3.8.5 \(\int \frac {\sqrt {c+d x}}{x^4 \sqrt {a+b x}} \, dx\) [705]

Optimal. Leaf size=191 \[ -\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a x^3}+\frac {(5 b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{12 a^2 c x^2}-\frac {(5 b c-3 a d) (3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{24 a^3 c^2 x}+\frac {(b c-a d) \left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{7/2} c^{5/2}} \]

[Out]

1/8*(-a*d+b*c)*(a^2*d^2+2*a*b*c*d+5*b^2*c^2)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(7/2)/c^(5
/2)-1/3*(b*x+a)^(1/2)*(d*x+c)^(1/2)/x^3/a+1/12*(-a*d+5*b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^2/c/x^2-1/24*(-3*a*d
+5*b*c)*(a*d+3*b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^3/c^2/x

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Rubi [A]
time = 0.10, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {101, 156, 12, 95, 214} \begin {gather*} -\frac {\sqrt {a+b x} \sqrt {c+d x} (5 b c-3 a d) (a d+3 b c)}{24 a^3 c^2 x}+\frac {\sqrt {a+b x} \sqrt {c+d x} (5 b c-a d)}{12 a^2 c x^2}+\frac {(b c-a d) \left (a^2 d^2+2 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{7/2} c^{5/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[c + d*x]/(x^4*Sqrt[a + b*x]),x]

[Out]

-1/3*(Sqrt[a + b*x]*Sqrt[c + d*x])/(a*x^3) + ((5*b*c - a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(12*a^2*c*x^2) - ((5*
b*c - 3*a*d)*(3*b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(24*a^3*c^2*x) + ((b*c - a*d)*(5*b^2*c^2 + 2*a*b*c*d +
a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(8*a^(7/2)*c^(5/2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 101

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

Rule 156

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rubi steps

\begin {align*} \int \frac {\sqrt {c+d x}}{x^4 \sqrt {a+b x}} \, dx &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a x^3}+\frac {\int \frac {\frac {1}{2} (-5 b c+a d)-2 b d x}{x^3 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{3 a}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a x^3}+\frac {(5 b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{12 a^2 c x^2}-\frac {\int \frac {-\frac {1}{4} (5 b c-3 a d) (3 b c+a d)-\frac {1}{2} b d (5 b c-a d) x}{x^2 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{6 a^2 c}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a x^3}+\frac {(5 b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{12 a^2 c x^2}-\frac {(5 b c-3 a d) (3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{24 a^3 c^2 x}+\frac {\int -\frac {3 (b c-a d) \left (5 b^2 c^2+2 a b c d+a^2 d^2\right )}{8 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{6 a^3 c^2}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a x^3}+\frac {(5 b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{12 a^2 c x^2}-\frac {(5 b c-3 a d) (3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{24 a^3 c^2 x}-\frac {\left ((b c-a d) \left (5 b^2 c^2+2 a b c d+a^2 d^2\right )\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{16 a^3 c^2}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a x^3}+\frac {(5 b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{12 a^2 c x^2}-\frac {(5 b c-3 a d) (3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{24 a^3 c^2 x}-\frac {\left ((b c-a d) \left (5 b^2 c^2+2 a b c d+a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{8 a^3 c^2}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a x^3}+\frac {(5 b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{12 a^2 c x^2}-\frac {(5 b c-3 a d) (3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{24 a^3 c^2 x}+\frac {(b c-a d) \left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{7/2} c^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 10.10, size = 160, normalized size = 0.84 \begin {gather*} \frac {\sqrt {a+b x} \sqrt {c+d x} \left (-15 b^2 c^2 x^2+2 a b c x (5 c+2 d x)+a^2 \left (-8 c^2-2 c d x+3 d^2 x^2\right )\right )}{24 a^3 c^2 x^3}-\frac {\left (-5 b^3 c^3+3 a b^2 c^2 d+a^2 b c d^2+a^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{7/2} c^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c + d*x]/(x^4*Sqrt[a + b*x]),x]

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(-15*b^2*c^2*x^2 + 2*a*b*c*x*(5*c + 2*d*x) + a^2*(-8*c^2 - 2*c*d*x + 3*d^2*x^2)))
/(24*a^3*c^2*x^3) - ((-5*b^3*c^3 + 3*a*b^2*c^2*d + a^2*b*c*d^2 + a^3*d^3)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqr
t[a]*Sqrt[c + d*x])])/(8*a^(7/2)*c^(5/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(407\) vs. \(2(159)=318\).
time = 0.08, size = 408, normalized size = 2.14

method	result	size
default	\(-\frac {\sqrt {d x +c}\, \sqrt {b x +a}\, \left (3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{3} d^{3} x^{3}+3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} b c \,d^{2} x^{3}+9 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a \,b^{2} c^{2} d \,x^{3}-15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) b^{3} c^{3} x^{3}-6 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} d^{2} x^{2}-8 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a b c d \,x^{2}+30 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{2} c^{2} x^{2}+4 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} c d x -20 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a b \,c^{2} x +16 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} c^{2} \sqrt {a c}\right )}{48 a^{3} c^{2} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, x^{3} \sqrt {a c}}\)	\(408\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(1/2)/x^4/(b*x+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/48*(d*x+c)^(1/2)*(b*x+a)^(1/2)/a^3/c^2*(3*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a
^3*d^3*x^3+3*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a^2*b*c*d^2*x^3+9*ln((a*d*x+b*c*x
+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a*b^2*c^2*d*x^3-15*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*
x+a))^(1/2)+2*a*c)/x)*b^3*c^3*x^3-6*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^2*d^2*x^2-8*(a*c)^(1/2)*((d*x+c)*(b*
x+a))^(1/2)*a*b*c*d*x^2+30*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*b^2*c^2*x^2+4*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/
2)*a^2*c*d*x-20*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a*b*c^2*x+16*((d*x+c)*(b*x+a))^(1/2)*a^2*c^2*(a*c)^(1/2))/
((d*x+c)*(b*x+a))^(1/2)/x^3/(a*c)^(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)^(1/2)/x^4/(b*x+a)^(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

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Fricas [A]
time = 1.12, size = 440, normalized size = 2.30 \begin {gather*} \left [-\frac {3 \, {\left (5 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {a c} x^{3} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (8 \, a^{3} c^{3} + {\left (15 \, a b^{2} c^{3} - 4 \, a^{2} b c^{2} d - 3 \, a^{3} c d^{2}\right )} x^{2} - 2 \, {\left (5 \, a^{2} b c^{3} - a^{3} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, a^{4} c^{3} x^{3}}, -\frac {3 \, {\left (5 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-a c} x^{3} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left (8 \, a^{3} c^{3} + {\left (15 \, a b^{2} c^{3} - 4 \, a^{2} b c^{2} d - 3 \, a^{3} c d^{2}\right )} x^{2} - 2 \, {\left (5 \, a^{2} b c^{3} - a^{3} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, a^{4} c^{3} x^{3}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2)/x^4/(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

[-1/96*(3*(5*b^3*c^3 - 3*a*b^2*c^2*d - a^2*b*c*d^2 - a^3*d^3)*sqrt(a*c)*x^3*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*
c*d + a^2*d^2)*x^2 - 4*(2*a*c + (b*c + a*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(a*b*c^2 + a^2*c*d)*x
)/x^2) + 4*(8*a^3*c^3 + (15*a*b^2*c^3 - 4*a^2*b*c^2*d - 3*a^3*c*d^2)*x^2 - 2*(5*a^2*b*c^3 - a^3*c^2*d)*x)*sqrt
(b*x + a)*sqrt(d*x + c))/(a^4*c^3*x^3), -1/48*(3*(5*b^3*c^3 - 3*a*b^2*c^2*d - a^2*b*c*d^2 - a^3*d^3)*sqrt(-a*c
)*x^3*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c)/(a*b*c*d*x^2 + a^2*c^2 + (a*b*
c^2 + a^2*c*d)*x)) + 2*(8*a^3*c^3 + (15*a*b^2*c^3 - 4*a^2*b*c^2*d - 3*a^3*c*d^2)*x^2 - 2*(5*a^2*b*c^3 - a^3*c^
2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^4*c^3*x^3)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c + d x}}{x^{4} \sqrt {a + b x}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**(1/2)/x**4/(b*x+a)**(1/2),x)

[Out]

Integral(sqrt(c + d*x)/(x**4*sqrt(a + b*x)), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2140 vs. \(2 (159) = 318\).
time = 6.53, size = 2140, normalized size = 11.20 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2)/x^4/(b*x+a)^(1/2),x, algorithm="giac")

[Out]

1/24*(3*(5*sqrt(b*d)*b^6*c^3 - 3*sqrt(b*d)*a*b^5*c^2*d - sqrt(b*d)*a^2*b^4*c*d^2 - sqrt(b*d)*a^3*b^3*d^3)*arct
an(-1/2*(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)/(sqrt(-a*b*c*d)*b)
)/(sqrt(-a*b*c*d)*a^3*b*c^2) - 2*(15*sqrt(b*d)*b^16*c^8 - 94*sqrt(b*d)*a*b^15*c^7*d + 246*sqrt(b*d)*a^2*b^14*c
^6*d^2 - 342*sqrt(b*d)*a^3*b^13*c^5*d^3 + 260*sqrt(b*d)*a^4*b^12*c^4*d^4 - 90*sqrt(b*d)*a^5*b^11*c^3*d^5 - 6*s
qrt(b*d)*a^6*b^10*c^2*d^6 + 14*sqrt(b*d)*a^7*b^9*c*d^7 - 3*sqrt(b*d)*a^8*b^8*d^8 - 75*sqrt(b*d)*(sqrt(b*d)*sqr
t(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^14*c^7 + 225*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b
^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^13*c^6*d - 111*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a
)*b*d - a*b*d))^2*a^2*b^12*c^5*d^2 - 291*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b
*d))^2*a^3*b^11*c^4*d^3 + 399*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^4*
b^10*c^3*d^4 - 141*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^5*b^9*c^2*d^5
- 21*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^6*b^8*c*d^6 + 15*sqrt(b*d)
*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^7*b^7*d^7 + 150*sqrt(b*d)*(sqrt(b*d)*sqrt
(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^12*c^6 - 120*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^
2*c + (b*x + a)*b*d - a*b*d))^4*a*b^11*c^5*d - 42*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*
b*d - a*b*d))^4*a^2*b^10*c^4*d^2 - 240*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d
))^4*a^3*b^9*c^3*d^3 + 306*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^4*b^8
*c^2*d^4 - 24*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^5*b^7*c*d^5 - 30*s
qrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^6*b^6*d^6 - 150*sqrt(b*d)*(sqrt(b
*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*b^10*c^5 - 50*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) -
sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a*b^9*c^4*d - 36*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x
+ a)*b*d - a*b*d))^6*a^2*b^8*c^3*d^2 - 108*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d -
a*b*d))^6*a^3*b^7*c^2*d^3 + 58*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^4
*b^6*c*d^4 + 30*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^5*b^5*d^5 + 75*s
qrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*b^8*c^4 + 30*sqrt(b*d)*(sqrt(b*d)*s
qrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a*b^7*c^3*d - 60*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sq
rt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a^2*b^6*c^2*d^2 - 30*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b
*x + a)*b*d - a*b*d))^8*a^3*b^5*c*d^3 - 15*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a
*b*d))^8*a^4*b^4*d^4 - 15*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*b^6*c^3
+ 9*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a*b^5*c^2*d + 3*sqrt(b*d)*(s
qrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^2*b^4*c*d^2 + 3*sqrt(b*d)*(sqrt(b*d)*sqrt(b
*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^3*b^3*d^3)/((b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2 - 2*(sqrt
(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^2*c - 2*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c +
(b*x + a)*b*d - a*b*d))^2*a*b*d + (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4)^3*a^3*c^
2))*abs(b)/b^3

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Mupad [B]
time = 91.00, size = 1574, normalized size = 8.24 \begin {gather*} \frac {\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4\,\left (\frac {5\,a^4\,b^2\,d^4}{64}+\frac {a^3\,b^3\,c\,d^3}{2}+\frac {7\,a^2\,b^4\,c^2\,d^2}{16}-\frac {23\,a\,b^5\,c^3\,d}{16}-\frac {13\,b^6\,c^4}{64}\right )}{a^4\,c^3\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}-\frac {b^6}{192\,a^2\,c\,d^3}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^5\,\left (-\frac {a^5\,b\,d^5}{64}+\frac {7\,a^4\,b^2\,c\,d^4}{16}+\frac {37\,a^3\,b^3\,c^2\,d^3}{32}-\frac {47\,a^2\,b^4\,c^3\,d^2}{32}-\frac {3\,a\,b^5\,c^4\,d}{2}+\frac {17\,b^6\,c^5}{64}\right )}{a^{9/2}\,c^{7/2}\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^5}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2\,\left (\frac {a^2\,b^4\,d^2}{64}+\frac {3\,a\,b^5\,c\,d}{32}-\frac {5\,b^6\,c^2}{64}\right )}{a^3\,c^2\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^7\,\left (\frac {3\,a^5\,d^5}{64}-\frac {13\,a^4\,b\,c\,d^4}{64}-\frac {49\,a^3\,b^2\,c^2\,d^3}{64}+\frac {69\,a^2\,b^3\,c^3\,d^2}{64}+\frac {45\,a\,b^4\,c^4\,d}{64}-\frac {27\,b^5\,c^5}{64}\right )}{a^{9/2}\,c^{7/2}\,d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^7}-\frac {\left (\frac {b^6\,c}{64}-\frac {a\,b^5\,d}{64}\right )\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{a^{5/2}\,c^{3/2}\,d^3\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3\,\left (-\frac {17\,a^3\,b^3\,d^3}{192}-\frac {9\,a^2\,b^4\,c\,d^2}{32}+\frac {3\,a\,b^5\,c^2\,d}{32}+\frac {73\,b^6\,c^3}{192}\right )}{a^{7/2}\,c^{5/2}\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^3}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6\,\left (-\frac {5\,a^6\,d^6}{192}+\frac {a^5\,b\,c\,d^5}{16}+a^4\,b^2\,c^2\,d^4+\frac {5\,a^3\,b^3\,c^3\,d^3}{32}-\frac {5\,a^2\,b^4\,c^4\,d^2}{2}+\frac {a\,b^5\,c^5\,d}{8}+\frac {37\,b^6\,c^6}{192}\right )}{a^5\,c^4\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^6}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^8\,\left (-\frac {a^4\,d^4}{64}+\frac {3\,a^3\,b\,c\,d^3}{32}+\frac {13\,a^2\,b^2\,c^2\,d^2}{64}-\frac {19\,a\,b^3\,c^3\,d}{32}+\frac {15\,b^4\,c^4}{64}\right )}{a^4\,c^3\,d\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^8}}{\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^9}{{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^9}+\frac {b^3\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3}{d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^3}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^7\,\left (3\,a^2\,d^2+9\,a\,b\,c\,d+3\,b^2\,c^2\right )}{a\,c\,d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^7}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6\,\left (a^3\,d^3+9\,a^2\,b\,c\,d^2+9\,a\,b^2\,c^2\,d+b^3\,c^3\right )}{a^{3/2}\,c^{3/2}\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^6}-\frac {\left (3\,a\,d+3\,b\,c\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^8}{\sqrt {a}\,\sqrt {c}\,d\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^8}-\frac {\left (3\,c\,b^3+3\,a\,d\,b^2\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{\sqrt {a}\,\sqrt {c}\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^5\,\left (3\,a^2\,b\,d^2+9\,a\,b^2\,c\,d+3\,b^3\,c^2\right )}{a\,c\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^5}}+\frac {\left (\frac {d^3}{32\,a^{3/2}\,c^{3/2}}-\frac {d^2\,\left (a\,d+b\,c\right )}{32\,a^{5/2}\,c^{3/2}}\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}+\frac {\left (\sqrt {a+b\,x}-\sqrt {a}\right )\,\left (\frac {d\,\left (3\,a^2\,d^2+8\,a\,b\,c\,d+3\,b^2\,c^2\right )}{32\,a^3\,c^2}-\frac {4\,a\,d^7-3\,b\,c\,d^6}{64\,a^2\,c^2\,d^4}+\frac {4\,\left (\frac {d^3}{16\,a^{3/2}\,c^{3/2}}-\frac {d^2\,\left (a\,d+b\,c\right )}{16\,a^{5/2}\,c^{3/2}}\right )\,\left (a\,d+b\,c\right )}{\sqrt {a}\,\sqrt {c}\,d}\right )}{\sqrt {c+d\,x}-\sqrt {c}}+\frac {\ln \left (\frac {\left (\sqrt {c}\,\sqrt {a+b\,x}-\sqrt {a}\,\sqrt {c+d\,x}\right )\,\left (b\,\sqrt {c}-\frac {\sqrt {a}\,d\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {c+d\,x}-\sqrt {c}}\right )}{\sqrt {c+d\,x}-\sqrt {c}}\right )\,\left (a^{7/2}\,\sqrt {c}\,d^3-5\,\sqrt {a}\,b^3\,c^{7/2}+3\,a^{3/2}\,b^2\,c^{5/2}\,d+a^{5/2}\,b\,c^{3/2}\,d^2\right )}{16\,a^4\,c^3}-\frac {\ln \left (\frac {\sqrt {a+b\,x}-\sqrt {a}}{\sqrt {c+d\,x}-\sqrt {c}}\right )\,\left (a^{7/2}\,\sqrt {c}\,d^3-5\,\sqrt {a}\,b^3\,c^{7/2}+3\,a^{3/2}\,b^2\,c^{5/2}\,d+a^{5/2}\,b\,c^{3/2}\,d^2\right )}{16\,a^4\,c^3}-\frac {d^3\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3}{192\,a^2\,c\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^(1/2)/(x^4*(a + b*x)^(1/2)),x)

[Out]

((((a + b*x)^(1/2) - a^(1/2))^4*((5*a^4*b^2*d^4)/64 - (13*b^6*c^4)/64 + (a^3*b^3*c*d^3)/2 + (7*a^2*b^4*c^2*d^2
)/16 - (23*a*b^5*c^3*d)/16))/(a^4*c^3*d^3*((c + d*x)^(1/2) - c^(1/2))^4) - b^6/(192*a^2*c*d^3) - (((a + b*x)^(
1/2) - a^(1/2))^5*((17*b^6*c^5)/64 - (a^5*b*d^5)/64 + (7*a^4*b^2*c*d^4)/16 - (47*a^2*b^4*c^3*d^2)/32 + (37*a^3
*b^3*c^2*d^3)/32 - (3*a*b^5*c^4*d)/2))/(a^(9/2)*c^(7/2)*d^3*((c + d*x)^(1/2) - c^(1/2))^5) + (((a + b*x)^(1/2)
- a^(1/2))^2*((a^2*b^4*d^2)/64 - (5*b^6*c^2)/64 + (3*a*b^5*c*d)/32))/(a^3*c^2*d^3*((c + d*x)^(1/2) - c^(1/2))
^2) + (((a + b*x)^(1/2) - a^(1/2))^7*((3*a^5*d^5)/64 - (27*b^5*c^5)/64 + (69*a^2*b^3*c^3*d^2)/64 - (49*a^3*b^2
*c^2*d^3)/64 + (45*a*b^4*c^4*d)/64 - (13*a^4*b*c*d^4)/64))/(a^(9/2)*c^(7/2)*d^2*((c + d*x)^(1/2) - c^(1/2))^7)
- (((b^6*c)/64 - (a*b^5*d)/64)*((a + b*x)^(1/2) - a^(1/2)))/(a^(5/2)*c^(3/2)*d^3*((c + d*x)^(1/2) - c^(1/2)))
+ (((a + b*x)^(1/2) - a^(1/2))^3*((73*b^6*c^3)/192 - (17*a^3*b^3*d^3)/192 - (9*a^2*b^4*c*d^2)/32 + (3*a*b^5*c
^2*d)/32))/(a^(7/2)*c^(5/2)*d^3*((c + d*x)^(1/2) - c^(1/2))^3) + (((a + b*x)^(1/2) - a^(1/2))^6*((37*b^6*c^6)/
192 - (5*a^6*d^6)/192 - (5*a^2*b^4*c^4*d^2)/2 + (5*a^3*b^3*c^3*d^3)/32 + a^4*b^2*c^2*d^4 + (a*b^5*c^5*d)/8 + (
a^5*b*c*d^5)/16))/(a^5*c^4*d^3*((c + d*x)^(1/2) - c^(1/2))^6) + (((a + b*x)^(1/2) - a^(1/2))^8*((15*b^4*c^4)/6
4 - (a^4*d^4)/64 + (13*a^2*b^2*c^2*d^2)/64 - (19*a*b^3*c^3*d)/32 + (3*a^3*b*c*d^3)/32))/(a^4*c^3*d*((c + d*x)^
(1/2) - c^(1/2))^8))/(((a + b*x)^(1/2) - a^(1/2))^9/((c + d*x)^(1/2) - c^(1/2))^9 + (b^3*((a + b*x)^(1/2) - a^
(1/2))^3)/(d^3*((c + d*x)^(1/2) - c^(1/2))^3) + (((a + b*x)^(1/2) - a^(1/2))^7*(3*a^2*d^2 + 3*b^2*c^2 + 9*a*b*
c*d))/(a*c*d^2*((c + d*x)^(1/2) - c^(1/2))^7) - (((a + b*x)^(1/2) - a^(1/2))^6*(a^3*d^3 + b^3*c^3 + 9*a*b^2*c^
2*d + 9*a^2*b*c*d^2))/(a^(3/2)*c^(3/2)*d^3*((c + d*x)^(1/2) - c^(1/2))^6) - ((3*a*d + 3*b*c)*((a + b*x)^(1/2)
- a^(1/2))^8)/(a^(1/2)*c^(1/2)*d*((c + d*x)^(1/2) - c^(1/2))^8) - ((3*b^3*c + 3*a*b^2*d)*((a + b*x)^(1/2) - a^
(1/2))^4)/(a^(1/2)*c^(1/2)*d^3*((c + d*x)^(1/2) - c^(1/2))^4) + (((a + b*x)^(1/2) - a^(1/2))^5*(3*b^3*c^2 + 3*
a^2*b*d^2 + 9*a*b^2*c*d))/(a*c*d^3*((c + d*x)^(1/2) - c^(1/2))^5)) + ((d^3/(32*a^(3/2)*c^(3/2)) - (d^2*(a*d +
b*c))/(32*a^(5/2)*c^(3/2)))*((a + b*x)^(1/2) - a^(1/2))^2)/((c + d*x)^(1/2) - c^(1/2))^2 + (((a + b*x)^(1/2) -
a^(1/2))*((d*(3*a^2*d^2 + 3*b^2*c^2 + 8*a*b*c*d))/(32*a^3*c^2) - (4*a*d^7 - 3*b*c*d^6)/(64*a^2*c^2*d^4) + (4*
(d^3/(16*a^(3/2)*c^(3/2)) - (d^2*(a*d + b*c))/(16*a^(5/2)*c^(3/2)))*(a*d + b*c))/(a^(1/2)*c^(1/2)*d)))/((c + d
*x)^(1/2) - c^(1/2)) + (log(((c^(1/2)*(a + b*x)^(1/2) - a^(1/2)*(c + d*x)^(1/2))*(b*c^(1/2) - (a^(1/2)*d*((a +
b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2))))/((c + d*x)^(1/2) - c^(1/2)))*(a^(7/2)*c^(1/2)*d^3 - 5*a^
(1/2)*b^3*c^(7/2) + 3*a^(3/2)*b^2*c^(5/2)*d + a^(5/2)*b*c^(3/2)*d^2))/(16*a^4*c^3) - (log(((a + b*x)^(1/2) - a
^(1/2))/((c + d*x)^(1/2) - c^(1/2)))*(a^(7/2)*c^(1/2)*d^3 - 5*a^(1/2)*b^3*c^(7/2) + 3*a^(3/2)*b^2*c^(5/2)*d +
a^(5/2)*b*c^(3/2)*d^2))/(16*a^4*c^3) - (d^3*((a + b*x)^(1/2) - a^(1/2))^3)/(192*a^2*c*((c + d*x)^(1/2) - c^(1/
2))^3)

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