3.845 \(\int \frac{15 d^2+20 d e x+8 e^2 x^2}{\sqrt{a+b x} \sqrt{d+e x}} \, dx\)

Optimal. Leaf size=122 \[ \frac{2 \left (3 a^2 e^2-8 a b d e+8 b^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{b^{5/2} \sqrt{e}}+\frac{4 e (a+b x)^{3/2} \sqrt{d+e x}}{b^2}+\frac{2 \sqrt{a+b x} \sqrt{d+e x} (7 b d-5 a e)}{b^2} \]

[Out]

(2*(7*b*d - 5*a*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/b^2 + (4*e*(a + b*x)^(3/2)*Sqrt[
d + e*x])/b^2 + (2*(8*b^2*d^2 - 8*a*b*d*e + 3*a^2*e^2)*ArcTanh[(Sqrt[e]*Sqrt[a +
b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(b^(5/2)*Sqrt[e])

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Rubi [A]  time = 0.287447, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{2 \left (3 a^2 e^2-8 a b d e+8 b^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{b^{5/2} \sqrt{e}}+\frac{4 e (a+b x)^{3/2} \sqrt{d+e x}}{b^2}+\frac{2 \sqrt{a+b x} \sqrt{d+e x} (7 b d-5 a e)}{b^2} \]

Antiderivative was successfully verified.

[In]  Int[(15*d^2 + 20*d*e*x + 8*e^2*x^2)/(Sqrt[a + b*x]*Sqrt[d + e*x]),x]

[Out]

(2*(7*b*d - 5*a*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/b^2 + (4*e*(a + b*x)^(3/2)*Sqrt[
d + e*x])/b^2 + (2*(8*b^2*d^2 - 8*a*b*d*e + 3*a^2*e^2)*ArcTanh[(Sqrt[e]*Sqrt[a +
b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(b^(5/2)*Sqrt[e])

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Rubi in Sympy [A]  time = 43.0614, size = 184, normalized size = 1.51 \[ \frac{20 d \sqrt{a + b x} \sqrt{d + e x}}{b} + \frac{4 e x \sqrt{a + b x} \sqrt{d + e x}}{b} - \frac{6 \sqrt{a + b x} \sqrt{d + e x} \left (a e + b d\right )}{b^{2}} - \frac{10 d \left (2 a e - b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{e} \sqrt{a + b x}} \right )}}{b^{\frac{3}{2}} \sqrt{e}} - \frac{8 \left (a b d e - \frac{3 \left (a e + b d\right )^{2}}{4}\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{e} \sqrt{a + b x}} \right )}}{b^{\frac{5}{2}} \sqrt{e}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((8*e**2*x**2+20*d*e*x+15*d**2)/(e*x+d)**(1/2)/(b*x+a)**(1/2),x)

[Out]

20*d*sqrt(a + b*x)*sqrt(d + e*x)/b + 4*e*x*sqrt(a + b*x)*sqrt(d + e*x)/b - 6*sqr
t(a + b*x)*sqrt(d + e*x)*(a*e + b*d)/b**2 - 10*d*(2*a*e - b*d)*atanh(sqrt(b)*sqr
t(d + e*x)/(sqrt(e)*sqrt(a + b*x)))/(b**(3/2)*sqrt(e)) - 8*(a*b*d*e - 3*(a*e + b
*d)**2/4)*atanh(sqrt(b)*sqrt(d + e*x)/(sqrt(e)*sqrt(a + b*x)))/(b**(5/2)*sqrt(e)
)

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Mathematica [A]  time = 0.108814, size = 115, normalized size = 0.94 \[ \frac{\left (3 a^2 e^2-8 a b d e+8 b^2 d^2\right ) \log \left (2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x}+a e+b d+2 b e x\right )}{b^{5/2} \sqrt{e}}+\frac{2 \sqrt{a+b x} \sqrt{d+e x} (-3 a e+7 b d+2 b e x)}{b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(15*d^2 + 20*d*e*x + 8*e^2*x^2)/(Sqrt[a + b*x]*Sqrt[d + e*x]),x]

[Out]

(2*Sqrt[a + b*x]*Sqrt[d + e*x]*(7*b*d - 3*a*e + 2*b*e*x))/b^2 + ((8*b^2*d^2 - 8*
a*b*d*e + 3*a^2*e^2)*Log[b*d + a*e + 2*b*e*x + 2*Sqrt[b]*Sqrt[e]*Sqrt[a + b*x]*S
qrt[d + e*x]])/(b^(5/2)*Sqrt[e])

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Maple [B]  time = 0.035, size = 247, normalized size = 2. \[{\frac{1}{{b}^{2}} \left ( 3\,\ln \left ( 1/2\,{\frac{2\,bex+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){a}^{2}{e}^{2}-8\,\ln \left ( 1/2\,{\frac{2\,bex+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ) abde+8\,\ln \left ( 1/2\,{\frac{2\,bex+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){b}^{2}{d}^{2}+4\,e\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }xb\sqrt{be}-6\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}ae+14\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}bd \right ) \sqrt{ex+d}\sqrt{bx+a}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }}}{\frac{1}{\sqrt{be}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((8*e^2*x^2+20*d*e*x+15*d^2)/(e*x+d)^(1/2)/(b*x+a)^(1/2),x)

[Out]

(3*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a
^2*e^2-8*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1
/2))*a*b*d*e+8*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b
*e)^(1/2))*b^2*d^2+4*e*((b*x+a)*(e*x+d))^(1/2)*x*b*(b*e)^(1/2)-6*((b*x+a)*(e*x+d
))^(1/2)*(b*e)^(1/2)*a*e+14*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*b*d)*(e*x+d)^(1/
2)*(b*x+a)^(1/2)/(b*e)^(1/2)/b^2/((b*x+a)*(e*x+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((8*e^2*x^2 + 20*d*e*x + 15*d^2)/(sqrt(b*x + a)*sqrt(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.317592, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (2 \, b e x + 7 \, b d - 3 \, a e\right )} \sqrt{b e} \sqrt{b x + a} \sqrt{e x + d} +{\left (8 \, b^{2} d^{2} - 8 \, a b d e + 3 \, a^{2} e^{2}\right )} \log \left (4 \,{\left (2 \, b^{2} e^{2} x + b^{2} d e + a b e^{2}\right )} \sqrt{b x + a} \sqrt{e x + d} +{\left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 8 \,{\left (b^{2} d e + a b e^{2}\right )} x\right )} \sqrt{b e}\right )}{2 \, \sqrt{b e} b^{2}}, \frac{2 \,{\left (2 \, b e x + 7 \, b d - 3 \, a e\right )} \sqrt{-b e} \sqrt{b x + a} \sqrt{e x + d} +{\left (8 \, b^{2} d^{2} - 8 \, a b d e + 3 \, a^{2} e^{2}\right )} \arctan \left (\frac{{\left (2 \, b e x + b d + a e\right )} \sqrt{-b e}}{2 \, \sqrt{b x + a} \sqrt{e x + d} b e}\right )}{\sqrt{-b e} b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((8*e^2*x^2 + 20*d*e*x + 15*d^2)/(sqrt(b*x + a)*sqrt(e*x + d)),x, algorithm="fricas")

[Out]

[1/2*(4*(2*b*e*x + 7*b*d - 3*a*e)*sqrt(b*e)*sqrt(b*x + a)*sqrt(e*x + d) + (8*b^2
*d^2 - 8*a*b*d*e + 3*a^2*e^2)*log(4*(2*b^2*e^2*x + b^2*d*e + a*b*e^2)*sqrt(b*x +
a)*sqrt(e*x + d) + (8*b^2*e^2*x^2 + b^2*d^2 + 6*a*b*d*e + a^2*e^2 + 8*(b^2*d*e
+ a*b*e^2)*x)*sqrt(b*e)))/(sqrt(b*e)*b^2), (2*(2*b*e*x + 7*b*d - 3*a*e)*sqrt(-b*
e)*sqrt(b*x + a)*sqrt(e*x + d) + (8*b^2*d^2 - 8*a*b*d*e + 3*a^2*e^2)*arctan(1/2*
(2*b*e*x + b*d + a*e)*sqrt(-b*e)/(sqrt(b*x + a)*sqrt(e*x + d)*b*e)))/(sqrt(-b*e)
*b^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{15 d^{2} + 20 d e x + 8 e^{2} x^{2}}{\sqrt{a + b x} \sqrt{d + e x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((8*e**2*x**2+20*d*e*x+15*d**2)/(e*x+d)**(1/2)/(b*x+a)**(1/2),x)

[Out]

Integral((15*d**2 + 20*d*e*x + 8*e**2*x**2)/(sqrt(a + b*x)*sqrt(d + e*x)), x)

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GIAC/XCAS [A]  time = 0.285379, size = 196, normalized size = 1.61 \[ \frac{2 \,{\left (\sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e} \sqrt{b x + a}{\left (\frac{2 \,{\left (b x + a\right )} e}{b^{3}} + \frac{{\left (7 \, b^{6} d e^{2} - 5 \, a b^{5} e^{3}\right )} e^{\left (-2\right )}}{b^{8}}\right )} - \frac{{\left (8 \, b^{2} d^{2} - 8 \, a b d e + 3 \, a^{2} e^{2}\right )} e^{\left (-\frac{1}{2}\right )}{\rm ln}\left ({\left | -\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} + \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e} \right |}\right )}{b^{\frac{5}{2}}}\right )} b}{{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((8*e^2*x^2 + 20*d*e*x + 15*d^2)/(sqrt(b*x + a)*sqrt(e*x + d)),x, algorithm="giac")

[Out]

2*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*e/b^3 + (7*b^6
*d*e^2 - 5*a*b^5*e^3)*e^(-2)/b^8) - (8*b^2*d^2 - 8*a*b*d*e + 3*a^2*e^2)*e^(-1/2)
*ln(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b
^(5/2))*b/abs(b)