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

Optimal. Leaf size=110 \[ \sqrt {a+b x} \sqrt {c+d x}-2 \sqrt {a} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {(a d+b c) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} \sqrt {d}} \]

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Rubi [A]  time = 0.07, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {101, 157, 63, 217, 206, 93, 208} \begin {gather*} \sqrt {a+b x} \sqrt {c+d x}-2 \sqrt {a} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {(a d+b c) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} \sqrt {d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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 101

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

Rule 157

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

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]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rubi steps

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

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Mathematica [A]  time = 0.61, size = 162, normalized size = 1.47 \begin {gather*} \frac {\sqrt {b c-a d} (a d+b c) \sqrt {\frac {b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )+b \sqrt {d} \left (\sqrt {a+b x} (c+d x)-2 \sqrt {a} \sqrt {c} \sqrt {c+d x} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )\right )}{b \sqrt {d} \sqrt {c+d x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[b*c - a*d]*(b*c + a*d)*Sqrt[(b*(c + d*x))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d]]
+ b*Sqrt[d]*(Sqrt[a + b*x]*(c + d*x) - 2*Sqrt[a]*Sqrt[c]*Sqrt[c + d*x]*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a
]*Sqrt[c + d*x])]))/(b*Sqrt[d]*Sqrt[c + d*x])

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IntegrateAlgebraic [A]  time = 0.26, size = 138, normalized size = 1.25 \begin {gather*} \frac {\sqrt {c+d x} (b c-a d)}{\sqrt {a+b x} \left (\frac {b (c+d x)}{a+b x}-d\right )}-2 \sqrt {a} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )+\frac {(a d+b c) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{\sqrt {b} \sqrt {d}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(Sqrt[a + b*x]*Sqrt[c + d*x])/x,x]

[Out]

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

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fricas [B]  time = 2.06, size = 822, normalized size = 7.47 \begin {gather*} \left [\frac {2 \, \sqrt {a c} b d \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 \, \sqrt {b x + a} \sqrt {d x + c} b d + {\left (b c + a d\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right )}{4 \, b d}, \frac {\sqrt {a c} b d \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 ) + 2 \, \sqrt {b x + a} \sqrt {d x + c} b d - {\left (b c + a d\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right )}{2 \, b d}, \frac {4 \, \sqrt {-a c} b d \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 ) + 4 \, \sqrt {b x + a} \sqrt {d x + c} b d + {\left (b c + a d\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right )}{4 \, b d}, \frac {2 \, \sqrt {-a c} b d \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 \, \sqrt {b x + a} \sqrt {d x + c} b d - {\left (b c + a d\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right )}{2 \, b d}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(2*sqrt(a*c)*b*d*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*sqrt(b*x + a)*sqrt(d*x + c)*b*d + (b*c + a*
d)*sqrt(b*d)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b*d*x + b*c + a*d)*sqrt(b*d)*sqrt(b*x +
a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x))/(b*d), 1/2*(sqrt(a*c)*b*d*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
) + 2*sqrt(b*x + a)*sqrt(d*x + c)*b*d - (b*c + a*d)*sqrt(-b*d)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)*sqr
t(b*x + a)*sqrt(d*x + c)/(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)))/(b*d), 1/4*(4*sqrt(-a*c)*b*d*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)) + 4*sqrt(b*x + a)*sqrt(d*x + c)*b*d + (b*c + a*d)*sqrt(b*d)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a
^2*d^2 + 4*(2*b*d*x + b*c + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x))/(b*d), 1/2*
(2*sqrt(-a*c)*b*d*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*sqrt(b*x + a)*sqrt(d*x + c)*b*d - (b*c + a*d)*sqrt(-b*d)*arctan(1/2*(2*b*d*
x + b*c + a*d)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c)/(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)))/(b*d)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:

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maple [B]  time = 0.02, size = 244, normalized size = 2.22 \begin {gather*} \frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (-2 \sqrt {b d}\, a c \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )+\sqrt {a c}\, a d \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+\sqrt {a c}\, b c \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\right )}{2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(ln(1/2*(2*b*d*x+a*d+b*c+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2))/(b*d)^
(1/2))*(a*c)^(1/2)*a*d+ln(1/2*(2*b*d*x+a*d+b*c+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2))/(b*d)^(1/2))*(a*
c)^(1/2)*b*c-2*a*c*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*(b*d)^(1/2)+2*(b*d*
x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*(a*c)^(1/2))/(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)/(b*d)^(1/2)/(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((b*x+a)^(1/2)*(d*x+c)^(1/2)/x,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 = 24.19, size = 4785, normalized size = 43.50

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(((2*a*d + 2*b*c)*((a + b*x)^(1/2) - a^(1/2))^3)/(d*((c + d*x)^(1/2) - c^(1/2))^3) + ((2*b^2*c + 2*a*b*d)*((a
+ b*x)^(1/2) - a^(1/2)))/(d^2*((c + d*x)^(1/2) - c^(1/2))) - (8*a^(1/2)*b*c^(1/2)*((a + b*x)^(1/2) - a^(1/2))^
2)/(d*((c + d*x)^(1/2) - c^(1/2))^2))/(((a + b*x)^(1/2) - a^(1/2))^4/((c + d*x)^(1/2) - c^(1/2))^4 + b^2/d^2 -
(2*b*((a + b*x)^(1/2) - a^(1/2))^2)/(d*((c + d*x)^(1/2) - c^(1/2))^2)) + a^(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)*c^(1/2)*log(((a + b*x)^(1/2) - a^(1/2))/((c + d*x)^(1/2) - c^
(1/2))) + (atan((((b*d)^(1/2)*(a*d + b*c)*(((b*d)^(1/2)*(a*d + b*c)*(((b*d)^(1/2)*(a*d + b*c)*((2*(a^3*b^9*c^8
*d + 9*a^4*b^8*c^7*d^2 - 10*a^5*b^7*c^6*d^3 - 10*a^6*b^6*c^5*d^4 + 9*a^7*b^5*c^4*d^5 + a^8*b^4*c^3*d^6))/(a^5*
c^5*d^10) + ((b*d)^(1/2)*((2*(2*a^(5/2)*b^9*c^(15/2)*d - 2*a^(9/2)*b^7*c^(11/2)*d^3 - 2*a^(11/2)*b^6*c^(9/2)*d
^4 + 2*a^(15/2)*b^4*c^(5/2)*d^6))/(a^5*c^5*d^10) - (2*((a + b*x)^(1/2) - a^(1/2))*(4*a^3*b^9*c^9*d - 15*a^4*b^
8*c^8*d^2 + 36*a^5*b^7*c^7*d^3 - 50*a^6*b^6*c^6*d^4 + 36*a^7*b^5*c^5*d^5 - 15*a^8*b^4*c^4*d^6 + 4*a^9*b^3*c^3*
d^7))/(a^6*c^6*d^10*((c + d*x)^(1/2) - c^(1/2))))*(a*d + b*c))/(b*d) - (2*((a + b*x)^(1/2) - a^(1/2))*(4*a^(7/
2)*b^9*c^(19/2)*d - 19*a^(9/2)*b^8*c^(17/2)*d^2 + 64*a^(11/2)*b^7*c^(15/2)*d^3 - 98*a^(13/2)*b^6*c^(13/2)*d^4
+ 64*a^(15/2)*b^5*c^(11/2)*d^5 - 19*a^(17/2)*b^4*c^(9/2)*d^6 + 4*a^(19/2)*b^3*c^(7/2)*d^7))/(a^6*c^6*d^10*((c
+ d*x)^(1/2) - c^(1/2)))))/(b*d) - (2*(2*a^(5/2)*b^10*c^(19/2) + a^(7/2)*b^9*c^(17/2)*d - 9*a^(9/2)*b^8*c^(15/
2)*d^2 + 6*a^(11/2)*b^7*c^(13/2)*d^3 + 6*a^(13/2)*b^6*c^(11/2)*d^4 - 9*a^(15/2)*b^5*c^(9/2)*d^5 + a^(17/2)*b^4
*c^(7/2)*d^6 + 2*a^(19/2)*b^3*c^(5/2)*d^7))/(a^5*c^5*d^10) + (2*((a + b*x)^(1/2) - a^(1/2))*(4*a^3*b^10*c^11 -
8*a^4*b^9*c^10*d + 9*a^5*b^8*c^9*d^2 - 16*a^6*b^7*c^8*d^3 + 22*a^7*b^6*c^7*d^4 - 16*a^8*b^5*c^6*d^5 + 9*a^9*b
^4*c^5*d^6 - 8*a^10*b^3*c^4*d^7 + 4*a^11*b^2*c^3*d^8))/(a^6*c^6*d^10*((c + d*x)^(1/2) - c^(1/2)))))/(b*d) - (2
*(a^4*b^9*c^9*d + 9*a^5*b^8*c^8*d^2 - 10*a^6*b^7*c^7*d^3 - 10*a^7*b^6*c^6*d^4 + 9*a^8*b^5*c^5*d^5 + a^9*b^4*c^
4*d^6))/(a^5*c^5*d^10) + (2*((a + b*x)^(1/2) - a^(1/2))*(4*a^(7/2)*b^10*c^(23/2) - 16*a^(9/2)*b^9*c^(21/2)*d +
21*a^(11/2)*b^8*c^(19/2)*d^2 + 20*a^(13/2)*b^7*c^(17/2)*d^3 - 58*a^(15/2)*b^6*c^(15/2)*d^4 + 20*a^(17/2)*b^5*
c^(13/2)*d^5 + 21*a^(19/2)*b^4*c^(11/2)*d^6 - 16*a^(21/2)*b^3*c^(9/2)*d^7 + 4*a^(23/2)*b^2*c^(7/2)*d^8))/(a^6*
c^6*d^10*((c + d*x)^(1/2) - c^(1/2))))*1i)/(b*d) - ((b*d)^(1/2)*(a*d + b*c)*((2*(a^4*b^9*c^9*d + 9*a^5*b^8*c^8
*d^2 - 10*a^6*b^7*c^7*d^3 - 10*a^7*b^6*c^6*d^4 + 9*a^8*b^5*c^5*d^5 + a^9*b^4*c^4*d^6))/(a^5*c^5*d^10) + ((b*d)
^(1/2)*(a*d + b*c)*(((b*d)^(1/2)*(a*d + b*c)*(((b*d)^(1/2)*((2*(2*a^(5/2)*b^9*c^(15/2)*d - 2*a^(9/2)*b^7*c^(11
/2)*d^3 - 2*a^(11/2)*b^6*c^(9/2)*d^4 + 2*a^(15/2)*b^4*c^(5/2)*d^6))/(a^5*c^5*d^10) - (2*((a + b*x)^(1/2) - a^(
1/2))*(4*a^3*b^9*c^9*d - 15*a^4*b^8*c^8*d^2 + 36*a^5*b^7*c^7*d^3 - 50*a^6*b^6*c^6*d^4 + 36*a^7*b^5*c^5*d^5 - 1
5*a^8*b^4*c^4*d^6 + 4*a^9*b^3*c^3*d^7))/(a^6*c^6*d^10*((c + d*x)^(1/2) - c^(1/2))))*(a*d + b*c))/(b*d) - (2*(a
^3*b^9*c^8*d + 9*a^4*b^8*c^7*d^2 - 10*a^5*b^7*c^6*d^3 - 10*a^6*b^6*c^5*d^4 + 9*a^7*b^5*c^4*d^5 + a^8*b^4*c^3*d
^6))/(a^5*c^5*d^10) + (2*((a + b*x)^(1/2) - a^(1/2))*(4*a^(7/2)*b^9*c^(19/2)*d - 19*a^(9/2)*b^8*c^(17/2)*d^2 +
64*a^(11/2)*b^7*c^(15/2)*d^3 - 98*a^(13/2)*b^6*c^(13/2)*d^4 + 64*a^(15/2)*b^5*c^(11/2)*d^5 - 19*a^(17/2)*b^4*
c^(9/2)*d^6 + 4*a^(19/2)*b^3*c^(7/2)*d^7))/(a^6*c^6*d^10*((c + d*x)^(1/2) - c^(1/2)))))/(b*d) - (2*(2*a^(5/2)*
b^10*c^(19/2) + a^(7/2)*b^9*c^(17/2)*d - 9*a^(9/2)*b^8*c^(15/2)*d^2 + 6*a^(11/2)*b^7*c^(13/2)*d^3 + 6*a^(13/2)
*b^6*c^(11/2)*d^4 - 9*a^(15/2)*b^5*c^(9/2)*d^5 + a^(17/2)*b^4*c^(7/2)*d^6 + 2*a^(19/2)*b^3*c^(5/2)*d^7))/(a^5*
c^5*d^10) + (2*((a + b*x)^(1/2) - a^(1/2))*(4*a^3*b^10*c^11 - 8*a^4*b^9*c^10*d + 9*a^5*b^8*c^9*d^2 - 16*a^6*b^
7*c^8*d^3 + 22*a^7*b^6*c^7*d^4 - 16*a^8*b^5*c^6*d^5 + 9*a^9*b^4*c^5*d^6 - 8*a^10*b^3*c^4*d^7 + 4*a^11*b^2*c^3*
d^8))/(a^6*c^6*d^10*((c + d*x)^(1/2) - c^(1/2)))))/(b*d) - (2*((a + b*x)^(1/2) - a^(1/2))*(4*a^(7/2)*b^10*c^(2
3/2) - 16*a^(9/2)*b^9*c^(21/2)*d + 21*a^(11/2)*b^8*c^(19/2)*d^2 + 20*a^(13/2)*b^7*c^(17/2)*d^3 - 58*a^(15/2)*b
^6*c^(15/2)*d^4 + 20*a^(17/2)*b^5*c^(13/2)*d^5 + 21*a^(19/2)*b^4*c^(11/2)*d^6 - 16*a^(21/2)*b^3*c^(9/2)*d^7 +
4*a^(23/2)*b^2*c^(7/2)*d^8))/(a^6*c^6*d^10*((c + d*x)^(1/2) - c^(1/2))))*1i)/(b*d))/((4*(2*a^(7/2)*b^10*c^(21/
2) - a^(9/2)*b^9*c^(19/2)*d - 9*a^(11/2)*b^8*c^(17/2)*d^2 + 8*a^(13/2)*b^7*c^(15/2)*d^3 + 8*a^(15/2)*b^6*c^(13
/2)*d^4 - 9*a^(17/2)*b^5*c^(11/2)*d^5 - a^(19/2)*b^4*c^(9/2)*d^6 + 2*a^(21/2)*b^3*c^(7/2)*d^7))/(a^5*c^5*d^10)
+ ((b*d)^(1/2)*(a*d + b*c)*(((b*d)^(1/2)*(a*d + b*c)*(((b*d)^(1/2)*(a*d + b*c)*((2*(a^3*b^9*c^8*d + 9*a^4*b^8
*c^7*d^2 - 10*a^5*b^7*c^6*d^3 - 10*a^6*b^6*c^5*d^4 + 9*a^7*b^5*c^4*d^5 + a^8*b^4*c^3*d^6))/(a^5*c^5*d^10) + ((
b*d)^(1/2)*((2*(2*a^(5/2)*b^9*c^(15/2)*d - 2*a^(9/2)*b^7*c^(11/2)*d^3 - 2*a^(11/2)*b^6*c^(9/2)*d^4 + 2*a^(15/2
)*b^4*c^(5/2)*d^6))/(a^5*c^5*d^10) - (2*((a + b*x)^(1/2) - a^(1/2))*(4*a^3*b^9*c^9*d - 15*a^4*b^8*c^8*d^2 + 36
*a^5*b^7*c^7*d^3 - 50*a^6*b^6*c^6*d^4 + 36*a^7*b^5*c^5*d^5 - 15*a^8*b^4*c^4*d^6 + 4*a^9*b^3*c^3*d^7))/(a^6*c^6
*d^10*((c + d*x)^(1/2) - c^(1/2))))*(a*d + b*c))/(b*d) - (2*((a + b*x)^(1/2) - a^(1/2))*(4*a^(7/2)*b^9*c^(19/2
)*d - 19*a^(9/2)*b^8*c^(17/2)*d^2 + 64*a^(11/2)*b^7*c^(15/2)*d^3 - 98*a^(13/2)*b^6*c^(13/2)*d^4 + 64*a^(15/2)*
b^5*c^(11/2)*d^5 - 19*a^(17/2)*b^4*c^(9/2)*d^6 + 4*a^(19/2)*b^3*c^(7/2)*d^7))/(a^6*c^6*d^10*((c + d*x)^(1/2) -
c^(1/2)))))/(b*d) - (2*(2*a^(5/2)*b^10*c^(19/2) + a^(7/2)*b^9*c^(17/2)*d - 9*a^(9/2)*b^8*c^(15/2)*d^2 + 6*a^(
11/2)*b^7*c^(13/2)*d^3 + 6*a^(13/2)*b^6*c^(11/2)*d^4 - 9*a^(15/2)*b^5*c^(9/2)*d^5 + a^(17/2)*b^4*c^(7/2)*d^6 +
2*a^(19/2)*b^3*c^(5/2)*d^7))/(a^5*c^5*d^10) + (2*((a + b*x)^(1/2) - a^(1/2))*(4*a^3*b^10*c^11 - 8*a^4*b^9*c^1
0*d + 9*a^5*b^8*c^9*d^2 - 16*a^6*b^7*c^8*d^3 + 22*a^7*b^6*c^7*d^4 - 16*a^8*b^5*c^6*d^5 + 9*a^9*b^4*c^5*d^6 - 8
*a^10*b^3*c^4*d^7 + 4*a^11*b^2*c^3*d^8))/(a^6*c^6*d^10*((c + d*x)^(1/2) - c^(1/2)))))/(b*d) - (2*(a^4*b^9*c^9*
d + 9*a^5*b^8*c^8*d^2 - 10*a^6*b^7*c^7*d^3 - 10*a^7*b^6*c^6*d^4 + 9*a^8*b^5*c^5*d^5 + a^9*b^4*c^4*d^6))/(a^5*c
^5*d^10) + (2*((a + b*x)^(1/2) - a^(1/2))*(4*a^(7/2)*b^10*c^(23/2) - 16*a^(9/2)*b^9*c^(21/2)*d + 21*a^(11/2)*b
^8*c^(19/2)*d^2 + 20*a^(13/2)*b^7*c^(17/2)*d^3 - 58*a^(15/2)*b^6*c^(15/2)*d^4 + 20*a^(17/2)*b^5*c^(13/2)*d^5 +
21*a^(19/2)*b^4*c^(11/2)*d^6 - 16*a^(21/2)*b^3*c^(9/2)*d^7 + 4*a^(23/2)*b^2*c^(7/2)*d^8))/(a^6*c^6*d^10*((c +
d*x)^(1/2) - c^(1/2)))))/(b*d) + ((b*d)^(1/2)*(a*d + b*c)*((2*(a^4*b^9*c^9*d + 9*a^5*b^8*c^8*d^2 - 10*a^6*b^7
*c^7*d^3 - 10*a^7*b^6*c^6*d^4 + 9*a^8*b^5*c^5*d^5 + a^9*b^4*c^4*d^6))/(a^5*c^5*d^10) + ((b*d)^(1/2)*(a*d + b*c
)*(((b*d)^(1/2)*(a*d + b*c)*(((b*d)^(1/2)*((2*(2*a^(5/2)*b^9*c^(15/2)*d - 2*a^(9/2)*b^7*c^(11/2)*d^3 - 2*a^(11
/2)*b^6*c^(9/2)*d^4 + 2*a^(15/2)*b^4*c^(5/2)*d^6))/(a^5*c^5*d^10) - (2*((a + b*x)^(1/2) - a^(1/2))*(4*a^3*b^9*
c^9*d - 15*a^4*b^8*c^8*d^2 + 36*a^5*b^7*c^7*d^3 - 50*a^6*b^6*c^6*d^4 + 36*a^7*b^5*c^5*d^5 - 15*a^8*b^4*c^4*d^6
+ 4*a^9*b^3*c^3*d^7))/(a^6*c^6*d^10*((c + d*x)^(1/2) - c^(1/2))))*(a*d + b*c))/(b*d) - (2*(a^3*b^9*c^8*d + 9*
a^4*b^8*c^7*d^2 - 10*a^5*b^7*c^6*d^3 - 10*a^6*b^6*c^5*d^4 + 9*a^7*b^5*c^4*d^5 + a^8*b^4*c^3*d^6))/(a^5*c^5*d^1
0) + (2*((a + b*x)^(1/2) - a^(1/2))*(4*a^(7/2)*b^9*c^(19/2)*d - 19*a^(9/2)*b^8*c^(17/2)*d^2 + 64*a^(11/2)*b^7*
c^(15/2)*d^3 - 98*a^(13/2)*b^6*c^(13/2)*d^4 + 64*a^(15/2)*b^5*c^(11/2)*d^5 - 19*a^(17/2)*b^4*c^(9/2)*d^6 + 4*a
^(19/2)*b^3*c^(7/2)*d^7))/(a^6*c^6*d^10*((c + d*x)^(1/2) - c^(1/2)))))/(b*d) - (2*(2*a^(5/2)*b^10*c^(19/2) + a
^(7/2)*b^9*c^(17/2)*d - 9*a^(9/2)*b^8*c^(15/2)*d^2 + 6*a^(11/2)*b^7*c^(13/2)*d^3 + 6*a^(13/2)*b^6*c^(11/2)*d^4
- 9*a^(15/2)*b^5*c^(9/2)*d^5 + a^(17/2)*b^4*c^(7/2)*d^6 + 2*a^(19/2)*b^3*c^(5/2)*d^7))/(a^5*c^5*d^10) + (2*((
a + b*x)^(1/2) - a^(1/2))*(4*a^3*b^10*c^11 - 8*a^4*b^9*c^10*d + 9*a^5*b^8*c^9*d^2 - 16*a^6*b^7*c^8*d^3 + 22*a^
7*b^6*c^7*d^4 - 16*a^8*b^5*c^6*d^5 + 9*a^9*b^4*c^5*d^6 - 8*a^10*b^3*c^4*d^7 + 4*a^11*b^2*c^3*d^8))/(a^6*c^6*d^
10*((c + d*x)^(1/2) - c^(1/2)))))/(b*d) - (2*((a + b*x)^(1/2) - a^(1/2))*(4*a^(7/2)*b^10*c^(23/2) - 16*a^(9/2)
*b^9*c^(21/2)*d + 21*a^(11/2)*b^8*c^(19/2)*d^2 + 20*a^(13/2)*b^7*c^(17/2)*d^3 - 58*a^(15/2)*b^6*c^(15/2)*d^4 +
20*a^(17/2)*b^5*c^(13/2)*d^5 + 21*a^(19/2)*b^4*c^(11/2)*d^6 - 16*a^(21/2)*b^3*c^(9/2)*d^7 + 4*a^(23/2)*b^2*c^
(7/2)*d^8))/(a^6*c^6*d^10*((c + d*x)^(1/2) - c^(1/2)))))/(b*d) + (4*((a + b*x)^(1/2) - a^(1/2))*(16*a^6*b^8*c^
10*d^2 - 8*a^5*b^9*c^11*d + 8*a^7*b^7*c^9*d^3 - 32*a^8*b^6*c^8*d^4 + 8*a^9*b^5*c^7*d^5 + 16*a^10*b^4*c^6*d^6 -
8*a^11*b^3*c^5*d^7))/(a^6*c^6*d^10*((c + d*x)^(1/2) - c^(1/2)))))*(b*d)^(1/2)*(a*d + b*c)*2i)/(b*d)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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