\(\int \frac {x \sqrt {a+b x}}{\sqrt {c+d x}} \, dx\) [578]

Optimal result

Integrand size = 20, antiderivative size = 125 \[ \int \frac {x \sqrt {a+b x}}{\sqrt {c+d x}} \, dx=-\frac {(3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b d^2}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 b d}+\frac {(b c-a d) (3 b c+a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{3/2} d^{5/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {81, 52, 65, 223, 212} \[ \int \frac {x \sqrt {a+b x}}{\sqrt {c+d x}} \, dx=\frac {(b c-a d) (a d+3 b c) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{3/2} d^{5/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+3 b c)}{4 b d^2}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 b d} \]

[In]

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

[Out]

-1/4*((3*b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(b*d^2) + ((a + b*x)^(3/2)*Sqrt[c + d*x])/(2*b*d) + ((b*c - a
*d)*(3*b*c + a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(4*b^(3/2)*d^(5/2))

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

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 81

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

Rule 212

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

Rule 223

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

Mathematica [A] (verified)

Time = 0.75 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.03 \[ \int \frac {x \sqrt {a+b x}}{\sqrt {c+d x}} \, dx=\frac {\sqrt {b} \sqrt {d} \sqrt {a+b x} \sqrt {c+d x} (-3 b c+a d+2 b d x)+\left (-6 b^2 c^2+4 a b c d+2 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{4 b^{3/2} d^{5/2}} \]

[In]

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

[Out]

(Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]*(-3*b*c + a*d + 2*b*d*x) + (-6*b^2*c^2 + 4*a*b*c*d + 2*a^2*d^2)*A
rcTanh[(Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*(Sqrt[a - (b*c)/d] - Sqrt[a + b*x]))])/(4*b^(3/2)*d^(5/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(249\) vs. \(2(99)=198\).

Time = 1.51 (sec) , antiderivative size = 250, normalized size of antiderivative = 2.00

method	result	size
default	\(-\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} d^{2}+2 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a b c d -3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{2} c^{2}-4 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b d x +6 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b c -2 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a d \right )}{8 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, d^{2} b \sqrt {b d}}\)	\(250\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 306, normalized size of antiderivative = 2.45 \[ \int \frac {x \sqrt {a+b x}}{\sqrt {c+d x}} \, dx=\left [-\frac {{\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (2 \, b^{2} d^{2} x - 3 \, b^{2} c d + a b d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, b^{2} d^{3}}, -\frac {{\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 2 \, {\left (2 \, b^{2} d^{2} x - 3 \, b^{2} c d + a b d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, b^{2} d^{3}}\right ] \]

[In]

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

[Out]

[-1/16*((3*b^2*c^2 - 2*a*b*c*d - a^2*d^2)*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) - 4*(2*b^2*d^2*x - 3*b^2*c*
d + a*b*d^2)*sqrt(b*x + a)*sqrt(d*x + c))/(b^2*d^3), -1/8*((3*b^2*c^2 - 2*a*b*c*d - a^2*d^2)*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)) - 2*(2*b^2*d^2*x - 3*b^2*c*d + a*b*d^2)*sqrt(b*x + a)*sqrt(d*x + c))/(b^2*d^3)]

Sympy [F]

\[ \int \frac {x \sqrt {a+b x}}{\sqrt {c+d x}} \, dx=\int \frac {x \sqrt {a + b x}}{\sqrt {c + d x}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {x \sqrt {a+b x}}{\sqrt {c+d x}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(x*(b*x+a)^(1/2)/(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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.19 \[ \int \frac {x \sqrt {a+b x}}{\sqrt {c+d x}} \, dx=\frac {{\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} {\left (\frac {2 \, {\left (b x + a\right )}}{b^{2} d} - \frac {3 \, b^{3} c d + a b^{2} d^{2}}{b^{4} d^{3}}\right )} - \frac {{\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b d^{2}}\right )} b}{4 \, {\left | b \right |}} \]

[In]

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

[Out]

1/4*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*(2*(b*x + a)/(b^2*d) - (3*b^3*c*d + a*b^2*d^2)/(b^4*d^3
)) - (3*b^2*c^2 - 2*a*b*c*d - a^2*d^2)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d))
)/(sqrt(b*d)*b*d^2))*b/abs(b)

Mupad [B] (verification not implemented)

Time = 19.23 (sec) , antiderivative size = 589, normalized size of antiderivative = 4.71 \[ \int \frac {x \sqrt {a+b x}}{\sqrt {c+d x}} \, dx=\frac {\frac {\left (\sqrt {a+b\,x}-\sqrt {a}\right )\,\left (\frac {a^2\,b^2\,d^2}{2}+a\,b^3\,c\,d-\frac {3\,b^4\,c^2}{2}\right )}{d^6\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3\,\left (\frac {7\,a^2\,b\,d^2}{2}+23\,a\,b^2\,c\,d+\frac {11\,b^3\,c^2}{2}\right )}{d^5\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^3}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^5\,\left (\frac {7\,a^2\,d^2}{2}+23\,a\,b\,c\,d+\frac {11\,b^2\,c^2}{2}\right )}{d^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^5}-\frac {8\,a^{3/2}\,\sqrt {c}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6}{d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^6}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^7\,\left (\frac {a^2\,d^2}{2}+a\,b\,c\,d-\frac {3\,b^2\,c^2}{2}\right )}{b\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^7}-\frac {\sqrt {a}\,\sqrt {c}\,\left (32\,c\,b^2+16\,a\,d\,b\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{d^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}-\frac {8\,a^{3/2}\,b^2\,\sqrt {c}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{d^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}}{\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^8}{{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^8}+\frac {b^4}{d^4}-\frac {4\,b^3\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}+\frac {6\,b^2\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}-\frac {4\,b\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6}{d\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^6}}-\frac {\mathrm {atanh}\left (\frac {\sqrt {d}\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {b}\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}\right )\,\left (a\,d-b\,c\right )\,\left (a\,d+3\,b\,c\right )}{2\,b^{3/2}\,d^{5/2}} \]

[In]

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

[Out]

((((a + b*x)^(1/2) - a^(1/2))*((a^2*b^2*d^2)/2 - (3*b^4*c^2)/2 + a*b^3*c*d))/(d^6*((c + d*x)^(1/2) - c^(1/2)))
+ (((a + b*x)^(1/2) - a^(1/2))^3*((11*b^3*c^2)/2 + (7*a^2*b*d^2)/2 + 23*a*b^2*c*d))/(d^5*((c + d*x)^(1/2) - c
^(1/2))^3) + (((a + b*x)^(1/2) - a^(1/2))^5*((7*a^2*d^2)/2 + (11*b^2*c^2)/2 + 23*a*b*c*d))/(d^4*((c + d*x)^(1/
2) - c^(1/2))^5) - (8*a^(3/2)*c^(1/2)*((a + b*x)^(1/2) - a^(1/2))^6)/(d^2*((c + d*x)^(1/2) - c^(1/2))^6) + (((
a + b*x)^(1/2) - a^(1/2))^7*((a^2*d^2)/2 - (3*b^2*c^2)/2 + a*b*c*d))/(b*d^3*((c + d*x)^(1/2) - c^(1/2))^7) - (
a^(1/2)*c^(1/2)*(32*b^2*c + 16*a*b*d)*((a + b*x)^(1/2) - a^(1/2))^4)/(d^4*((c + d*x)^(1/2) - c^(1/2))^4) - (8*
a^(3/2)*b^2*c^(1/2)*((a + b*x)^(1/2) - a^(1/2))^2)/(d^4*((c + d*x)^(1/2) - c^(1/2))^2))/(((a + b*x)^(1/2) - a^
(1/2))^8/((c + d*x)^(1/2) - c^(1/2))^8 + b^4/d^4 - (4*b^3*((a + b*x)^(1/2) - a^(1/2))^2)/(d^3*((c + d*x)^(1/2)
- c^(1/2))^2) + (6*b^2*((a + b*x)^(1/2) - a^(1/2))^4)/(d^2*((c + d*x)^(1/2) - c^(1/2))^4) - (4*b*((a + b*x)^(
1/2) - a^(1/2))^6)/(d*((c + d*x)^(1/2) - c^(1/2))^6)) - (atanh((d^(1/2)*((a + b*x)^(1/2) - a^(1/2)))/(b^(1/2)*
((c + d*x)^(1/2) - c^(1/2))))*(a*d - b*c)*(a*d + 3*b*c))/(2*b^(3/2)*d^(5/2))