3.271 \(\int \frac{x^3}{\left (\sqrt{a+b x}+\sqrt{a+c x}\right )^2} \, dx\)
Optimal. Leaf size=195 \[ -\frac{a^3 (b+c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{b} \sqrt{a+c x}}\right )}{4 b^{5/2} c^{5/2}}+\frac{a^2 (b+c) \sqrt{a+b x} \sqrt{a+c x}}{4 b^2 c^2 (b-c)}+\frac{a (b+c) (a+b x)^{3/2} \sqrt{a+c x}}{2 b^2 c (b-c)^2}+\frac{a x^2}{(b-c)^2}-\frac{2 (a+b x)^{3/2} (a+c x)^{3/2}}{3 b c (b-c)^2}+\frac{x^3 (b+c)}{3 (b-c)^2} \]
[Out]
(a*x^2)/(b - c)^2 + ((b + c)*x^3)/(3*(b - c)^2) + (a^2*(b + c)*Sqrt[a + b*x]*Sqr
t[a + c*x])/(4*b^2*(b - c)*c^2) + (a*(b + c)*(a + b*x)^(3/2)*Sqrt[a + c*x])/(2*b
^2*(b - c)^2*c) - (2*(a + b*x)^(3/2)*(a + c*x)^(3/2))/(3*b*(b - c)^2*c) - (a^3*(
b + c)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[a + c*x])])/(4*b^(5/2)*c^(5
/2))
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Rubi [A] time = 0.644019, antiderivative size = 195, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2
\[ -\frac{a^3 (b+c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{b} \sqrt{a+c x}}\right )}{4 b^{5/2} c^{5/2}}+\frac{a^2 (b+c) \sqrt{a+b x} \sqrt{a+c x}}{4 b^2 c^2 (b-c)}+\frac{a (b+c) (a+b x)^{3/2} \sqrt{a+c x}}{2 b^2 c (b-c)^2}+\frac{a x^2}{(b-c)^2}-\frac{2 (a+b x)^{3/2} (a+c x)^{3/2}}{3 b c (b-c)^2}+\frac{x^3 (b+c)}{3 (b-c)^2} \]
Antiderivative was successfully verified.
[In] Int[x^3/(Sqrt[a + b*x] + Sqrt[a + c*x])^2,x]
[Out]
(a*x^2)/(b - c)^2 + ((b + c)*x^3)/(3*(b - c)^2) + (a^2*(b + c)*Sqrt[a + b*x]*Sqr
t[a + c*x])/(4*b^2*(b - c)*c^2) + (a*(b + c)*(a + b*x)^(3/2)*Sqrt[a + c*x])/(2*b
^2*(b - c)^2*c) - (2*(a + b*x)^(3/2)*(a + c*x)^(3/2))/(3*b*(b - c)^2*c) - (a^3*(
b + c)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[a + c*x])])/(4*b^(5/2)*c^(5
/2))
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{a^{3} \left (b + c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{b} \sqrt{a + c x}} \right )}}{4 b^{\frac{5}{2}} c^{\frac{5}{2}}} + \frac{a^{2} \sqrt{a + b x} \sqrt{a + c x} \left (b + c\right )}{4 b^{2} c^{2} \left (b - c\right )} + \frac{2 a \int x\, dx}{\left (b - c\right )^{2}} + \frac{a \left (a + b x\right )^{\frac{3}{2}} \sqrt{a + c x} \left (b + c\right )}{2 b^{2} c \left (b - c\right )^{2}} + \frac{x^{3} \left (b + c\right )}{3 \left (b - c\right )^{2}} - \frac{2 \left (a + b x\right )^{\frac{3}{2}} \left (a + c x\right )^{\frac{3}{2}}}{3 b c \left (b - c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3/((b*x+a)**(1/2)+(c*x+a)**(1/2))**2,x)
[Out]
-a**3*(b + c)*atanh(sqrt(c)*sqrt(a + b*x)/(sqrt(b)*sqrt(a + c*x)))/(4*b**(5/2)*c
**(5/2)) + a**2*sqrt(a + b*x)*sqrt(a + c*x)*(b + c)/(4*b**2*c**2*(b - c)) + 2*a*
Integral(x, x)/(b - c)**2 + a*(a + b*x)**(3/2)*sqrt(a + c*x)*(b + c)/(2*b**2*c*(
b - c)**2) + x**3*(b + c)/(3*(b - c)**2) - 2*(a + b*x)**(3/2)*(a + c*x)**(3/2)/(
3*b*c*(b - c)**2)
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Mathematica [A] time = 0.161263, size = 168, normalized size = 0.86 \[ -\frac{a^3 (b+c) \log \left (2 \sqrt{b} \sqrt{c} \sqrt{a+b x} \sqrt{a+c x}+a b+a c+2 b c x\right )}{8 b^{5/2} c^{5/2}}+\frac{\sqrt{a+b x} \sqrt{a+c x} \left (a^2 \left (3 b^2-2 b c+3 c^2\right )-2 a b c x (b+c)-8 b^2 c^2 x^2\right )}{12 b^2 c^2 (b-c)^2}+\frac{a x^2}{(b-c)^2}+\frac{x^3 (b+c)}{3 (b-c)^2} \]
Antiderivative was successfully verified.
[In] Integrate[x^3/(Sqrt[a + b*x] + Sqrt[a + c*x])^2,x]
[Out]
(a*x^2)/(b - c)^2 + ((b + c)*x^3)/(3*(b - c)^2) + (Sqrt[a + b*x]*Sqrt[a + c*x]*(
a^2*(3*b^2 - 2*b*c + 3*c^2) - 2*a*b*c*(b + c)*x - 8*b^2*c^2*x^2))/(12*b^2*(b - c
)^2*c^2) - (a^3*(b + c)*Log[a*b + a*c + 2*b*c*x + 2*Sqrt[b]*Sqrt[c]*Sqrt[a + b*x
]*Sqrt[a + c*x]])/(8*b^(5/2)*c^(5/2))
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Maple [B] time = 0.019, size = 517, normalized size = 2.7 \[{\frac{b{x}^{3}}{3\, \left ( b-c \right ) ^{2}}}+{\frac{c{x}^{3}}{3\, \left ( b-c \right ) ^{2}}}+{\frac{a{x}^{2}}{ \left ( b-c \right ) ^{2}}}-{\frac{1}{24\, \left ( b-c \right ) ^{2}{b}^{2}{c}^{2}}\sqrt{bx+a}\sqrt{cx+a} \left ( 16\,{x}^{2}{b}^{2}{c}^{2}\sqrt{bc{x}^{2}+abx+acx+{a}^{2}}\sqrt{bc}+3\,\ln \left ( 1/2\,{\frac{2\,bcx+2\,\sqrt{bc{x}^{2}+abx+acx+{a}^{2}}\sqrt{bc}+ab+ac}{\sqrt{bc}}} \right ){a}^{3}{b}^{3}-3\,\ln \left ( 1/2\,{\frac{2\,bcx+2\,\sqrt{bc{x}^{2}+abx+acx+{a}^{2}}\sqrt{bc}+ab+ac}{\sqrt{bc}}} \right ){a}^{3}{b}^{2}c-3\,\ln \left ( 1/2\,{\frac{2\,bcx+2\,\sqrt{bc{x}^{2}+abx+acx+{a}^{2}}\sqrt{bc}+ab+ac}{\sqrt{bc}}} \right ){a}^{3}b{c}^{2}+3\,\ln \left ( 1/2\,{\frac{2\,bcx+2\,\sqrt{bc{x}^{2}+abx+acx+{a}^{2}}\sqrt{bc}+ab+ac}{\sqrt{bc}}} \right ){a}^{3}{c}^{3}+4\,\sqrt{bc}\sqrt{bc{x}^{2}+abx+acx+{a}^{2}}xa{b}^{2}c+4\,\sqrt{bc}\sqrt{bc{x}^{2}+abx+acx+{a}^{2}}xab{c}^{2}-6\,\sqrt{bc}\sqrt{bc{x}^{2}+abx+acx+{a}^{2}}{a}^{2}{b}^{2}+4\,\sqrt{bc}\sqrt{bc{x}^{2}+abx+acx+{a}^{2}}{a}^{2}bc-6\,\sqrt{bc}\sqrt{bc{x}^{2}+abx+acx+{a}^{2}}{a}^{2}{c}^{2} \right ){\frac{1}{\sqrt{bc{x}^{2}+abx+acx+{a}^{2}}}}{\frac{1}{\sqrt{bc}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3/((b*x+a)^(1/2)+(c*x+a)^(1/2))^2,x)
[Out]
1/3*x^3/(b-c)^2*b+1/3*x^3/(b-c)^2*c+a*x^2/(b-c)^2-1/24/(b-c)^2*(b*x+a)^(1/2)*(c*
x+a)^(1/2)*(16*x^2*b^2*c^2*(b*c*x^2+a*b*x+a*c*x+a^2)^(1/2)*(b*c)^(1/2)+3*ln(1/2*
(2*b*c*x+2*(b*c*x^2+a*b*x+a*c*x+a^2)^(1/2)*(b*c)^(1/2)+a*b+a*c)/(b*c)^(1/2))*a^3
*b^3-3*ln(1/2*(2*b*c*x+2*(b*c*x^2+a*b*x+a*c*x+a^2)^(1/2)*(b*c)^(1/2)+a*b+a*c)/(b
*c)^(1/2))*a^3*b^2*c-3*ln(1/2*(2*b*c*x+2*(b*c*x^2+a*b*x+a*c*x+a^2)^(1/2)*(b*c)^(
1/2)+a*b+a*c)/(b*c)^(1/2))*a^3*b*c^2+3*ln(1/2*(2*b*c*x+2*(b*c*x^2+a*b*x+a*c*x+a^
2)^(1/2)*(b*c)^(1/2)+a*b+a*c)/(b*c)^(1/2))*a^3*c^3+4*(b*c)^(1/2)*(b*c*x^2+a*b*x+
a*c*x+a^2)^(1/2)*x*a*b^2*c+4*(b*c)^(1/2)*(b*c*x^2+a*b*x+a*c*x+a^2)^(1/2)*x*a*b*c
^2-6*(b*c)^(1/2)*(b*c*x^2+a*b*x+a*c*x+a^2)^(1/2)*a^2*b^2+4*(b*c)^(1/2)*(b*c*x^2+
a*b*x+a*c*x+a^2)^(1/2)*a^2*b*c-6*(b*c)^(1/2)*(b*c*x^2+a*b*x+a*c*x+a^2)^(1/2)*a^2
*c^2)/(b*c*x^2+a*b*x+a*c*x+a^2)^(1/2)/b^2/c^2/(b*c)^(1/2)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{{\left (\sqrt{b x + a} + \sqrt{c x + a}\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(sqrt(b*x + a) + sqrt(c*x + a))^2,x, algorithm="maxima")
[Out]
integrate(x^3/(sqrt(b*x + a) + sqrt(c*x + a))^2, x)
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Fricas [A] time = 0.324604, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(sqrt(b*x + a) + sqrt(c*x + a))^2,x, algorithm="fricas")
[Out]
[1/24*(2*(32*(b^5*c^2 + 7*b^4*c^3 + 7*b^3*c^4 + b^2*c^5)*x^5 + 2*(a*b^5*c + 188*
a*b^4*c^2 + 518*a*b^3*c^3 + 188*a*b^2*c^4 + a*b*c^5)*x^4 - (3*a^2*b^5 + 7*a^2*b^
4*c - 1034*a^2*b^3*c^2 - 1034*a^2*b^2*c^3 + 7*a^2*b*c^4 + 3*a^2*c^5)*x^3 - 12*(3
*a^3*b^4 - 70*a^3*b^2*c^2 + 3*a^3*c^4)*x^2 - 48*(a^4*b^3 - a^4*b^2*c - a^4*b*c^2
+ a^4*c^3)*x)*sqrt(b*c)*sqrt(b*x + a)*sqrt(c*x + a) - 3*(32*a^6*b^3 - 32*a^6*b^
2*c - 32*a^6*b*c^2 + 32*a^6*c^3 + (a^3*b^6 + 14*a^3*b^5*c - a^3*b^4*c^2 - 28*a^3
*b^3*c^3 - a^3*b^2*c^4 + 14*a^3*b*c^5 + a^3*c^6)*x^3 + 6*(3*a^4*b^5 + 7*a^4*b^4*
c - 10*a^4*b^3*c^2 - 10*a^4*b^2*c^3 + 7*a^4*b*c^4 + 3*a^4*c^5)*x^2 - 2*(16*a^5*b
^3 - 16*a^5*b^2*c - 16*a^5*b*c^2 + 16*a^5*c^3 + (3*a^3*b^5 + 7*a^3*b^4*c - 10*a^
3*b^3*c^2 - 10*a^3*b^2*c^3 + 7*a^3*b*c^4 + 3*a^3*c^5)*x^2 + 16*(a^4*b^4 - 2*a^4*
b^2*c^2 + a^4*c^4)*x)*sqrt(b*x + a)*sqrt(c*x + a) + 48*(a^5*b^4 - 2*a^5*b^2*c^2
+ a^5*c^4)*x)*log((2*a*b*c*x - 2*(b*c*x + sqrt(b*c)*a)*sqrt(b*x + a)*sqrt(c*x +
a) + (2*b*c*x^2 + 2*a^2 + (a*b + a*c)*x)*sqrt(b*c))/((b + c)*x - 2*sqrt(b*x + a)
*sqrt(c*x + a) + 2*a)) - 2*(4*(b^6*c^2 + 28*b^5*c^3 + 70*b^4*c^4 + 28*b^3*c^5 +
b^2*c^6)*x^6 + 144*(a*b^5*c^2 + 7*a*b^4*c^3 + 7*a*b^3*c^4 + a*b^2*c^5)*x^5 - 6*(
a^2*b^5*c - 132*a^2*b^4*c^2 - 378*a^2*b^3*c^3 - 132*a^2*b^2*c^4 + a^2*b*c^5)*x^4
- (15*a^3*b^5 + 43*a^3*b^4*c - 1466*a^3*b^3*c^2 - 1466*a^3*b^2*c^3 + 43*a^3*b*c
^4 + 15*a^3*c^5)*x^3 - 12*(5*a^4*b^4 - 74*a^4*b^2*c^2 + 5*a^4*c^4)*x^2 - 48*(a^5
*b^3 - a^5*b^2*c - a^5*b*c^2 + a^5*c^3)*x)*sqrt(b*c))/(2*(16*a^2*b^4*c^2 - 32*a^
2*b^3*c^3 + 16*a^2*b^2*c^4 + (3*b^6*c^2 + 4*b^5*c^3 - 14*b^4*c^4 + 4*b^3*c^5 + 3
*b^2*c^6)*x^2 + 16*(a*b^5*c^2 - a*b^4*c^3 - a*b^3*c^4 + a*b^2*c^5)*x)*sqrt(b*c)*
sqrt(b*x + a)*sqrt(c*x + a) - (32*a^3*b^4*c^2 - 64*a^3*b^3*c^3 + 32*a^3*b^2*c^4
+ (b^7*c^2 + 13*b^6*c^3 - 14*b^5*c^4 - 14*b^4*c^5 + 13*b^3*c^6 + b^2*c^7)*x^3 +
6*(3*a*b^6*c^2 + 4*a*b^5*c^3 - 14*a*b^4*c^4 + 4*a*b^3*c^5 + 3*a*b^2*c^6)*x^2 + 4
8*(a^2*b^5*c^2 - a^2*b^4*c^3 - a^2*b^3*c^4 + a^2*b^2*c^5)*x)*sqrt(b*c)), 1/12*((
32*(b^5*c^2 + 7*b^4*c^3 + 7*b^3*c^4 + b^2*c^5)*x^5 + 2*(a*b^5*c + 188*a*b^4*c^2
+ 518*a*b^3*c^3 + 188*a*b^2*c^4 + a*b*c^5)*x^4 - (3*a^2*b^5 + 7*a^2*b^4*c - 1034
*a^2*b^3*c^2 - 1034*a^2*b^2*c^3 + 7*a^2*b*c^4 + 3*a^2*c^5)*x^3 - 12*(3*a^3*b^4 -
70*a^3*b^2*c^2 + 3*a^3*c^4)*x^2 - 48*(a^4*b^3 - a^4*b^2*c - a^4*b*c^2 + a^4*c^3
)*x)*sqrt(-b*c)*sqrt(b*x + a)*sqrt(c*x + a) + 3*(32*a^6*b^3 - 32*a^6*b^2*c - 32*
a^6*b*c^2 + 32*a^6*c^3 + (a^3*b^6 + 14*a^3*b^5*c - a^3*b^4*c^2 - 28*a^3*b^3*c^3
- a^3*b^2*c^4 + 14*a^3*b*c^5 + a^3*c^6)*x^3 + 6*(3*a^4*b^5 + 7*a^4*b^4*c - 10*a^
4*b^3*c^2 - 10*a^4*b^2*c^3 + 7*a^4*b*c^4 + 3*a^4*c^5)*x^2 - 2*(16*a^5*b^3 - 16*a
^5*b^2*c - 16*a^5*b*c^2 + 16*a^5*c^3 + (3*a^3*b^5 + 7*a^3*b^4*c - 10*a^3*b^3*c^2
- 10*a^3*b^2*c^3 + 7*a^3*b*c^4 + 3*a^3*c^5)*x^2 + 16*(a^4*b^4 - 2*a^4*b^2*c^2 +
a^4*c^4)*x)*sqrt(b*x + a)*sqrt(c*x + a) + 48*(a^5*b^4 - 2*a^5*b^2*c^2 + a^5*c^4
)*x)*arctan((sqrt(-b*c)*sqrt(b*x + a)*sqrt(c*x + a) - sqrt(-b*c)*a)/(b*c*x)) - (
4*(b^6*c^2 + 28*b^5*c^3 + 70*b^4*c^4 + 28*b^3*c^5 + b^2*c^6)*x^6 + 144*(a*b^5*c^
2 + 7*a*b^4*c^3 + 7*a*b^3*c^4 + a*b^2*c^5)*x^5 - 6*(a^2*b^5*c - 132*a^2*b^4*c^2
- 378*a^2*b^3*c^3 - 132*a^2*b^2*c^4 + a^2*b*c^5)*x^4 - (15*a^3*b^5 + 43*a^3*b^4*
c - 1466*a^3*b^3*c^2 - 1466*a^3*b^2*c^3 + 43*a^3*b*c^4 + 15*a^3*c^5)*x^3 - 12*(5
*a^4*b^4 - 74*a^4*b^2*c^2 + 5*a^4*c^4)*x^2 - 48*(a^5*b^3 - a^5*b^2*c - a^5*b*c^2
+ a^5*c^3)*x)*sqrt(-b*c))/(2*(16*a^2*b^4*c^2 - 32*a^2*b^3*c^3 + 16*a^2*b^2*c^4
+ (3*b^6*c^2 + 4*b^5*c^3 - 14*b^4*c^4 + 4*b^3*c^5 + 3*b^2*c^6)*x^2 + 16*(a*b^5*c
^2 - a*b^4*c^3 - a*b^3*c^4 + a*b^2*c^5)*x)*sqrt(-b*c)*sqrt(b*x + a)*sqrt(c*x + a
) - (32*a^3*b^4*c^2 - 64*a^3*b^3*c^3 + 32*a^3*b^2*c^4 + (b^7*c^2 + 13*b^6*c^3 -
14*b^5*c^4 - 14*b^4*c^5 + 13*b^3*c^6 + b^2*c^7)*x^3 + 6*(3*a*b^6*c^2 + 4*a*b^5*c
^3 - 14*a*b^4*c^4 + 4*a*b^3*c^5 + 3*a*b^2*c^6)*x^2 + 48*(a^2*b^5*c^2 - a^2*b^4*c
^3 - a^2*b^3*c^4 + a^2*b^2*c^5)*x)*sqrt(-b*c))]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\left (\sqrt{a + b x} + \sqrt{a + c x}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3/((b*x+a)**(1/2)+(c*x+a)**(1/2))**2,x)
[Out]
Integral(x**3/(sqrt(a + b*x) + sqrt(a + c*x))**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^3/(sqrt(b*x + a) + sqrt(c*x + a))^2,x, algorithm="giac")
[Out]
Timed out