3.537 \(\int x \sqrt{a+b x} \sqrt{c+d x} \, dx\)

Optimal. Leaf size=163 \[ \frac{1}{8} \sqrt{a+b x} \sqrt{c+d x} \left (\frac{a^2}{b^2}-\frac{c^2}{d^2}\right )+\frac{(a d+b c) (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{5/2} d^{5/2}}-\frac{(a+b x)^{3/2} \sqrt{c+d x} (a d+b c)}{4 b^2 d}+\frac{(a+b x)^{3/2} (c+d x)^{3/2}}{3 b d} \]

[Out]

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

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Rubi [A]  time = 0.2346, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{1}{8} \sqrt{a+b x} \sqrt{c+d x} \left (\frac{a^2}{b^2}-\frac{c^2}{d^2}\right )+\frac{(a d+b c) (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{5/2} d^{5/2}}-\frac{(a+b x)^{3/2} \sqrt{c+d x} (a d+b c)}{4 b^2 d}+\frac{(a+b x)^{3/2} (c+d x)^{3/2}}{3 b d} \]

Antiderivative was successfully verified.

[In]  Int[x*Sqrt[a + b*x]*Sqrt[c + d*x],x]

[Out]

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

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Rubi in Sympy [A]  time = 23.935, size = 141, normalized size = 0.87 \[ \sqrt{a + b x} \sqrt{c + d x} \left (\frac{a^{2}}{8 b^{2}} - \frac{c^{2}}{8 d^{2}}\right ) + \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}}}{3 b d} - \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d + b c\right )}{4 b^{2} d} + \frac{\left (a d - b c\right )^{2} \left (a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{8 b^{\frac{5}{2}} d^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(a + b*x)*sqrt(c + d*x)*(a**2/(8*b**2) - c**2/(8*d**2)) + (a + b*x)**(3/2)*(
c + d*x)**(3/2)/(3*b*d) - (a + b*x)**(3/2)*sqrt(c + d*x)*(a*d + b*c)/(4*b**2*d)
+ (a*d - b*c)**2*(a*d + b*c)*atanh(sqrt(d)*sqrt(a + b*x)/(sqrt(b)*sqrt(c + d*x))
)/(8*b**(5/2)*d**(5/2))

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Mathematica [A]  time = 0.11901, size = 145, normalized size = 0.89 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (-3 a^2 d^2+2 a b d (c+d x)+b^2 \left (-3 c^2+2 c d x+8 d^2 x^2\right )\right )}{24 b^2 d^2}+\frac{(a d+b c) (b c-a d)^2 \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{16 b^{5/2} d^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x*Sqrt[a + b*x]*Sqrt[c + d*x],x]

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(-3*a^2*d^2 + 2*a*b*d*(c + d*x) + b^2*(-3*c^2 + 2*c
*d*x + 8*d^2*x^2)))/(24*b^2*d^2) + ((b*c - a*d)^2*(b*c + a*d)*Log[b*c + a*d + 2*
b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(16*b^(5/2)*d^(5/2))

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Maple [B]  time = 0.017, size = 472, normalized size = 2.9 \[{\frac{1}{48\,{b}^{2}{d}^{2}}\sqrt{bx+a}\sqrt{dx+c} \left ( 16\,{x}^{2}{b}^{2}{d}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{3}{d}^{3}-3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}bc{d}^{2}-3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) a{b}^{2}{c}^{2}d+3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){b}^{3}{c}^{3}+4\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}xab{d}^{2}+4\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}x{b}^{2}cd-6\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{2}{d}^{2}+4\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}abcd-6\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{b}^{2}{c}^{2} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(16*x^2*b^2*d^2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)
*(b*d)^(1/2)+3*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d
+b*c)/(b*d)^(1/2))*a^3*d^3-3*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(
b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^2*b*c*d^2-3*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+
b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a*b^2*c^2*d+3*ln(1/2*(2*b*d*x
+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*b^3*c^3+4*(
b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a*b*d^2+4*(b*d)^(1/2)*(b*d*x^2+a*d*
x+b*c*x+a*c)^(1/2)*x*b^2*c*d-6*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^2*d
^2+4*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a*b*c*d-6*(b*d)^(1/2)*(b*d*x^2+
a*d*x+b*c*x+a*c)^(1/2)*b^2*c^2)/(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)/b^2/d^2/(b*d)^(1
/2)

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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(sqrt(b*x + a)*sqrt(d*x + c)*x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.277964, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (8 \, b^{2} d^{2} x^{2} - 3 \, b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2} + 2 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} + 3 \,{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} \log \left (4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{96 \, \sqrt{b d} b^{2} d^{2}}, \frac{2 \,{\left (8 \, b^{2} d^{2} x^{2} - 3 \, b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2} + 2 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} + 3 \,{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{48 \, \sqrt{-b d} b^{2} d^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/96*(4*(8*b^2*d^2*x^2 - 3*b^2*c^2 + 2*a*b*c*d - 3*a^2*d^2 + 2*(b^2*c*d + a*b*d
^2)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 3*(b^3*c^3 - a*b^2*c^2*d - a^2*b*
c*d^2 + a^3*d^3)*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), 1/48*(2*(8*b^2*d^2*x^2 - 3*b^2*c^2 + 2*a*b*c*d
- 3*a^2*d^2 + 2*(b^2*c*d + a*b*d^2)*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c) +
3*(b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*arctan(1/2*(2*b*d*x + b*c + a
*d)*sqrt(-b*d)/(sqrt(b*x + a)*sqrt(d*x + c)*b*d)))/(sqrt(-b*d)*b^2*d^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.231326, size = 259, normalized size = 1.59 \[ \frac{{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a}{\left (2 \,{\left (b x + a\right )}{\left (\frac{4 \,{\left (b x + a\right )}}{b^{6} d^{2}} + \frac{b c d^{3} - 7 \, a d^{4}}{b^{6} d^{6}}\right )} - \frac{3 \,{\left (b^{2} c^{2} d^{2} - a^{2} d^{4}\right )}}{b^{6} d^{6}}\right )} - \frac{3 \,{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} b^{5} d^{4}}\right )}{\left | b \right |}}{1920 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1920*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x +
a)/(b^6*d^2) + (b*c*d^3 - 7*a*d^4)/(b^6*d^6)) - 3*(b^2*c^2*d^2 - a^2*d^4)/(b^6*
d^6)) - 3*(b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*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^5*d^4))*abs(b)/b^
4