3.582 \(\int \frac {\sqrt {a+b x}}{x^3 \sqrt {c+d x}} \, dx\)

Optimal. Leaf size=131 \[ \frac {(b c-a d) (3 a d+b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{3/2} c^{5/2}}+\frac {\sqrt {a+b x} \sqrt {c+d x} (3 a d+b c)}{4 a c^2 x}-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 a c x^2} \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {96, 94, 93, 208} \[ \frac {(b c-a d) (3 a d+b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{3/2} c^{5/2}}+\frac {\sqrt {a+b x} \sqrt {c+d x} (3 a d+b c)}{4 a c^2 x}-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 a c x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b*c + 3*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(4*a*c^2*x) - ((a + b*x)^(3/2)*Sqrt[c + d*x])/(2*a*c*x^2) + ((b*c
- a*d)*(b*c + 3*a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(4*a^(3/2)*c^(5/2))

Rule 93

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 94

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[(n*(d*e - c*f))/((m + 1)*(b*e - a*
f)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] &&  !(SumSimplerQ[p, 1] &&  !SumSimplerQ[m, 1])

Rule 96

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(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[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 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, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

Rule 208

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

Rubi steps

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

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Mathematica [A]  time = 0.09, size = 113, normalized size = 0.86 \[ \frac {\left (-3 a^2 d^2+2 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{3/2} c^{5/2}}+\frac {\sqrt {a+b x} \sqrt {c+d x} (-2 a c+3 a d x-b c x)}{4 a c^2 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(-2*a*c - b*c*x + 3*a*d*x))/(4*a*c^2*x^2) + ((b^2*c^2 + 2*a*b*c*d - 3*a^2*d^2)*Ar
cTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(4*a^(3/2)*c^(5/2))

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fricas [A]  time = 1.18, size = 330, normalized size = 2.52 \[ \left [-\frac {{\left (b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \sqrt {a c} x^{2} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (2 \, a^{2} c^{2} + {\left (a b c^{2} - 3 \, a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, a^{2} c^{3} x^{2}}, -\frac {{\left (b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \sqrt {-a c} x^{2} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left (2 \, a^{2} c^{2} + {\left (a b c^{2} - 3 \, a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, a^{2} c^{3} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(1/2)/x^3/(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

[-1/16*((b^2*c^2 + 2*a*b*c*d - 3*a^2*d^2)*sqrt(a*c)*x^2*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*(2*a^2*c^
2 + (a*b*c^2 - 3*a^2*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*c^3*x^2), -1/8*((b^2*c^2 + 2*a*b*c*d - 3*a^2*d^
2)*sqrt(-a*c)*x^2*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*(2*a^2*c^2 + (a*b*c^2 - 3*a^2*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*c^3
*x^2)]

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giac [B]  time = 4.00, size = 1043, normalized size = 7.96 \[ \frac {b {\left (\frac {{\left (\sqrt {b d} b^{3} c^{2} + 2 \, \sqrt {b d} a b^{2} c d - 3 \, \sqrt {b d} a^{2} b d^{2}\right )} \arctan \left (-\frac {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}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} a b c^{2}} - \frac {2 \, {\left (\sqrt {b d} b^{9} c^{5} - 7 \, \sqrt {b d} a b^{8} c^{4} d + 18 \, \sqrt {b d} a^{2} b^{7} c^{3} d^{2} - 22 \, \sqrt {b d} a^{3} b^{6} c^{2} d^{3} + 13 \, \sqrt {b d} a^{4} b^{5} c d^{4} - 3 \, \sqrt {b d} a^{5} b^{4} d^{5} - 3 \, \sqrt {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} b^{7} c^{4} + 16 \, \sqrt {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} a b^{6} c^{3} d - 14 \, \sqrt {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} a^{2} b^{5} c^{2} d^{2} - 8 \, \sqrt {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} a^{3} b^{4} c d^{3} + 9 \, \sqrt {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} a^{4} b^{3} d^{4} + 3 \, \sqrt {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 )}^{4} b^{5} c^{3} - 7 \, \sqrt {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 )}^{4} a b^{4} c^{2} d - 3 \, \sqrt {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 )}^{4} a^{2} b^{3} c d^{2} - 9 \, \sqrt {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 )}^{4} a^{3} b^{2} d^{3} - \sqrt {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 )}^{6} b^{3} c^{2} - 2 \, \sqrt {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 )}^{6} a b^{2} c d + 3 \, \sqrt {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 )}^{6} a^{2} b d^{2}\right )}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\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^{2} c - 2 \, {\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 d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4}\right )}^{2} a c^{2}}\right )}}{4 \, {\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(1/2)/x^3/(d*x+c)^(1/2),x, algorithm="giac")

[Out]

1/4*b*((sqrt(b*d)*b^3*c^2 + 2*sqrt(b*d)*a*b^2*c*d - 3*sqrt(b*d)*a^2*b*d^2)*arctan(-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*b*c^2) - 2*
(sqrt(b*d)*b^9*c^5 - 7*sqrt(b*d)*a*b^8*c^4*d + 18*sqrt(b*d)*a^2*b^7*c^3*d^2 - 22*sqrt(b*d)*a^3*b^6*c^2*d^3 + 1
3*sqrt(b*d)*a^4*b^5*c*d^4 - 3*sqrt(b*d)*a^5*b^4*d^5 - 3*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x
+ a)*b*d - a*b*d))^2*b^7*c^4 + 16*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^6*c^3*d - 14*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^5*c^2*d^2
- 8*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^4*c*d^3 + 9*sqrt(b*d)*(s
qrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^4*b^3*d^4 + 3*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x
+ a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^5*c^3 - 7*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^4*c^2*d - 3*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^3*c*d^2 - 9*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^2*d^
3 - sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*b^3*c^2 - 2*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^2*c*d + 3*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - s
qrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^2*b*d^2)/((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)^2*a*c^2))/abs(b)

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maple [B]  time = 0.02, size = 258, normalized size = 1.97 \[ -\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (3 a^{2} d^{2} x^{2} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-2 a b c d \,x^{2} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-b^{2} c^{2} x^{2} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-6 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, b c x +4 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a c \right )}{8 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a \,c^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(1/2)/x^3/(d*x+c)^(1/2),x)

[Out]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(1/2)/x^3/(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 details)Is a*d-b*c zero or nonzero?

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mupad [B]  time = 19.77, size = 893, normalized size = 6.82 \[ \frac {\ln \left (\frac {\sqrt {a+b\,x}-\sqrt {a}}{\sqrt {c+d\,x}-\sqrt {c}}\right )\,\left (\sqrt {a}\,b^2\,c^{5/2}-3\,a^{5/2}\,\sqrt {c}\,d^2+2\,a^{3/2}\,b\,c^{3/2}\,d\right )}{8\,a^2\,c^3}-\frac {\left (\sqrt {a+b\,x}-\sqrt {a}\right )\,\left (\frac {b\,d}{4\,a\,c}-\frac {3\,d\,\left (a\,d+b\,c\right )}{16\,a\,c^2}\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 (\sqrt {a}\,b^2\,c^{5/2}-3\,a^{5/2}\,\sqrt {c}\,d^2+2\,a^{3/2}\,b\,c^{3/2}\,d\right )}{8\,a^2\,c^3}-\frac {\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^5\,\left (\frac {a^2\,d^2}{4}-\frac {11\,a\,b\,c\,d}{16}+\frac {5\,b^2\,c^2}{16}\right )}{a\,c^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^5}-\frac {b^4}{32\,\sqrt {a}\,c^{3/2}\,d^2}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2\,\left (\frac {11\,a^2\,b^2\,d^2}{32}+\frac {a\,b^3\,c\,d}{8}-\frac {5\,b^4\,c^2}{32}\right )}{a^{3/2}\,c^{5/2}\,d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3\,\left (\frac {a^3\,b\,d^3}{16}-\frac {9\,a^2\,b^2\,c\,d^2}{8}+\frac {3\,a\,b^3\,c^2\,d}{8}+\frac {b^4\,c^3}{16}\right )}{a^2\,c^3\,d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^3}+\frac {\left (\frac {b^4\,c}{8}-\frac {a\,b^3\,d}{8}\right )\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{a\,c^2\,d^2\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4\,\left (-\frac {7\,a^4\,d^4}{32}+\frac {a^3\,b\,c\,d^3}{2}+\frac {21\,a^2\,b^2\,c^2\,d^2}{32}-\frac {a\,b^3\,c^3\,d}{2}+\frac {b^4\,c^4}{32}\right )}{a^{5/2}\,c^{7/2}\,d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}}{\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6}{{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^6}+\frac {b^2\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4\,\left (a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )}{a\,c\,d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}-\frac {\left (2\,c\,b^2+2\,a\,d\,b\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3}{\sqrt {a}\,\sqrt {c}\,d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^3}-\frac {\left (2\,a\,d+2\,b\,c\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^5}{\sqrt {a}\,\sqrt {c}\,d\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^5}}+\frac {d^2\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{32\,\sqrt {a}\,c^{3/2}\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^(1/2)/(x^3*(c + d*x)^(1/2)),x)

[Out]

(log(((a + b*x)^(1/2) - a^(1/2))/((c + d*x)^(1/2) - c^(1/2)))*(a^(1/2)*b^2*c^(5/2) - 3*a^(5/2)*c^(1/2)*d^2 + 2
*a^(3/2)*b*c^(3/2)*d))/(8*a^2*c^3) - (((a + b*x)^(1/2) - a^(1/2))*((b*d)/(4*a*c) - (3*d*(a*d + b*c))/(16*a*c^2
)))/((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^(1/2)*b^2*c^(
5/2) - 3*a^(5/2)*c^(1/2)*d^2 + 2*a^(3/2)*b*c^(3/2)*d))/(8*a^2*c^3) - ((((a + b*x)^(1/2) - a^(1/2))^5*((a^2*d^2
)/4 + (5*b^2*c^2)/16 - (11*a*b*c*d)/16))/(a*c^3*((c + d*x)^(1/2) - c^(1/2))^5) - b^4/(32*a^(1/2)*c^(3/2)*d^2)
+ (((a + b*x)^(1/2) - a^(1/2))^2*((11*a^2*b^2*d^2)/32 - (5*b^4*c^2)/32 + (a*b^3*c*d)/8))/(a^(3/2)*c^(5/2)*d^2*
((c + d*x)^(1/2) - c^(1/2))^2) + (((a + b*x)^(1/2) - a^(1/2))^3*((b^4*c^3)/16 + (a^3*b*d^3)/16 - (9*a^2*b^2*c*
d^2)/8 + (3*a*b^3*c^2*d)/8))/(a^2*c^3*d^2*((c + d*x)^(1/2) - c^(1/2))^3) + (((b^4*c)/8 - (a*b^3*d)/8)*((a + b*
x)^(1/2) - a^(1/2)))/(a*c^2*d^2*((c + d*x)^(1/2) - c^(1/2))) + (((a + b*x)^(1/2) - a^(1/2))^4*((b^4*c^4)/32 -
(7*a^4*d^4)/32 + (21*a^2*b^2*c^2*d^2)/32 - (a*b^3*c^3*d)/2 + (a^3*b*c*d^3)/2))/(a^(5/2)*c^(7/2)*d^2*((c + d*x)
^(1/2) - c^(1/2))^4))/(((a + b*x)^(1/2) - a^(1/2))^6/((c + d*x)^(1/2) - c^(1/2))^6 + (b^2*((a + b*x)^(1/2) - a
^(1/2))^2)/(d^2*((c + d*x)^(1/2) - c^(1/2))^2) + (((a + b*x)^(1/2) - a^(1/2))^4*(a^2*d^2 + b^2*c^2 + 4*a*b*c*d
))/(a*c*d^2*((c + d*x)^(1/2) - c^(1/2))^4) - ((2*b^2*c + 2*a*b*d)*((a + b*x)^(1/2) - a^(1/2))^3)/(a^(1/2)*c^(1
/2)*d^2*((c + d*x)^(1/2) - c^(1/2))^3) - ((2*a*d + 2*b*c)*((a + b*x)^(1/2) - a^(1/2))^5)/(a^(1/2)*c^(1/2)*d*((
c + d*x)^(1/2) - c^(1/2))^5)) + (d^2*((a + b*x)^(1/2) - a^(1/2))^2)/(32*a^(1/2)*c^(3/2)*((c + d*x)^(1/2) - c^(
1/2))^2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a + b x}}{x^{3} \sqrt {c + d x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(1/2)/x**3/(d*x+c)**(1/2),x)

[Out]

Integral(sqrt(a + b*x)/(x**3*sqrt(c + d*x)), x)

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