3.245 \(\int \frac{x^2}{\left (\sqrt{a+b x}+\sqrt{c+b x}\right )^2} \, dx\)

Optimal. Leaf size=228 \[ \frac{5 (a+c) (a+b x)^{3/2} (b x+c)^{3/2}}{12 b^3 (a-c)^2}+\frac{\left (4 a c-5 (a+c)^2\right ) (a+b x)^{3/2} \sqrt{b x+c}}{16 b^3 (a-c)^2}-\frac{\left (4 a c-5 (a+c)^2\right ) \sqrt{a+b x} \sqrt{b x+c}}{32 b^3 (a-c)}-\frac{\left (4 a c-5 (a+c)^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{b x+c}}\right )}{32 b^3}-\frac{x (a+b x)^{3/2} (b x+c)^{3/2}}{2 b^2 (a-c)^2}+\frac{b x^4}{2 (a-c)^2}+\frac{x^3 (a+c)}{3 (a-c)^2} \]

[Out]

((a + c)*x^3)/(3*(a - c)^2) + (b*x^4)/(2*(a - c)^2) - ((4*a*c - 5*(a + c)^2)*Sqr
t[a + b*x]*Sqrt[c + b*x])/(32*b^3*(a - c)) + ((4*a*c - 5*(a + c)^2)*(a + b*x)^(3
/2)*Sqrt[c + b*x])/(16*b^3*(a - c)^2) + (5*(a + c)*(a + b*x)^(3/2)*(c + b*x)^(3/
2))/(12*b^3*(a - c)^2) - (x*(a + b*x)^(3/2)*(c + b*x)^(3/2))/(2*b^2*(a - c)^2) -
((4*a*c - 5*(a + c)^2)*ArcTanh[Sqrt[a + b*x]/Sqrt[c + b*x]])/(32*b^3)

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Rubi [A]  time = 0.751523, antiderivative size = 228, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28 \[ \frac{5 (a+c) (a+b x)^{3/2} (b x+c)^{3/2}}{12 b^3 (a-c)^2}+\frac{\left (4 a c-5 (a+c)^2\right ) (a+b x)^{3/2} \sqrt{b x+c}}{16 b^3 (a-c)^2}-\frac{\left (4 a c-5 (a+c)^2\right ) \sqrt{a+b x} \sqrt{b x+c}}{32 b^3 (a-c)}-\frac{\left (4 a c-5 (a+c)^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{b x+c}}\right )}{32 b^3}-\frac{x (a+b x)^{3/2} (b x+c)^{3/2}}{2 b^2 (a-c)^2}+\frac{b x^4}{2 (a-c)^2}+\frac{x^3 (a+c)}{3 (a-c)^2} \]

Antiderivative was successfully verified.

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

[Out]

((a + c)*x^3)/(3*(a - c)^2) + (b*x^4)/(2*(a - c)^2) - ((4*a*c - 5*(a + c)^2)*Sqr
t[a + b*x]*Sqrt[c + b*x])/(32*b^3*(a - c)) + ((4*a*c - 5*(a + c)^2)*(a + b*x)^(3
/2)*Sqrt[c + b*x])/(16*b^3*(a - c)^2) + (5*(a + c)*(a + b*x)^(3/2)*(c + b*x)^(3/
2))/(12*b^3*(a - c)^2) - (x*(a + b*x)^(3/2)*(c + b*x)^(3/2))/(2*b^2*(a - c)^2) -
((4*a*c - 5*(a + c)^2)*ArcTanh[Sqrt[a + b*x]/Sqrt[c + b*x]])/(32*b^3)

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Rubi in Sympy [A]  time = 61.8664, size = 197, normalized size = 0.86 \[ \frac{b x^{4}}{2 \left (a - c\right )^{2}} + \frac{x^{3} \left (a + c\right )}{3 \left (a - c\right )^{2}} - \frac{x \left (a + b x\right )^{\frac{3}{2}} \left (b x + c\right )^{\frac{3}{2}}}{2 b^{2} \left (a - c\right )^{2}} - \frac{\left (a c - \frac{5 \left (a + c\right )^{2}}{4}\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{b x + c}} \right )}}{8 b^{3}} + \frac{\sqrt{a + b x} \left (4 a c - 5 \left (a + c\right )^{2}\right ) \sqrt{b x + c}}{32 b^{3} \left (a - c\right )} + \frac{5 \left (a + c\right ) \left (a + b x\right )^{\frac{3}{2}} \left (b x + c\right )^{\frac{3}{2}}}{12 b^{3} \left (a - c\right )^{2}} + \frac{\sqrt{a + b x} \left (a c - \frac{5 \left (a + c\right )^{2}}{4}\right ) \left (b x + c\right )^{\frac{3}{2}}}{4 b^{3} \left (a - c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*x**4/(2*(a - c)**2) + x**3*(a + c)/(3*(a - c)**2) - x*(a + b*x)**(3/2)*(b*x +
c)**(3/2)/(2*b**2*(a - c)**2) - (a*c - 5*(a + c)**2/4)*atanh(sqrt(a + b*x)/sqrt(
b*x + c))/(8*b**3) + sqrt(a + b*x)*(4*a*c - 5*(a + c)**2)*sqrt(b*x + c)/(32*b**3
*(a - c)) + 5*(a + c)*(a + b*x)**(3/2)*(b*x + c)**(3/2)/(12*b**3*(a - c)**2) + s
qrt(a + b*x)*(a*c - 5*(a + c)**2/4)*(b*x + c)**(3/2)/(4*b**3*(a - c)**2)

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Mathematica [A]  time = 0.184766, size = 167, normalized size = 0.73 \[ \frac{3 (a-c)^2 \left (5 a^2+6 a c+5 c^2\right ) \log \left (2 \sqrt{a+b x} \sqrt{b x+c}+a+2 b x+c\right )-2 \sqrt{a+b x} \sqrt{b x+c} \left (15 a^3-2 b x \left (5 a^2-2 a c+5 c^2\right )-7 a^2 c+8 b^2 x^2 (a+c)-7 a c^2+48 b^3 x^3+15 c^3\right )+64 b^3 x^3 (a+c)+96 b^4 x^4}{192 b^3 (a-c)^2} \]

Antiderivative was successfully verified.

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

[Out]

(64*b^3*(a + c)*x^3 + 96*b^4*x^4 - 2*Sqrt[a + b*x]*Sqrt[c + b*x]*(15*a^3 - 7*a^2
*c - 7*a*c^2 + 15*c^3 - 2*b*(5*a^2 - 2*a*c + 5*c^2)*x + 8*b^2*(a + c)*x^2 + 48*b
^3*x^3) + 3*(a - c)^2*(5*a^2 + 6*a*c + 5*c^2)*Log[a + c + 2*b*x + 2*Sqrt[a + b*x
]*Sqrt[c + b*x]])/(192*b^3*(a - c)^2)

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Maple [C]  time = 0.027, size = 604, normalized size = 2.7 \[{\frac{a{x}^{3}}{3\, \left ( a-c \right ) ^{2}}}+{\frac{c{x}^{3}}{3\, \left ( a-c \right ) ^{2}}}+{\frac{b{x}^{4}}{2\, \left ( a-c \right ) ^{2}}}-{\frac{{\it csgn} \left ( b \right ) }{192\, \left ( a-c \right ) ^{2}{b}^{3}}\sqrt{bx+a}\sqrt{bx+c} \left ( 96\,{\it csgn} \left ( b \right ){x}^{3}{b}^{3}\sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}+16\,{\it csgn} \left ( b \right ){x}^{2}a{b}^{2}\sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}+16\,{\it csgn} \left ( b \right ){x}^{2}{b}^{2}c\sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}-20\,{\it csgn} \left ( b \right ) \sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}x{a}^{2}b+8\,{\it csgn} \left ( b \right ) \sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}xabc-20\,{\it csgn} \left ( b \right ) \sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}xb{c}^{2}+30\,{\it csgn} \left ( b \right ) \sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}{a}^{3}-14\,{\it csgn} \left ( b \right ) \sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}{a}^{2}c-14\,{\it csgn} \left ( b \right ) \sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}a{c}^{2}+30\,{\it csgn} \left ( b \right ) \sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}{c}^{3}-15\,\ln \left ( 1/2\, \left ( 2\,{\it csgn} \left ( b \right ) \sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}+2\,bx+a+c \right ){\it csgn} \left ( b \right ) \right ){a}^{4}+12\,\ln \left ( 1/2\, \left ( 2\,{\it csgn} \left ( b \right ) \sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}+2\,bx+a+c \right ){\it csgn} \left ( b \right ) \right ){a}^{3}c+6\,\ln \left ( 1/2\, \left ( 2\,{\it csgn} \left ( b \right ) \sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}+2\,bx+a+c \right ){\it csgn} \left ( b \right ) \right ){a}^{2}{c}^{2}+12\,\ln \left ( 1/2\, \left ( 2\,{\it csgn} \left ( b \right ) \sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}+2\,bx+a+c \right ){\it csgn} \left ( b \right ) \right ) a{c}^{3}-15\,\ln \left ( 1/2\, \left ( 2\,{\it csgn} \left ( b \right ) \sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}+2\,bx+a+c \right ){\it csgn} \left ( b \right ) \right ){c}^{4} \right ){\frac{1}{\sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*x^3/(a-c)^2*a+1/3*x^3/(a-c)^2*c+1/2*b*x^4/(a-c)^2-1/192/(a-c)^2*(b*x+a)^(1/2
)*(b*x+c)^(1/2)*(96*csgn(b)*x^3*b^3*(b^2*x^2+a*b*x+b*c*x+a*c)^(1/2)+16*csgn(b)*x
^2*a*b^2*(b^2*x^2+a*b*x+b*c*x+a*c)^(1/2)+16*csgn(b)*x^2*b^2*c*(b^2*x^2+a*b*x+b*c
*x+a*c)^(1/2)-20*csgn(b)*(b^2*x^2+a*b*x+b*c*x+a*c)^(1/2)*x*a^2*b+8*csgn(b)*(b^2*
x^2+a*b*x+b*c*x+a*c)^(1/2)*x*a*b*c-20*csgn(b)*(b^2*x^2+a*b*x+b*c*x+a*c)^(1/2)*x*
b*c^2+30*csgn(b)*(b^2*x^2+a*b*x+b*c*x+a*c)^(1/2)*a^3-14*csgn(b)*(b^2*x^2+a*b*x+b
*c*x+a*c)^(1/2)*a^2*c-14*csgn(b)*(b^2*x^2+a*b*x+b*c*x+a*c)^(1/2)*a*c^2+30*csgn(b
)*(b^2*x^2+a*b*x+b*c*x+a*c)^(1/2)*c^3-15*ln(1/2*(2*csgn(b)*(b^2*x^2+a*b*x+b*c*x+
a*c)^(1/2)+2*b*x+a+c)*csgn(b))*a^4+12*ln(1/2*(2*csgn(b)*(b^2*x^2+a*b*x+b*c*x+a*c
)^(1/2)+2*b*x+a+c)*csgn(b))*a^3*c+6*ln(1/2*(2*csgn(b)*(b^2*x^2+a*b*x+b*c*x+a*c)^
(1/2)+2*b*x+a+c)*csgn(b))*a^2*c^2+12*ln(1/2*(2*csgn(b)*(b^2*x^2+a*b*x+b*c*x+a*c)
^(1/2)+2*b*x+a+c)*csgn(b))*a*c^3-15*ln(1/2*(2*csgn(b)*(b^2*x^2+a*b*x+b*c*x+a*c)^
(1/2)+2*b*x+a+c)*csgn(b))*c^4)*csgn(b)/b^3/(b^2*x^2+a*b*x+b*c*x+a*c)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (\sqrt{b x + a} + \sqrt{b x + c}\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^2/(sqrt(b*x + a) + sqrt(b*x + c))^2, x)

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Fricas [A]  time = 0.309121, size = 2361, normalized size = 10.36 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1536*(196608*b^8*x^8 - 59*a^8 + 224*a^7*c + 7212*a^6*c^2 + 2336*a^5*c^3 - 9186
*a^4*c^4 + 2336*a^3*c^5 + 7212*a^2*c^6 + 224*a*c^7 - 59*c^8 + 524288*(a*b^7 + b^
7*c)*x^7 + 8192*(59*a^2*b^6 + 154*a*b^6*c + 59*b^6*c^2)*x^6 + 8192*(23*a^3*b^5 +
121*a^2*b^5*c + 121*a*b^5*c^2 + 23*b^5*c^3)*x^5 + 128*(471*a^4*b^4 + 2188*a^3*b
^4*c + 4490*a^2*b^4*c^2 + 2188*a*b^4*c^3 + 471*b^4*c^4)*x^4 + 256*(147*a^5*b^3 +
303*a^4*b^3*c + 142*a^3*b^3*c^2 + 142*a^2*b^3*c^3 + 303*a*b^3*c^4 + 147*b^3*c^5
)*x^3 + 32*(325*a^6*b^2 + 2114*a^5*b^2*c - 37*a^4*b^2*c^2 - 2884*a^3*b^2*c^3 - 3
7*a^2*b^2*c^4 + 2114*a*b^2*c^5 + 325*b^2*c^6)*x^2 - 8*(24576*b^7*x^7 - 29*a^7 +
369*a^6*c + 1003*a^5*c^2 - 703*a^4*c^3 - 703*a^3*c^4 + 1003*a^2*c^5 + 369*a*c^6
- 29*c^7 + 53248*(a*b^6 + b^6*c)*x^6 + 12288*(3*a^2*b^5 + 8*a*b^5*c + 3*b^5*c^2)
*x^5 + 10240*(a^3*b^4 + 5*a^2*b^4*c + 5*a*b^4*c^2 + b^4*c^3)*x^4 + 16*(291*a^4*b
^3 + 412*a^3*b^3*c + 722*a^2*b^3*c^2 + 412*a*b^3*c^3 + 291*b^3*c^4)*x^3 + 24*(11
5*a^5*b^2 + 207*a^4*b^2*c - 242*a^3*b^2*c^2 - 242*a^2*b^2*c^3 + 207*a*b^2*c^4 +
115*b^2*c^5)*x^2 + 2*(175*a^6*b + 1814*a^5*b*c + 209*a^4*b*c^2 - 2476*a^3*b*c^3
+ 209*a^2*b*c^4 + 1814*a*b*c^5 + 175*b*c^6)*x)*sqrt(b*x + a)*sqrt(b*x + c) + 32*
(a^7*b + 555*a^6*b*c + 1033*a^5*b*c^2 - 949*a^4*b*c^3 - 949*a^3*b*c^4 + 1033*a^2
*b*c^5 + 555*a*b*c^6 + b*c^7)*x - 24*(5*a^8 + 136*a^7*c + 236*a^6*c^2 - 200*a^5*
c^3 - 354*a^4*c^4 - 200*a^3*c^5 + 236*a^2*c^6 + 136*a*c^7 + 5*c^8 + 128*(5*a^4*b
^4 - 4*a^3*b^4*c - 2*a^2*b^4*c^2 - 4*a*b^4*c^3 + 5*b^4*c^4)*x^4 + 256*(5*a^5*b^3
+ a^4*b^3*c - 6*a^3*b^3*c^2 - 6*a^2*b^3*c^3 + a*b^3*c^4 + 5*b^3*c^5)*x^3 + 32*(
25*a^6*b^2 + 50*a^5*b^2*c - 41*a^4*b^2*c^2 - 68*a^3*b^2*c^3 - 41*a^2*b^2*c^4 + 5
0*a*b^2*c^5 + 25*b^2*c^6)*x^2 - 8*(5*a^7 + 31*a^6*c + 5*a^5*c^2 - 41*a^4*c^3 - 4
1*a^3*c^4 + 5*a^2*c^5 + 31*a*c^6 + 5*c^7 + 16*(5*a^4*b^3 - 4*a^3*b^3*c - 2*a^2*b
^3*c^2 - 4*a*b^3*c^3 + 5*b^3*c^4)*x^3 + 24*(5*a^5*b^2 + a^4*b^2*c - 6*a^3*b^2*c^
2 - 6*a^2*b^2*c^3 + a*b^2*c^4 + 5*b^2*c^5)*x^2 + 2*(25*a^6*b + 50*a^5*b*c - 41*a
^4*b*c^2 - 68*a^3*b*c^3 - 41*a^2*b*c^4 + 50*a*b*c^5 + 25*b*c^6)*x)*sqrt(b*x + a)
*sqrt(b*x + c) + 32*(5*a^7*b + 31*a^6*b*c + 5*a^5*b*c^2 - 41*a^4*b*c^3 - 41*a^3*
b*c^4 + 5*a^2*b*c^5 + 31*a*b*c^6 + 5*b*c^7)*x)*log(-2*b*x + 2*sqrt(b*x + a)*sqrt
(b*x + c) - a - c))/(a^6*b^3 + 26*a^5*b^3*c + 15*a^4*b^3*c^2 - 84*a^3*b^3*c^3 +
15*a^2*b^3*c^4 + 26*a*b^3*c^5 + b^3*c^6 + 128*(a^2*b^7 - 2*a*b^7*c + b^7*c^2)*x^
4 + 256*(a^3*b^6 - a^2*b^6*c - a*b^6*c^2 + b^6*c^3)*x^3 + 32*(5*a^4*b^5 + 4*a^3*
b^5*c - 18*a^2*b^5*c^2 + 4*a*b^5*c^3 + 5*b^5*c^4)*x^2 - 8*(a^5*b^3 + 5*a^4*b^3*c
- 6*a^3*b^3*c^2 - 6*a^2*b^3*c^3 + 5*a*b^3*c^4 + b^3*c^5 + 16*(a^2*b^6 - 2*a*b^6
*c + b^6*c^2)*x^3 + 24*(a^3*b^5 - a^2*b^5*c - a*b^5*c^2 + b^5*c^3)*x^2 + 2*(5*a^
4*b^4 + 4*a^3*b^4*c - 18*a^2*b^4*c^2 + 4*a*b^4*c^3 + 5*b^4*c^4)*x)*sqrt(b*x + a)
*sqrt(b*x + c) + 32*(a^5*b^4 + 5*a^4*b^4*c - 6*a^3*b^4*c^2 - 6*a^2*b^4*c^3 + 5*a
*b^4*c^4 + b^4*c^5)*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out