3.1751 \(\int \frac{(a+b x) \sqrt{e+f x}}{c+d x} \, dx\)

Optimal. Leaf size=98 \[ \frac{2 (b c-a d) \sqrt{d e-c f} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{5/2}}-\frac{2 \sqrt{e+f x} (b c-a d)}{d^2}+\frac{2 b (e+f x)^{3/2}}{3 d f} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.165123, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{2 (b c-a d) \sqrt{d e-c f} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{5/2}}-\frac{2 \sqrt{e+f x} (b c-a d)}{d^2}+\frac{2 b (e+f x)^{3/2}}{3 d f} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 17.8235, size = 85, normalized size = 0.87 \[ \frac{2 b \left (e + f x\right )^{\frac{3}{2}}}{3 d f} + \frac{2 \sqrt{e + f x} \left (a d - b c\right )}{d^{2}} - \frac{2 \left (a d - b c\right ) \sqrt{c f - d e} \operatorname{atan}{\left (\frac{\sqrt{d} \sqrt{e + f x}}{\sqrt{c f - d e}} \right )}}{d^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b*(e + f*x)**(3/2)/(3*d*f) + 2*sqrt(e + f*x)*(a*d - b*c)/d**2 - 2*(a*d - b*c)*
sqrt(c*f - d*e)*atan(sqrt(d)*sqrt(e + f*x)/sqrt(c*f - d*e))/d**(5/2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.213632, size = 94, normalized size = 0.96 \[ \frac{2 (b c-a d) \sqrt{d e-c f} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{5/2}}+\frac{2 \sqrt{e+f x} (3 a d f-3 b c f+b d (e+f x))}{3 d^2 f} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.013, size = 211, normalized size = 2.2 \[{\frac{2\,b}{3\,df} \left ( fx+e \right ) ^{{\frac{3}{2}}}}+2\,{\frac{a\sqrt{fx+e}}{d}}-2\,{\frac{bc\sqrt{fx+e}}{{d}^{2}}}-2\,{\frac{acf}{d\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }+2\,{\frac{ae}{\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }+2\,{\frac{b{c}^{2}f}{{d}^{2}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }-2\,{\frac{bce}{d\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*b*(f*x+e)^(3/2)/d/f+2/d*a*(f*x+e)^(1/2)-2/d^2*b*c*(f*x+e)^(1/2)-2*f/d/((c*f-
d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)^(1/2))*a*c+2/((c*f-d*e)*d)^(1
/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)^(1/2))*a*e+2*f/d^2/((c*f-d*e)*d)^(1/2)*
arctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)^(1/2))*b*c^2-2/d/((c*f-d*e)*d)^(1/2)*arctan
((f*x+e)^(1/2)*d/((c*f-d*e)*d)^(1/2))*b*c*e

_______________________________________________________________________________________

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(f*x + e)/(d*x + c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.22108, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (b c - a d\right )} f \sqrt{\frac{d e - c f}{d}} \log \left (\frac{d f x + 2 \, d e - c f - 2 \, \sqrt{f x + e} d \sqrt{\frac{d e - c f}{d}}}{d x + c}\right ) - 2 \,{\left (b d f x + b d e - 3 \,{\left (b c - a d\right )} f\right )} \sqrt{f x + e}}{3 \, d^{2} f}, \frac{2 \,{\left (3 \,{\left (b c - a d\right )} f \sqrt{-\frac{d e - c f}{d}} \arctan \left (\frac{\sqrt{f x + e}}{\sqrt{-\frac{d e - c f}{d}}}\right ) +{\left (b d f x + b d e - 3 \,{\left (b c - a d\right )} f\right )} \sqrt{f x + e}\right )}}{3 \, d^{2} f}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/3*(3*(b*c - a*d)*f*sqrt((d*e - c*f)/d)*log((d*f*x + 2*d*e - c*f - 2*sqrt(f*x
+ e)*d*sqrt((d*e - c*f)/d))/(d*x + c)) - 2*(b*d*f*x + b*d*e - 3*(b*c - a*d)*f)*
sqrt(f*x + e))/(d^2*f), 2/3*(3*(b*c - a*d)*f*sqrt(-(d*e - c*f)/d)*arctan(sqrt(f*
x + e)/sqrt(-(d*e - c*f)/d)) + (b*d*f*x + b*d*e - 3*(b*c - a*d)*f)*sqrt(f*x + e)
)/(d^2*f)]

_______________________________________________________________________________________

Sympy [A]  time = 15.6016, size = 212, normalized size = 2.16 \[ \frac{2 \left (\frac{b \left (e + f x\right )^{\frac{3}{2}}}{3 d} - \frac{f \left (a d - b c\right ) \left (c f - d e\right ) \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{\sqrt{e + f x}}{\sqrt{\frac{c f - d e}{d}}} \right )}}{d \sqrt{\frac{c f - d e}{d}}} & \text{for}\: \frac{c f - d e}{d} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{\sqrt{e + f x}}{\sqrt{\frac{- c f + d e}{d}}} \right )}}{d \sqrt{\frac{- c f + d e}{d}}} & \text{for}\: e + f x > \frac{- c f + d e}{d} \wedge \frac{c f - d e}{d} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{\sqrt{e + f x}}{\sqrt{\frac{- c f + d e}{d}}} \right )}}{d \sqrt{\frac{- c f + d e}{d}}} & \text{for}\: \frac{c f - d e}{d} < 0 \wedge e + f x < \frac{- c f + d e}{d} \end{cases}\right )}{d^{2}} + \frac{\sqrt{e + f x} \left (a d f - b c f\right )}{d^{2}}\right )}{f} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(b*(e + f*x)**(3/2)/(3*d) - f*(a*d - b*c)*(c*f - d*e)*Piecewise((atan(sqrt(e +
f*x)/sqrt((c*f - d*e)/d))/(d*sqrt((c*f - d*e)/d)), (c*f - d*e)/d > 0), (-acoth(
sqrt(e + f*x)/sqrt((-c*f + d*e)/d))/(d*sqrt((-c*f + d*e)/d)), ((c*f - d*e)/d < 0
) & (e + f*x > (-c*f + d*e)/d)), (-atanh(sqrt(e + f*x)/sqrt((-c*f + d*e)/d))/(d*
sqrt((-c*f + d*e)/d)), ((c*f - d*e)/d < 0) & (e + f*x < (-c*f + d*e)/d)))/d**2 +
sqrt(e + f*x)*(a*d*f - b*c*f)/d**2)/f

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.216843, size = 176, normalized size = 1.8 \[ \frac{2 \,{\left (b c^{2} f - a c d f - b c d e + a d^{2} e\right )} \arctan \left (\frac{\sqrt{f x + e} d}{\sqrt{c d f - d^{2} e}}\right )}{\sqrt{c d f - d^{2} e} d^{2}} + \frac{2 \,{\left ({\left (f x + e\right )}^{\frac{3}{2}} b d^{2} f^{2} - 3 \, \sqrt{f x + e} b c d f^{3} + 3 \, \sqrt{f x + e} a d^{2} f^{3}\right )}}{3 \, d^{3} f^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(b*c^2*f - a*c*d*f - b*c*d*e + a*d^2*e)*arctan(sqrt(f*x + e)*d/sqrt(c*d*f - d^
2*e))/(sqrt(c*d*f - d^2*e)*d^2) + 2/3*((f*x + e)^(3/2)*b*d^2*f^2 - 3*sqrt(f*x +
e)*b*c*d*f^3 + 3*sqrt(f*x + e)*a*d^2*f^3)/(d^3*f^3)