\(\int \frac {x^2}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx\) [742]

Optimal result

Integrand size = 22, antiderivative size = 112 \[ \int \frac {x^2}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx=\frac {2 c^2 \sqrt {a+b x}}{d^2 (b c-a d) \sqrt {c+d x}}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b d^2}-\frac {(3 b c+a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2} d^{5/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {91, 81, 65, 223, 212} \[ \int \frac {x^2}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx=-\frac {(a d+3 b c) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2} d^{5/2}}+\frac {2 c^2 \sqrt {a+b x}}{d^2 \sqrt {c+d x} (b c-a d)}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b d^2} \]

[In]

Int[x^2/(Sqrt[a + b*x]*(c + d*x)^(3/2)),x]

[Out]

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

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 91

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c - a*d
)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d*e - c*f)*(n + 1))), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

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

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.20 \[ \int \frac {x^2}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx=\frac {\sqrt {b} \sqrt {d} \sqrt {a+b x} (a d (c+d x)-b c (3 c+d x))+\left (3 b^2 c^2-2 a b c d-a^2 d^2\right ) \sqrt {c+d x} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2} d^{5/2} (-b c+a d) \sqrt {c+d x}} \]

[In]

Integrate[x^2/(Sqrt[a + b*x]*(c + d*x)^(3/2)),x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(438\) vs. \(2(92)=184\).

Time = 0.58 (sec) , antiderivative size = 439, normalized size of antiderivative = 3.92

method	result	size
default	\(-\frac {\sqrt {b x +a}\, \left (\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} d^{3} x +2 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a b c \,d^{2} x -3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{2} c^{2} d x +\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} c \,d^{2}+2 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a b \,c^{2} d -3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{2} c^{3}-2 a \,d^{2} x \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 b c d x \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}-2 a c d \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+6 b \,c^{2} \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\right )}{2 \sqrt {b d}\, b \left (a d -b c \right ) \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, d^{2} \sqrt {d x +c}}\)	\(439\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 227 vs. \(2 (92) = 184\).

Time = 0.27 (sec) , antiderivative size = 468, normalized size of antiderivative = 4.18 \[ \int \frac {x^2}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx=\left [\frac {{\left (3 \, b^{2} c^{3} - 2 \, a b c^{2} d - a^{2} c d^{2} + {\left (3 \, b^{2} c^{2} d - 2 \, a b c d^{2} - a^{2} d^{3}\right )} x\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (3 \, b^{2} c^{2} d - a b c d^{2} + {\left (b^{2} c d^{2} - a b d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{4 \, {\left (b^{3} c^{2} d^{3} - a b^{2} c d^{4} + {\left (b^{3} c d^{4} - a b^{2} d^{5}\right )} x\right )}}, \frac {{\left (3 \, b^{2} c^{3} - 2 \, a b c^{2} d - a^{2} c d^{2} + {\left (3 \, b^{2} c^{2} d - 2 \, a b c d^{2} - a^{2} d^{3}\right )} x\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (3 \, b^{2} c^{2} d - a b c d^{2} + {\left (b^{2} c d^{2} - a b d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{3} c^{2} d^{3} - a b^{2} c d^{4} + {\left (b^{3} c d^{4} - a b^{2} d^{5}\right )} x\right )}}\right ] \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {x^2}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx=\int \frac {x^{2}}{\sqrt {a + b x} \left (c + d x\right )^{\frac {3}{2}}}\, dx \]

[In]

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

[Out]

Integral(x**2/(sqrt(a + b*x)*(c + d*x)**(3/2)), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^2}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(x^2/(d*x+c)^(3/2)/(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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 193 vs. \(2 (92) = 184\).

Time = 0.33 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.72 \[ \int \frac {x^2}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx=\frac {\sqrt {b x + a} {\left (\frac {{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} {\left (b x + a\right )}}{b^{3} c d^{3} {\left | b \right |} - a b^{2} d^{4} {\left | b \right |}} + \frac {3 \, b^{4} c^{2} d - 2 \, a b^{3} c d^{2} + a^{2} b^{2} d^{3}}{b^{3} c d^{3} {\left | b \right |} - a b^{2} d^{4} {\left | b \right |}}\right )}}{\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}} + \frac {{\left (3 \, b c + a d\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} d^{2} {\left | b \right |}} \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx=\int \frac {x^2}{\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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