3.2221 \(\int \frac{(A+B x) \sqrt{d+e x}}{\sqrt{a+b x}} \, dx\)

Optimal. Leaf size=140 \[ -\frac{(b d-a e) (3 a B e-4 A b e+b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{4 b^{5/2} e^{3/2}}-\frac{\sqrt{a+b x} \sqrt{d+e x} (3 a B e-4 A b e+b B d)}{4 b^2 e}+\frac{B \sqrt{a+b x} (d+e x)^{3/2}}{2 b e} \]

[Out]

-((b*B*d - 4*A*b*e + 3*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(4*b^2*e) + (B*Sqrt[a
+ b*x]*(d + e*x)^(3/2))/(2*b*e) - ((b*d - a*e)*(b*B*d - 4*A*b*e + 3*a*B*e)*ArcT
anh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(4*b^(5/2)*e^(3/2))

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Rubi [A]  time = 0.274312, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{(b d-a e) (3 a B e-4 A b e+b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{4 b^{5/2} e^{3/2}}-\frac{\sqrt{a+b x} \sqrt{d+e x} (3 a B e-4 A b e+b B d)}{4 b^2 e}+\frac{B \sqrt{a+b x} (d+e x)^{3/2}}{2 b e} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*Sqrt[d + e*x])/Sqrt[a + b*x],x]

[Out]

-((b*B*d - 4*A*b*e + 3*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(4*b^2*e) + (B*Sqrt[a
+ b*x]*(d + e*x)^(3/2))/(2*b*e) - ((b*d - a*e)*(b*B*d - 4*A*b*e + 3*a*B*e)*ArcT
anh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(4*b^(5/2)*e^(3/2))

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Rubi in Sympy [A]  time = 18.7734, size = 131, normalized size = 0.94 \[ \frac{B \sqrt{a + b x} \left (d + e x\right )^{\frac{3}{2}}}{2 b e} + \frac{\sqrt{a + b x} \sqrt{d + e x} \left (4 A b e - 3 B a e - B b d\right )}{4 b^{2} e} - \frac{\left (a e - b d\right ) \left (4 A b e - 3 B a e - B b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a + b x}}{\sqrt{b} \sqrt{d + e x}} \right )}}{4 b^{\frac{5}{2}} e^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.131295, size = 129, normalized size = 0.92 \[ \frac{\sqrt{a+b x} \sqrt{d+e x} (-3 a B e+4 A b e+b B (d+2 e x))}{4 b^2 e}-\frac{(b d-a e) (3 a B e-4 A b e+b B d) \log \left (2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x}+a e+b d+2 b e x\right )}{8 b^{5/2} e^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*Sqrt[d + e*x])/Sqrt[a + b*x],x]

[Out]

(Sqrt[a + b*x]*Sqrt[d + e*x]*(4*A*b*e - 3*a*B*e + b*B*(d + 2*e*x)))/(4*b^2*e) -
((b*d - a*e)*(b*B*d - 4*A*b*e + 3*a*B*e)*Log[b*d + a*e + 2*b*e*x + 2*Sqrt[b]*Sqr
t[e]*Sqrt[a + b*x]*Sqrt[d + e*x]])/(8*b^(5/2)*e^(3/2))

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Maple [B]  time = 0.025, size = 375, normalized size = 2.7 \[ -{\frac{1}{8\,{b}^{2}e}\sqrt{ex+d}\sqrt{bx+a} \left ( 4\,\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ) aA{e}^{2}b-4\,\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){b}^{2}dAe-3\,B\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){a}^{2}{e}^{2}+2\,\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ) aBdeb+\ln \left ({\frac{1}{2} \left ( 2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd \right ){\frac{1}{\sqrt{be}}}} \right ){b}^{2}{d}^{2}B-4\,B\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }xeb\sqrt{be}-8\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }Aeb\sqrt{be}+6\,B\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }ae\sqrt{be}-2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }Bdb\sqrt{be} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }}}{\frac{1}{\sqrt{be}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.429015, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (2 \, B b e x + B b d -{\left (3 \, B a - 4 \, A b\right )} e\right )} \sqrt{b e} \sqrt{b x + a} \sqrt{e x + d} +{\left (B b^{2} d^{2} + 2 \,{\left (B a b - 2 \, A b^{2}\right )} d e -{\left (3 \, B a^{2} - 4 \, A a b\right )} e^{2}\right )} \log \left (-4 \,{\left (2 \, b^{2} e^{2} x + b^{2} d e + a b e^{2}\right )} \sqrt{b x + a} \sqrt{e x + d} +{\left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 8 \,{\left (b^{2} d e + a b e^{2}\right )} x\right )} \sqrt{b e}\right )}{16 \, \sqrt{b e} b^{2} e}, \frac{2 \,{\left (2 \, B b e x + B b d -{\left (3 \, B a - 4 \, A b\right )} e\right )} \sqrt{-b e} \sqrt{b x + a} \sqrt{e x + d} -{\left (B b^{2} d^{2} + 2 \,{\left (B a b - 2 \, A b^{2}\right )} d e -{\left (3 \, B a^{2} - 4 \, A a b\right )} e^{2}\right )} \arctan \left (\frac{{\left (2 \, b e x + b d + a e\right )} \sqrt{-b e}}{2 \, \sqrt{b x + a} \sqrt{e x + d} b e}\right )}{8 \, \sqrt{-b e} b^{2} e}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*sqrt(e*x + d)/sqrt(b*x + a),x, algorithm="fricas")

[Out]

[1/16*(4*(2*B*b*e*x + B*b*d - (3*B*a - 4*A*b)*e)*sqrt(b*e)*sqrt(b*x + a)*sqrt(e*
x + d) + (B*b^2*d^2 + 2*(B*a*b - 2*A*b^2)*d*e - (3*B*a^2 - 4*A*a*b)*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*e), 1/8*(2*(2*B*b*e*x + B*b*d - (3*B*a - 4*A*b)*e)*sqrt(-b*e)*sqrt(b*x + a)
*sqrt(e*x + d) - (B*b^2*d^2 + 2*(B*a*b - 2*A*b^2)*d*e - (3*B*a^2 - 4*A*a*b)*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*e)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.250857, size = 328, normalized size = 2.34 \[ -\frac{\frac{48 \,{\left (\frac{{\left (b^{2} d - a b e\right )} e^{\left (-\frac{1}{2}\right )}{\rm ln}\left ({\left | -\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} + \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e} \right |}\right )}{\sqrt{b}} - \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e} \sqrt{b x + a}\right )} A{\left | b \right |}}{b^{2}} - \frac{{\left (\sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e} \sqrt{b x + a}{\left (\frac{2 \,{\left (b x + a\right )} e^{\left (-2\right )}}{b^{4}} + \frac{{\left (b d e - 5 \, a e^{2}\right )} e^{\left (-4\right )}}{b^{4}}\right )} + \frac{{\left (b^{2} d^{2} + 2 \, a b d e - 3 \, a^{2} e^{2}\right )} e^{\left (-\frac{7}{2}\right )}{\rm ln}\left ({\left | -\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} + \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e} \right |}\right )}{b^{\frac{7}{2}}}\right )} B{\left | b \right |}}{b^{3}}}{48 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*sqrt(e*x + d)/sqrt(b*x + a),x, algorithm="giac")

[Out]

-1/48*(48*((b^2*d - a*b*e)*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)))/sqrt(b) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*
sqrt(b*x + a))*A*abs(b)/b^2 - (sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)
*(2*(b*x + a)*e^(-2)/b^4 + (b*d*e - 5*a*e^2)*e^(-4)/b^4) + (b^2*d^2 + 2*a*b*d*e
- 3*a^2*e^2)*e^(-7/2)*ln(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x
+ a)*b*e - a*b*e)))/b^(7/2))*B*abs(b)/b^3)/b