3.2187 \(\int \frac{\sqrt{a+b x} (A+B x)}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=148 \[ -\frac{(-a B e-2 A b e+3 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{\sqrt{b} e^{5/2}}+\frac{\sqrt{a+b x} \sqrt{d+e x} (-a B e-2 A b e+3 b B d)}{e^2 (b d-a e)}-\frac{2 (a+b x)^{3/2} (B d-A e)}{e \sqrt{d+e x} (b d-a e)} \]

[Out]

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

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Rubi [A]  time = 0.303695, antiderivative size = 148, 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 \[ -\frac{(-a B e-2 A b e+3 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{\sqrt{b} e^{5/2}}+\frac{\sqrt{a+b x} \sqrt{d+e x} (-a B e-2 A b e+3 b B d)}{e^2 (b d-a e)}-\frac{2 (a+b x)^{3/2} (B d-A e)}{e \sqrt{d+e x} (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[a + b*x]*(A + B*x))/(d + e*x)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 25.832, size = 136, normalized size = 0.92 \[ - \frac{2 \left (a + b x\right )^{\frac{3}{2}} \left (A e - B d\right )}{e \sqrt{d + e x} \left (a e - b d\right )} + \frac{2 \sqrt{a + b x} \sqrt{d + e x} \left (A b e + \frac{B \left (a e - 3 b d\right )}{2}\right )}{e^{2} \left (a e - b d\right )} + \frac{2 \left (A b e + \frac{B \left (a e - 3 b d\right )}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a + b x}}{\sqrt{b} \sqrt{d + e x}} \right )}}{\sqrt{b} e^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(b*x+a)**(1/2)/(e*x+d)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.123756, size = 108, normalized size = 0.73 \[ \frac{(a B e+2 A b e-3 b B d) \log \left (2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x}+a e+b d+2 b e x\right )}{2 \sqrt{b} e^{5/2}}+\frac{\sqrt{a+b x} (-2 A e+3 B d+B e x)}{e^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[a + b*x]*(A + B*x))/(d + e*x)^(3/2),x]

[Out]

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

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Maple [B]  time = 0.039, size = 386, normalized size = 2.6 \[{\frac{1}{2\,{e}^{2}}\sqrt{bx+a} \left ( 2\,A\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ) xb{e}^{2}+B\ln \left ({\frac{1}{2} \left ( 2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd \right ){\frac{1}{\sqrt{be}}}} \right ) xa{e}^{2}-3\,B\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ) xbde+2\,A\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ) bde+B\ln \left ({\frac{1}{2} \left ( 2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd \right ){\frac{1}{\sqrt{be}}}} \right ) ade-3\,B\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ) b{d}^{2}+2\,Bxe\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}-4\,Ae\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+6\,Bd\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }}}{\frac{1}{\sqrt{be}}}{\frac{1}{\sqrt{ex+d}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(b*x+a)^(1/2)/(e*x+d)^(3/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.593956, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (B e x + 3 \, B d - 2 \, A e\right )} \sqrt{b e} \sqrt{b x + a} \sqrt{e x + d} -{\left (3 \, B b d^{2} -{\left (B a + 2 \, A b\right )} d e +{\left (3 \, B b d e -{\left (B a + 2 \, A b\right )} e^{2}\right )} x\right )} \log \left (4 \,{\left (2 \, b^{2} e^{2} x + b^{2} d e + a b e^{2}\right )} \sqrt{b x + a} \sqrt{e x + d} +{\left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 8 \,{\left (b^{2} d e + a b e^{2}\right )} x\right )} \sqrt{b e}\right )}{4 \,{\left (e^{3} x + d e^{2}\right )} \sqrt{b e}}, \frac{2 \,{\left (B e x + 3 \, B d - 2 \, A e\right )} \sqrt{-b e} \sqrt{b x + a} \sqrt{e x + d} -{\left (3 \, B b d^{2} -{\left (B a + 2 \, A b\right )} d e +{\left (3 \, B b d e -{\left (B a + 2 \, A b\right )} e^{2}\right )} x\right )} \arctan \left (\frac{{\left (2 \, b e x + b d + a e\right )} \sqrt{-b e}}{2 \, \sqrt{b x + a} \sqrt{e x + d} b e}\right )}{2 \,{\left (e^{3} x + d e^{2}\right )} \sqrt{-b e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*sqrt(b*x + a)/(e*x + d)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \sqrt{a + b x}}{\left (d + e x\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(b*x+a)**(1/2)/(e*x+d)**(3/2),x)

[Out]

Integral((A + B*x)*sqrt(a + b*x)/(d + e*x)**(3/2), x)

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GIAC/XCAS [A]  time = 0.241987, size = 274, normalized size = 1.85 \[ \frac{{\left (3 \, B b d{\left | b \right |} - B a{\left | b \right |} e - 2 \, A b{\left | b \right |} e\right )} \sqrt{b} e^{\frac{1}{2}}{\rm ln}\left ({\left | -\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} + \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e} \right |}\right )}{32 \,{\left (b^{6} d e^{4} - a b^{5} e^{5}\right )}} + \frac{{\left (\frac{{\left (b x + a\right )} B b{\left | b \right |} e^{2}}{b^{6} d e^{4} - a b^{5} e^{5}} + \frac{3 \, B b^{2} d{\left | b \right |} e - B a b{\left | b \right |} e^{2} - 2 \, A b^{2}{\left | b \right |} e^{2}}{b^{6} d e^{4} - a b^{5} e^{5}}\right )} \sqrt{b x + a}}{32 \, \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*sqrt(b*x + a)/(e*x + d)^(3/2),x, algorithm="giac")

[Out]

1/32*(3*B*b*d*abs(b) - B*a*abs(b)*e - 2*A*b*abs(b)*e)*sqrt(b)*e^(1/2)*ln(abs(-sq
rt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/(b^6*d*e^4 -
a*b^5*e^5) + 1/32*((b*x + a)*B*b*abs(b)*e^2/(b^6*d*e^4 - a*b^5*e^5) + (3*B*b^2*
d*abs(b)*e - B*a*b*abs(b)*e^2 - 2*A*b^2*abs(b)*e^2)/(b^6*d*e^4 - a*b^5*e^5))*sqr
t(b*x + a)/sqrt(b^2*d + (b*x + a)*b*e - a*b*e)