3.2186 \(\int \frac{\sqrt{a+b x} (A+B x)}{\sqrt{d+e x}} \, dx\)

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

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.272019, 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 \[ \frac{(b d-a e) (a B e-4 A b e+3 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{4 b^{3/2} e^{5/2}}-\frac{\sqrt{a+b x} \sqrt{d+e x} (a B e-4 A b e+3 b B d)}{4 b e^2}+\frac{B (a+b x)^{3/2} \sqrt{d+e x}}{2 b e} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 18.2487, size = 131, normalized size = 0.94 \[ \frac{B \left (a + b x\right )^{\frac{3}{2}} \sqrt{d + e x}}{2 b e} + \frac{\sqrt{a + b x} \sqrt{d + e x} \left (4 A b e - B a e - 3 B b d\right )}{4 b e^{2}} + \frac{\left (a e - b d\right ) \left (4 A b e - B a e - 3 B b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{e} \sqrt{a + b x}} \right )}}{4 b^{\frac{3}{2}} e^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.122692, size = 130, normalized size = 0.93 \[ \frac{(b d-a e) (a B e-4 A b e+3 b B d) \log \left (2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x}+a e+b d+2 b e x\right )}{8 b^{3/2} e^{5/2}}+\frac{\sqrt{a+b x} \sqrt{d+e x} (a B e+b (4 A e-3 B d+2 B e x))}{4 b e^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.028, size = 376, normalized size = 2.7 \[{\frac{1}{8\,{e}^{2}b}\sqrt{bx+a}\sqrt{ex+d} \left ( 4\,\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ) aA{e}^{2}b-4\,\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){b}^{2}dAe-B\ln \left ({\frac{1}{2} \left ( 2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd \right ){\frac{1}{\sqrt{be}}}} \right ){a}^{2}{e}^{2}-2\,\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ) aBdeb+3\,\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){b}^{2}{d}^{2}B+4\,B\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }xeb\sqrt{be}+8\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }Aeb\sqrt{be}-6\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }Bdb\sqrt{be}+2\,B\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }ae\sqrt{be} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }}}{\frac{1}{\sqrt{be}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(b*x+a)^(1/2)*(e*x+d)^(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-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+3*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*x+a)*(e*x+d))^(1/2)*B*d*b*(b*e)^(1/2)+2*B*((b*x+a)*(e*x+d))^(1/2)*a*e*(b*
e)^(1/2))/((b*x+a)*(e*x+d))^(1/2)/e^2/b/(b*e)^(1/2)

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.436242, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (2 \, B b e x - 3 \, B b d +{\left (B a + 4 \, A b\right )} e\right )} \sqrt{b e} \sqrt{b x + a} \sqrt{e x + d} +{\left (3 \, B b^{2} d^{2} - 2 \,{\left (B a b + 2 \, A b^{2}\right )} d e -{\left (B a^{2} - 4 \, A a b\right )} e^{2}\right )} \log \left (4 \,{\left (2 \, b^{2} e^{2} x + b^{2} d e + a b e^{2}\right )} \sqrt{b x + a} \sqrt{e x + d} +{\left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 8 \,{\left (b^{2} d e + a b e^{2}\right )} x\right )} \sqrt{b e}\right )}{16 \, \sqrt{b e} b e^{2}}, \frac{2 \,{\left (2 \, B b e x - 3 \, B b d +{\left (B a + 4 \, A b\right )} e\right )} \sqrt{-b e} \sqrt{b x + a} \sqrt{e x + d} +{\left (3 \, B b^{2} d^{2} - 2 \,{\left (B a b + 2 \, A b^{2}\right )} d e -{\left (B a^{2} - 4 \, A a b\right )} e^{2}\right )} \arctan \left (\frac{{\left (2 \, b e x + b d + a e\right )} \sqrt{-b e}}{2 \, \sqrt{b x + a} \sqrt{e x + d} b e}\right )}{8 \, \sqrt{-b e} b e^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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)*(b*x+a)**(1/2)/(e*x+d)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.232362, size = 236, normalized size = 1.69 \[ \frac{{\left (\sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e} \sqrt{b x + a}{\left (\frac{2 \,{\left (b x + a\right )} B e^{\left (-1\right )}}{b^{2}} - \frac{{\left (3 \, B b^{3} d e + B a b^{2} e^{2} - 4 \, A b^{3} e^{2}\right )} e^{\left (-3\right )}}{b^{4}}\right )} - \frac{{\left (3 \, B b^{2} d^{2} - 2 \, B a b d e - 4 \, A b^{2} d e - B a^{2} e^{2} + 4 \, A a b e^{2}\right )} e^{\left (-\frac{5}{2}\right )}{\rm ln}\left ({\left | -\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} + \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e} \right |}\right )}{b^{\frac{3}{2}}}\right )} b}{4 \,{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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