3.24.96 \(\int \frac {x^2}{\sqrt {\frac {b+a x}{d+c x}}} \, dx\) [2396]

Optimal. Leaf size=193 \[ \frac {\sqrt {\frac {b+a x}{d+c x}} \left (15 b^2 c^2 d-4 a b c d^2-3 a^2 d^3+15 b^2 c^3 x-14 a b c^2 d x-a^2 c d^2 x-10 a b c^3 x^2+10 a^2 c^2 d x^2+8 a^2 c^3 x^3\right )}{24 a^3 c^2}+\frac {\left (-5 b^3 c^3+3 a b^2 c^2 d+a^2 b c d^2+a^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {b+a x}{d+c x}}}{\sqrt {a}}\right )}{8 a^{7/2} c^{5/2}} \]

[Out]

1/24*((a*x+b)/(c*x+d))^(1/2)*(8*a^2*c^3*x^3+10*a^2*c^2*d*x^2-10*a*b*c^3*x^2-a^2*c*d^2*x-14*a*b*c^2*d*x+15*b^2*
c^3*x-3*a^2*d^3-4*a*b*c*d^2+15*b^2*c^2*d)/a^3/c^2+1/8*(a^3*d^3+a^2*b*c*d^2+3*a*b^2*c^2*d-5*b^3*c^3)*arctanh(c^
(1/2)*((a*x+b)/(c*x+d))^(1/2)/a^(1/2))/a^(7/2)/c^(5/2)

________________________________________________________________________________________

Rubi [A]
time = 0.18, antiderivative size = 266, normalized size of antiderivative = 1.38, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {1980, 424, 393, 205, 214} \begin {gather*} -\frac {(3 a d+5 b c) (b c-a d)^2 \sqrt {\frac {a x+b}{c x+d}}}{12 a^2 c^2 \left (a-\frac {c (a x+b)}{c x+d}\right )^2}-\frac {\left (a^2 d^2+2 a b c d+5 b^2 c^2\right ) (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {a x+b}{c x+d}}}{\sqrt {a}}\right )}{8 a^{7/2} c^{5/2}}+\frac {(c x+d) \left (a^2 d^2+2 a b c d+5 b^2 c^2\right ) \sqrt {\frac {a x+b}{c x+d}}}{8 a^3 c^2}-\frac {(b c-a d)^2 \sqrt {\frac {a x+b}{c x+d}} \left (b-\frac {d (a x+b)}{c x+d}\right )}{3 a c \left (a-\frac {c (a x+b)}{c x+d}\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/Sqrt[(b + a*x)/(d + c*x)],x]

[Out]

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

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

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]

Rule 393

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

Rule 424

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

Rule 1980

Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)))/((c_) + (d_.)*(x_)))^(p_), x_Symbol] :> With[{q = Denominator[p]}
, Dist[q*e*(b*c - a*d), Subst[Int[x^(q*(p + 1) - 1)*(((-a)*e + c*x^q)^m/(b*e - d*x^q)^(m + 2)), x], x, (e*((a
+ b*x)/(c + d*x)))^(1/q)], x]] /; FreeQ[{a, b, c, d, e, m}, x] && FractionQ[p] && IntegerQ[m]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.51, size = 184, normalized size = 0.95 \begin {gather*} \frac {a (b+a x) \sqrt {d+c x} \left (15 b^2 c^2-2 a b c (2 d+5 c x)+a^2 \left (-3 d^2+2 c d x+8 c^2 x^2\right )\right )-3 \sqrt {\frac {a}{c}} \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 {b+a x} \log \left (\sqrt {b+a x}-\sqrt {\frac {a}{c}} \sqrt {d+c x}\right )}{24 a^4 c^2 \sqrt {\frac {b+a x}{d+c x}} \sqrt {d+c x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(b + a*x)*Sqrt[d + c*x]*(15*b^2*c^2 - 2*a*b*c*(2*d + 5*c*x) + a^2*(-3*d^2 + 2*c*d*x + 8*c^2*x^2)) - 3*Sqrt[
a/c]*(-5*b^3*c^3 + 3*a*b^2*c^2*d + a^2*b*c*d^2 + a^3*d^3)*Sqrt[b + a*x]*Log[Sqrt[b + a*x] - Sqrt[a/c]*Sqrt[d +
c*x]])/(24*a^4*c^2*Sqrt[(b + a*x)/(d + c*x)]*Sqrt[d + c*x])

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(587\) vs. \(2(177)=354\).
time = 0.09, size = 588, normalized size = 3.05

method	result	size
default	\(\frac {\left (a x +b \right ) \left (3 \ln \left (\frac {2 a c x +2 \sqrt {a c \,x^{2}+a d x +c x b +b d}\, \sqrt {a c}+a d +b c}{2 \sqrt {a c}}\right ) a^{3} d^{3}+3 \ln \left (\frac {2 a c x +2 \sqrt {a c \,x^{2}+a d x +c x b +b d}\, \sqrt {a c}+a d +b c}{2 \sqrt {a c}}\right ) a^{2} b c \,d^{2}-15 \ln \left (\frac {2 a c x +2 \sqrt {a c \,x^{2}+a d x +c x b +b d}\, \sqrt {a c}+a d +b c}{2 \sqrt {a c}}\right ) a \,b^{2} c^{2} d +9 \ln \left (\frac {2 a c x +2 \sqrt {a c \,x^{2}+a d x +c x b +b d}\, \sqrt {a c}+a d +b c}{2 \sqrt {a c}}\right ) b^{3} c^{3}+24 \ln \left (\frac {2 a c x +2 \sqrt {\left (c x +d \right ) \left (a x +b \right )}\, \sqrt {a c}+a d +b c}{2 \sqrt {a c}}\right ) a \,b^{2} c^{2} d -24 \ln \left (\frac {2 a c x +2 \sqrt {\left (c x +d \right ) \left (a x +b \right )}\, \sqrt {a c}+a d +b c}{2 \sqrt {a c}}\right ) b^{3} c^{3}-12 \sqrt {a c}\, \sqrt {a c \,x^{2}+a d x +c x b +b d}\, a^{2} c d x -36 \sqrt {a c}\, \sqrt {a c \,x^{2}+a d x +c x b +b d}\, a b \,c^{2} x +48 \sqrt {\left (c x +d \right ) \left (a x +b \right )}\, \sqrt {a c}\, b^{2} c^{2}+16 \left (a c \,x^{2}+a d x +c x b +b d \right )^{\frac {3}{2}} a c \sqrt {a c}-6 \sqrt {a c}\, \sqrt {a c \,x^{2}+a d x +c x b +b d}\, a^{2} d^{2}-24 \sqrt {a c}\, \sqrt {a c \,x^{2}+a d x +c x b +b d}\, a b c d -18 \sqrt {a c}\, \sqrt {a c \,x^{2}+a d x +c x b +b d}\, b^{2} c^{2}\right )}{48 a^{3} \sqrt {\frac {a x +b}{c x +d}}\, \sqrt {\left (c x +d \right ) \left (a x +b \right )}\, c^{2} \sqrt {a c}}\)	\(588\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*(a*x+b)/a^3*(3*ln(1/2*(2*a*c*x+2*(a*c*x^2+a*d*x+b*c*x+b*d)^(1/2)*(a*c)^(1/2)+a*d+b*c)/(a*c)^(1/2))*a^3*d^
3+3*ln(1/2*(2*a*c*x+2*(a*c*x^2+a*d*x+b*c*x+b*d)^(1/2)*(a*c)^(1/2)+a*d+b*c)/(a*c)^(1/2))*a^2*b*c*d^2-15*ln(1/2*
(2*a*c*x+2*(a*c*x^2+a*d*x+b*c*x+b*d)^(1/2)*(a*c)^(1/2)+a*d+b*c)/(a*c)^(1/2))*a*b^2*c^2*d+9*ln(1/2*(2*a*c*x+2*(
a*c*x^2+a*d*x+b*c*x+b*d)^(1/2)*(a*c)^(1/2)+a*d+b*c)/(a*c)^(1/2))*b^3*c^3+24*ln(1/2*(2*a*c*x+2*((c*x+d)*(a*x+b)
)^(1/2)*(a*c)^(1/2)+a*d+b*c)/(a*c)^(1/2))*a*b^2*c^2*d-24*ln(1/2*(2*a*c*x+2*((c*x+d)*(a*x+b))^(1/2)*(a*c)^(1/2)
+a*d+b*c)/(a*c)^(1/2))*b^3*c^3-12*(a*c)^(1/2)*(a*c*x^2+a*d*x+b*c*x+b*d)^(1/2)*a^2*c*d*x-36*(a*c)^(1/2)*(a*c*x^
2+a*d*x+b*c*x+b*d)^(1/2)*a*b*c^2*x+48*((c*x+d)*(a*x+b))^(1/2)*(a*c)^(1/2)*b^2*c^2+16*(a*c*x^2+a*d*x+b*c*x+b*d)
^(3/2)*a*c*(a*c)^(1/2)-6*(a*c)^(1/2)*(a*c*x^2+a*d*x+b*c*x+b*d)^(1/2)*a^2*d^2-24*(a*c)^(1/2)*(a*c*x^2+a*d*x+b*c
*x+b*d)^(1/2)*a*b*c*d-18*(a*c)^(1/2)*(a*c*x^2+a*d*x+b*c*x+b*d)^(1/2)*b^2*c^2)/((a*x+b)/(c*x+d))^(1/2)/((c*x+d)
*(a*x+b))^(1/2)/c^2/(a*c)^(1/2)

________________________________________________________________________________________

Maxima [A]
time = 0.48, size = 355, normalized size = 1.84 \begin {gather*} -\frac {3 \, {\left (5 \, b^{3} c^{5} - 3 \, a b^{2} c^{4} d - a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}\right )} \left (\frac {a x + b}{c x + d}\right )^{\frac {5}{2}} - 8 \, {\left (5 \, a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d - 3 \, a^{3} b c^{2} d^{2} + a^{4} c d^{3}\right )} \left (\frac {a x + b}{c x + d}\right )^{\frac {3}{2}} + 3 \, {\left (11 \, a^{2} b^{3} c^{3} - 13 \, a^{3} b^{2} c^{2} d + a^{4} b c d^{2} + a^{5} d^{3}\right )} \sqrt {\frac {a x + b}{c x + d}}}{24 \, {\left (a^{6} c^{2} - \frac {3 \, {\left (a x + b\right )} a^{5} c^{3}}{c x + d} + \frac {3 \, {\left (a x + b\right )}^{2} a^{4} c^{4}}{{\left (c x + d\right )}^{2}} - \frac {{\left (a x + b\right )}^{3} a^{3} c^{5}}{{\left (c x + d\right )}^{3}}\right )}} + \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 )} \log \left (\frac {c \sqrt {\frac {a x + b}{c x + d}} - \sqrt {a c}}{c \sqrt {\frac {a x + b}{c x + d}} + \sqrt {a c}}\right )}{16 \, \sqrt {a c} a^{3} c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(3*(5*b^3*c^5 - 3*a*b^2*c^4*d - a^2*b*c^3*d^2 - a^3*c^2*d^3)*((a*x + b)/(c*x + d))^(5/2) - 8*(5*a*b^3*c^
4 - 3*a^2*b^2*c^3*d - 3*a^3*b*c^2*d^2 + a^4*c*d^3)*((a*x + b)/(c*x + d))^(3/2) + 3*(11*a^2*b^3*c^3 - 13*a^3*b^
2*c^2*d + a^4*b*c*d^2 + a^5*d^3)*sqrt((a*x + b)/(c*x + d)))/(a^6*c^2 - 3*(a*x + b)*a^5*c^3/(c*x + d) + 3*(a*x
+ b)^2*a^4*c^4/(c*x + d)^2 - (a*x + b)^3*a^3*c^5/(c*x + d)^3) + 1/16*(5*b^3*c^3 - 3*a*b^2*c^2*d - a^2*b*c*d^2
- a^3*d^3)*log((c*sqrt((a*x + b)/(c*x + d)) - sqrt(a*c))/(c*sqrt((a*x + b)/(c*x + d)) + sqrt(a*c)))/(sqrt(a*c)
*a^3*c^2)

________________________________________________________________________________________

Fricas [A]
time = 0.53, size = 419, normalized size = 2.17 \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} \log \left (-2 \, a c x - b c - a d - 2 \, \sqrt {a c} {\left (c x + d\right )} \sqrt {\frac {a x + b}{c x + d}}\right ) - 2 \, {\left (8 \, a^{3} c^{4} x^{3} + 15 \, a b^{2} c^{3} d - 4 \, a^{2} b c^{2} d^{2} - 3 \, a^{3} c d^{3} - 10 \, {\left (a^{2} b c^{4} - a^{3} c^{3} d\right )} x^{2} + {\left (15 \, a b^{2} c^{4} - 14 \, a^{2} b c^{3} d - a^{3} c^{2} d^{2}\right )} x\right )} \sqrt {\frac {a x + b}{c x + d}}}{48 \, a^{4} c^{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} \arctan \left (\frac {\sqrt {-a c} {\left (c x + d\right )} \sqrt {\frac {a x + b}{c x + d}}}{a c x + b c}\right ) + {\left (8 \, a^{3} c^{4} x^{3} + 15 \, a b^{2} c^{3} d - 4 \, a^{2} b c^{2} d^{2} - 3 \, a^{3} c d^{3} - 10 \, {\left (a^{2} b c^{4} - a^{3} c^{3} d\right )} x^{2} + {\left (15 \, a b^{2} c^{4} - 14 \, a^{2} b c^{3} d - a^{3} c^{2} d^{2}\right )} x\right )} \sqrt {\frac {a x + b}{c x + d}}}{24 \, a^{4} c^{3}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-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)*log(-2*a*c*x - b*c - a*d - 2*sqrt(a*c)
*(c*x + d)*sqrt((a*x + b)/(c*x + d))) - 2*(8*a^3*c^4*x^3 + 15*a*b^2*c^3*d - 4*a^2*b*c^2*d^2 - 3*a^3*c*d^3 - 10
*(a^2*b*c^4 - a^3*c^3*d)*x^2 + (15*a*b^2*c^4 - 14*a^2*b*c^3*d - a^3*c^2*d^2)*x)*sqrt((a*x + b)/(c*x + d)))/(a^
4*c^3), 1/24*(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)*arctan(sqrt(-a*c)*(c*x + d)*sqr
t((a*x + b)/(c*x + d))/(a*c*x + b*c)) + (8*a^3*c^4*x^3 + 15*a*b^2*c^3*d - 4*a^2*b*c^2*d^2 - 3*a^3*c*d^3 - 10*(
a^2*b*c^4 - a^3*c^3*d)*x^2 + (15*a*b^2*c^4 - 14*a^2*b*c^3*d - a^3*c^2*d^2)*x)*sqrt((a*x + b)/(c*x + d)))/(a^4*
c^3)]

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\sqrt {\frac {a x + b}{c x + d}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2/sqrt((a*x + b)/(c*x + d)), x)

________________________________________________________________________________________

Giac [A]
time = 0.43, size = 231, normalized size = 1.20 \begin {gather*} \frac {1}{24} \, \sqrt {a c x^{2} + b c x + a d x + b d} {\left (2 \, x {\left (\frac {4 \, x}{a \mathrm {sgn}\left (c x + d\right )} - \frac {5 \, a b c^{2} \mathrm {sgn}\left (c x + d\right ) - a^{2} c d \mathrm {sgn}\left (c x + d\right )}{a^{3} c^{2}}\right )} + \frac {15 \, b^{2} c^{2} \mathrm {sgn}\left (c x + d\right ) - 4 \, a b c d \mathrm {sgn}\left (c x + d\right ) - 3 \, a^{2} d^{2} \mathrm {sgn}\left (c x + d\right )}{a^{3} c^{2}}\right )} + \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 )} \log \left ({\left | -b c - a d - 2 \, \sqrt {a c} {\left (\sqrt {a c} x - \sqrt {a c x^{2} + b c x + a d x + b d}\right )} \right |}\right )}{16 \, \sqrt {a c} a^{3} c^{2} \mathrm {sgn}\left (c x + d\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*sqrt(a*c*x^2 + b*c*x + a*d*x + b*d)*(2*x*(4*x/(a*sgn(c*x + d)) - (5*a*b*c^2*sgn(c*x + d) - a^2*c*d*sgn(c*
x + d))/(a^3*c^2)) + (15*b^2*c^2*sgn(c*x + d) - 4*a*b*c*d*sgn(c*x + d) - 3*a^2*d^2*sgn(c*x + d))/(a^3*c^2)) +
1/16*(5*b^3*c^3 - 3*a*b^2*c^2*d - a^2*b*c*d^2 - a^3*d^3)*log(abs(-b*c - a*d - 2*sqrt(a*c)*(sqrt(a*c)*x - sqrt(
a*c*x^2 + b*c*x + a*d*x + b*d))))/(sqrt(a*c)*a^3*c^2*sgn(c*x + d))

________________________________________________________________________________________

Mupad [B]
time = 0.86, size = 305, normalized size = 1.58 \begin {gather*} \frac {\frac {{\left (\frac {b+a\,x}{d+c\,x}\right )}^{5/2}\,\left (\frac {a^3\,d^3}{8}+\frac {a^2\,b\,c\,d^2}{8}+\frac {3\,a\,b^2\,c^2\,d}{8}-\frac {5\,b^3\,c^3}{8}\right )}{a^6}+\frac {{\left (\frac {b+a\,x}{d+c\,x}\right )}^{3/2}\,\left (\frac {a^3\,d^3}{3}-a^2\,b\,c\,d^2-a\,b^2\,c^2\,d+\frac {5\,b^3\,c^3}{3}\right )}{a^5\,c}-\frac {\sqrt {\frac {b+a\,x}{d+c\,x}}\,\left (\frac {a^3\,d^3}{8}+\frac {a^2\,b\,c\,d^2}{8}-\frac {13\,a\,b^2\,c^2\,d}{8}+\frac {11\,b^3\,c^3}{8}\right )}{a^4\,c^2}}{\frac {3\,c^2\,{\left (b+a\,x\right )}^2}{a^2\,{\left (d+c\,x\right )}^2}-\frac {c^3\,{\left (b+a\,x\right )}^3}{a^3\,{\left (d+c\,x\right )}^3}-\frac {3\,c\,\left (b+a\,x\right )}{a\,\left (d+c\,x\right )}+1}+\frac {\mathrm {atanh}\left (\frac {\sqrt {c}\,\sqrt {\frac {b+a\,x}{d+c\,x}}}{\sqrt {a}}\right )\,\left (a\,d-b\,c\right )\,\left (a^2\,d^2+2\,a\,b\,c\,d+5\,b^2\,c^2\right )}{8\,a^{7/2}\,c^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((((b + a*x)/(d + c*x))^(5/2)*((a^3*d^3)/8 - (5*b^3*c^3)/8 + (3*a*b^2*c^2*d)/8 + (a^2*b*c*d^2)/8))/a^6 + (((b
+ a*x)/(d + c*x))^(3/2)*((a^3*d^3)/3 + (5*b^3*c^3)/3 - a*b^2*c^2*d - a^2*b*c*d^2))/(a^5*c) - (((b + a*x)/(d +
c*x))^(1/2)*((a^3*d^3)/8 + (11*b^3*c^3)/8 - (13*a*b^2*c^2*d)/8 + (a^2*b*c*d^2)/8))/(a^4*c^2))/((3*c^2*(b + a*x
)^2)/(a^2*(d + c*x)^2) - (c^3*(b + a*x)^3)/(a^3*(d + c*x)^3) - (3*c*(b + a*x))/(a*(d + c*x)) + 1) + (atanh((c^
(1/2)*((b + a*x)/(d + c*x))^(1/2))/a^(1/2))*(a*d - b*c)*(a^2*d^2 + 5*b^2*c^2 + 2*a*b*c*d))/(8*a^(7/2)*c^(5/2))

________________________________________________________________________________________