3.722 \(\int \frac{x^2}{\sqrt{a+b x} \sqrt{c+d x}} \, dx\)

Optimal. Leaf size=127 \[ -\frac{\left (4 a b c d-3 (a d+b c)^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 b^{5/2} d^{5/2}}-\frac{3 \sqrt{a+b x} \sqrt{c+d x} (a d+b c)}{4 b^2 d^2}+\frac{x \sqrt{a+b x} \sqrt{c+d x}}{2 b d} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.255239, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{\left (4 a b c d-3 (a d+b c)^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 b^{5/2} d^{5/2}}-\frac{3 \sqrt{a+b x} \sqrt{c+d x} (a d+b c)}{4 b^2 d^2}+\frac{x \sqrt{a+b x} \sqrt{c+d x}}{2 b d} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 18.7848, size = 114, normalized size = 0.9 \[ \frac{x \sqrt{a + b x} \sqrt{c + d x}}{2 b d} - \frac{3 \sqrt{a + b x} \sqrt{c + d x} \left (a d + b c\right )}{4 b^{2} d^{2}} - \frac{\left (a b c d - \frac{3 \left (a d + b c\right )^{2}}{4}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{b^{\frac{5}{2}} d^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(b*x+a)**(1/2)/(d*x+c)**(1/2),x)

[Out]

x*sqrt(a + b*x)*sqrt(c + d*x)/(2*b*d) - 3*sqrt(a + b*x)*sqrt(c + d*x)*(a*d + b*c
)/(4*b**2*d**2) - (a*b*c*d - 3*(a*d + b*c)**2/4)*atanh(sqrt(d)*sqrt(a + b*x)/(sq
rt(b)*sqrt(c + d*x)))/(b**(5/2)*d**(5/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.145982, size = 123, normalized size = 0.97 \[ \frac{\left (3 a^2 d^2+2 a b c d+3 b^2 c^2\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{8 b^{5/2} d^{5/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} (-3 a d-3 b c+2 b d x)}{4 b^2 d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.032, size = 251, normalized size = 2. \[{\frac{1}{8\,{b}^{2}{d}^{2}} \left ( 3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}{d}^{2}+2\,c\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) adb+3\,{c}^{2}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){b}^{2}+4\,x\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }db\sqrt{bd}-6\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }ad\sqrt{bd}-6\,c\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }b\sqrt{bd} \right ) \sqrt{bx+a}\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(sqrt(b*x + a)*sqrt(d*x + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.254542, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (2 \, b d x - 3 \, b c - 3 \, a d\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} +{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, a^{2} d^{2}\right )} \log \left (4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{16 \, \sqrt{b d} b^{2} d^{2}}, \frac{2 \,{\left (2 \, b d x - 3 \, b c - 3 \, a d\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} +{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, a^{2} d^{2}\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{8 \, \sqrt{-b d} b^{2} d^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(sqrt(b*x + a)*sqrt(d*x + c)),x, algorithm="fricas")

[Out]

[1/16*(4*(2*b*d*x - 3*b*c - 3*a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + (3*b^
2*c^2 + 2*a*b*c*d + 3*a^2*d^2)*log(4*(2*b^2*d^2*x + b^2*c*d + a*b*d^2)*sqrt(b*x
+ a)*sqrt(d*x + c) + (8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 8*(b^2*c*d
+ a*b*d^2)*x)*sqrt(b*d)))/(sqrt(b*d)*b^2*d^2), 1/8*(2*(2*b*d*x - 3*b*c - 3*a*d)
*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c) + (3*b^2*c^2 + 2*a*b*c*d + 3*a^2*d^2)*ar
ctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)/(sqrt(b*x + a)*sqrt(d*x + c)*b*d)))/(s
qrt(-b*d)*b^2*d^2)]

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.240261, size = 203, normalized size = 1.6 \[ \frac{{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a}{\left (\frac{2 \,{\left (b x + a\right )}}{b^{3} d} - \frac{3 \, b^{6} c d + 5 \, a b^{5} d^{2}}{b^{8} d^{3}}\right )} - \frac{{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, a^{2} d^{2}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} b^{2} d^{2}}\right )} b}{4 \,{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(sqrt(b*x + a)*sqrt(d*x + c)),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^3*d) - (3
*b^6*c*d + 5*a*b^5*d^2)/(b^8*d^3)) - (3*b^2*c^2 + 2*a*b*c*d + 3*a^2*d^2)*ln(abs(
-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b^2*
d^2))*b/abs(b)