a^3*x^2*(a + b*x)^(1/2))/b^3 + (5*a*x^4*(a + b*x)^(1/2))/b + (5*a^4*x*(a + b*
x)^(1/2))/b^4)

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