3.21 \(\int \frac{\sqrt{a+b x} (e+f x)}{x (c+d x)^3} \, dx\)

Optimal. Leaf size=205 \[ -\frac{\left (-8 a^2 d^3 e+12 a b c d^2 e-b^2 c^2 (c f+3 d e)\right ) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{4 c^3 d^{3/2} (b c-a d)^{3/2}}-\frac{2 \sqrt{a} e \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{c^3}-\frac{\sqrt{a+b x} \left (4 a d^2 e-b c (c f+3 d e)\right )}{4 c^2 d (c+d x) (b c-a d)}+\frac{\sqrt{a+b x} (d e-c f)}{2 c d (c+d x)^2} \]

[Out]

((d*e - c*f)*Sqrt[a + b*x])/(2*c*d*(c + d*x)^2) - ((4*a*d^2*e - b*c*(3*d*e + c*f
))*Sqrt[a + b*x])/(4*c^2*d*(b*c - a*d)*(c + d*x)) - ((12*a*b*c*d^2*e - 8*a^2*d^3
*e - b^2*c^2*(3*d*e + c*f))*ArcTan[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d]])/(4*
c^3*d^(3/2)*(b*c - a*d)^(3/2)) - (2*Sqrt[a]*e*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/c^
3

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Rubi [A]  time = 0.814517, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ -\frac{\left (-8 a^2 d^3 e+12 a b c d^2 e-b^2 c^2 (c f+3 d e)\right ) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{4 c^3 d^{3/2} (b c-a d)^{3/2}}-\frac{2 \sqrt{a} e \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{c^3}-\frac{\sqrt{a+b x} \left (4 a d^2 e-b c (c f+3 d e)\right )}{4 c^2 d (c+d x) (b c-a d)}+\frac{\sqrt{a+b x} (d e-c f)}{2 c d (c+d x)^2} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[a + b*x]*(e + f*x))/(x*(c + d*x)^3),x]

[Out]

((d*e - c*f)*Sqrt[a + b*x])/(2*c*d*(c + d*x)^2) - ((4*a*d^2*e - b*c*(3*d*e + c*f
))*Sqrt[a + b*x])/(4*c^2*d*(b*c - a*d)*(c + d*x)) - ((12*a*b*c*d^2*e - 8*a^2*d^3
*e - b^2*c^2*(3*d*e + c*f))*ArcTan[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d]])/(4*
c^3*d^(3/2)*(b*c - a*d)^(3/2)) - (2*Sqrt[a]*e*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/c^
3

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Rubi in Sympy [A]  time = 84.9888, size = 189, normalized size = 0.92 \[ - \frac{2 \sqrt{a} e \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{c^{3}} - \frac{\sqrt{a + b x} \left (c f - d e\right )}{2 c d \left (c + d x\right )^{2}} + \frac{\sqrt{a + b x} \left (4 a d^{2} e - b c \left (c f + 3 d e\right )\right )}{4 c^{2} d \left (c + d x\right ) \left (a d - b c\right )} + \frac{\left (8 a^{2} d^{3} e - 12 a b c d^{2} e + b^{2} c^{3} f + 3 b^{2} c^{2} d e\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{a d - b c}} \right )}}{4 c^{3} d^{\frac{3}{2}} \left (a d - b c\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(a)*e*atanh(sqrt(a + b*x)/sqrt(a))/c**3 - sqrt(a + b*x)*(c*f - d*e)/(2*c*
d*(c + d*x)**2) + sqrt(a + b*x)*(4*a*d**2*e - b*c*(c*f + 3*d*e))/(4*c**2*d*(c +
d*x)*(a*d - b*c)) + (8*a**2*d**3*e - 12*a*b*c*d**2*e + b**2*c**3*f + 3*b**2*c**2
*d*e)*atanh(sqrt(d)*sqrt(a + b*x)/sqrt(a*d - b*c))/(4*c**3*d**(3/2)*(a*d - b*c)*
*(3/2))

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Mathematica [A]  time = 0.562576, size = 180, normalized size = 0.88 \[ \frac{\frac{\left (8 a^2 d^3 e-12 a b c d^2 e+b^2 c^2 (c f+3 d e)\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )}{d^{3/2} (a d-b c)^{3/2}}+\frac{c \sqrt{a+b x} \left (\frac{(c+d x) \left (b c (c f+3 d e)-4 a d^2 e\right )}{b c-a d}+2 c (d e-c f)\right )}{d (c+d x)^2}-8 \sqrt{a} e \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 c^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[a + b*x]*(e + f*x))/(x*(c + d*x)^3),x]

[Out]

((c*Sqrt[a + b*x]*(2*c*(d*e - c*f) + ((-4*a*d^2*e + b*c*(3*d*e + c*f))*(c + d*x)
)/(b*c - a*d)))/(d*(c + d*x)^2) - 8*Sqrt[a]*e*ArcTanh[Sqrt[a + b*x]/Sqrt[a]] + (
(-12*a*b*c*d^2*e + 8*a^2*d^3*e + b^2*c^2*(3*d*e + c*f))*ArcTanh[(Sqrt[d]*Sqrt[a
+ b*x])/Sqrt[-(b*c) + a*d]])/(d^(3/2)*(-(b*c) + a*d)^(3/2)))/(4*c^3)

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Maple [A]  time = 0.024, size = 221, normalized size = 1.1 \[ 2\,{b}^{2} \left ( -{\frac{e\sqrt{a}}{{c}^{3}{b}^{2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) }-{\frac{1}{{c}^{3}{b}^{2}} \left ({\frac{1}{ \left ( \left ( bx+a \right ) d-ad+bc \right ) ^{2}} \left ( -1/8\,{\frac{bc \left ( 4\,a{d}^{2}e-b{c}^{2}f-3\,bcde \right ) \left ( bx+a \right ) ^{3/2}}{ad-bc}}+1/8\,{\frac{ \left ( 4\,a{d}^{2}e+b{c}^{2}f-5\,bcde \right ) bc\sqrt{bx+a}}{d}} \right ) }-1/8\,{\frac{8\,{a}^{2}{d}^{3}e-12\,abc{d}^{2}e+{b}^{2}{c}^{3}f+3\,{b}^{2}{c}^{2}de}{ \left ( ad-bc \right ) d\sqrt{ \left ( ad-bc \right ) d}}{\it Artanh} \left ({\frac{\sqrt{bx+a}d}{\sqrt{ \left ( ad-bc \right ) d}}} \right ) } \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b^2*(-1/b^2*e*a^(1/2)/c^3*arctanh((b*x+a)^(1/2)/a^(1/2))-1/b^2/c^3*((-1/8*b*c*
(4*a*d^2*e-b*c^2*f-3*b*c*d*e)/(a*d-b*c)*(b*x+a)^(3/2)+1/8*(4*a*d^2*e+b*c^2*f-5*b
*c*d*e)*b*c/d*(b*x+a)^(1/2))/((b*x+a)*d-a*d+b*c)^2-1/8*(8*a^2*d^3*e-12*a*b*c*d^2
*e+b^2*c^3*f+3*b^2*c^2*d*e)/(a*d-b*c)/d/((a*d-b*c)*d)^(1/2)*arctanh((b*x+a)^(1/2
)*d/((a*d-b*c)*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(sqrt(b*x + a)*(f*x + e)/((d*x + c)^3*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.937992, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(8*((b*c*d^3 - a*d^4)*e*x^2 + 2*(b*c^2*d^2 - a*c*d^3)*e*x + (b*c^3*d - a*c^
2*d^2)*e)*sqrt(-b*c*d + a*d^2)*sqrt(a)*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)
/x) + 2*sqrt(-b*c*d + a*d^2)*((5*b*c^3*d - 6*a*c^2*d^2)*e - (b*c^4 - 2*a*c^3*d)*
f + (b*c^3*d*f + (3*b*c^2*d^2 - 4*a*c*d^3)*e)*x)*sqrt(b*x + a) - (b^2*c^5*f + (b
^2*c^3*d^2*f + (3*b^2*c^2*d^3 - 12*a*b*c*d^4 + 8*a^2*d^5)*e)*x^2 + (3*b^2*c^4*d
- 12*a*b*c^3*d^2 + 8*a^2*c^2*d^3)*e + 2*(b^2*c^4*d*f + (3*b^2*c^3*d^2 - 12*a*b*c
^2*d^3 + 8*a^2*c*d^4)*e)*x)*log((sqrt(-b*c*d + a*d^2)*(b*d*x - b*c + 2*a*d) - 2*
(b*c*d - a*d^2)*sqrt(b*x + a))/(d*x + c)))/((b*c^6*d - a*c^5*d^2 + (b*c^4*d^3 -
a*c^3*d^4)*x^2 + 2*(b*c^5*d^2 - a*c^4*d^3)*x)*sqrt(-b*c*d + a*d^2)), -1/8*(16*((
b*c*d^3 - a*d^4)*e*x^2 + 2*(b*c^2*d^2 - a*c*d^3)*e*x + (b*c^3*d - a*c^2*d^2)*e)*
sqrt(-b*c*d + a*d^2)*sqrt(-a)*arctan(sqrt(b*x + a)/sqrt(-a)) - 2*sqrt(-b*c*d + a
*d^2)*((5*b*c^3*d - 6*a*c^2*d^2)*e - (b*c^4 - 2*a*c^3*d)*f + (b*c^3*d*f + (3*b*c
^2*d^2 - 4*a*c*d^3)*e)*x)*sqrt(b*x + a) + (b^2*c^5*f + (b^2*c^3*d^2*f + (3*b^2*c
^2*d^3 - 12*a*b*c*d^4 + 8*a^2*d^5)*e)*x^2 + (3*b^2*c^4*d - 12*a*b*c^3*d^2 + 8*a^
2*c^2*d^3)*e + 2*(b^2*c^4*d*f + (3*b^2*c^3*d^2 - 12*a*b*c^2*d^3 + 8*a^2*c*d^4)*e
)*x)*log((sqrt(-b*c*d + a*d^2)*(b*d*x - b*c + 2*a*d) - 2*(b*c*d - a*d^2)*sqrt(b*
x + a))/(d*x + c)))/((b*c^6*d - a*c^5*d^2 + (b*c^4*d^3 - a*c^3*d^4)*x^2 + 2*(b*c
^5*d^2 - a*c^4*d^3)*x)*sqrt(-b*c*d + a*d^2)), 1/4*(4*((b*c*d^3 - a*d^4)*e*x^2 +
2*(b*c^2*d^2 - a*c*d^3)*e*x + (b*c^3*d - a*c^2*d^2)*e)*sqrt(b*c*d - a*d^2)*sqrt(
a)*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + sqrt(b*c*d - a*d^2)*((5*b*c^3*
d - 6*a*c^2*d^2)*e - (b*c^4 - 2*a*c^3*d)*f + (b*c^3*d*f + (3*b*c^2*d^2 - 4*a*c*d
^3)*e)*x)*sqrt(b*x + a) + (b^2*c^5*f + (b^2*c^3*d^2*f + (3*b^2*c^2*d^3 - 12*a*b*
c*d^4 + 8*a^2*d^5)*e)*x^2 + (3*b^2*c^4*d - 12*a*b*c^3*d^2 + 8*a^2*c^2*d^3)*e + 2
*(b^2*c^4*d*f + (3*b^2*c^3*d^2 - 12*a*b*c^2*d^3 + 8*a^2*c*d^4)*e)*x)*arctan(-(b*
c - a*d)/(sqrt(b*c*d - a*d^2)*sqrt(b*x + a))))/((b*c^6*d - a*c^5*d^2 + (b*c^4*d^
3 - a*c^3*d^4)*x^2 + 2*(b*c^5*d^2 - a*c^4*d^3)*x)*sqrt(b*c*d - a*d^2)), -1/4*(8*
((b*c*d^3 - a*d^4)*e*x^2 + 2*(b*c^2*d^2 - a*c*d^3)*e*x + (b*c^3*d - a*c^2*d^2)*e
)*sqrt(b*c*d - a*d^2)*sqrt(-a)*arctan(sqrt(b*x + a)/sqrt(-a)) - sqrt(b*c*d - a*d
^2)*((5*b*c^3*d - 6*a*c^2*d^2)*e - (b*c^4 - 2*a*c^3*d)*f + (b*c^3*d*f + (3*b*c^2
*d^2 - 4*a*c*d^3)*e)*x)*sqrt(b*x + a) - (b^2*c^5*f + (b^2*c^3*d^2*f + (3*b^2*c^2
*d^3 - 12*a*b*c*d^4 + 8*a^2*d^5)*e)*x^2 + (3*b^2*c^4*d - 12*a*b*c^3*d^2 + 8*a^2*
c^2*d^3)*e + 2*(b^2*c^4*d*f + (3*b^2*c^3*d^2 - 12*a*b*c^2*d^3 + 8*a^2*c*d^4)*e)*
x)*arctan(-(b*c - a*d)/(sqrt(b*c*d - a*d^2)*sqrt(b*x + a))))/((b*c^6*d - a*c^5*d
^2 + (b*c^4*d^3 - a*c^3*d^4)*x^2 + 2*(b*c^5*d^2 - a*c^4*d^3)*x)*sqrt(b*c*d - a*d
^2))]

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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((f*x+e)*(b*x+a)**(1/2)/x/(d*x+c)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.222982, size = 406, normalized size = 1.98 \[ \frac{{\left (b^{2} c^{3} f + 3 \, b^{2} c^{2} d e - 12 \, a b c d^{2} e + 8 \, a^{2} d^{3} e\right )} \arctan \left (\frac{\sqrt{b x + a} d}{\sqrt{b c d - a d^{2}}}\right )}{4 \,{\left (b c^{4} d - a c^{3} d^{2}\right )} \sqrt{b c d - a d^{2}}} + \frac{2 \, a \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ) e}{\sqrt{-a} c^{3}} - \frac{\sqrt{b x + a} b^{3} c^{3} f -{\left (b x + a\right )}^{\frac{3}{2}} b^{2} c^{2} d f - \sqrt{b x + a} a b^{2} c^{2} d f - 5 \, \sqrt{b x + a} b^{3} c^{2} d e - 3 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{2} c d^{2} e + 9 \, \sqrt{b x + a} a b^{2} c d^{2} e + 4 \,{\left (b x + a\right )}^{\frac{3}{2}} a b d^{3} e - 4 \, \sqrt{b x + a} a^{2} b d^{3} e}{4 \,{\left (b c^{3} d - a c^{2} d^{2}\right )}{\left (b c +{\left (b x + a\right )} d - a d\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(b^2*c^3*f + 3*b^2*c^2*d*e - 12*a*b*c*d^2*e + 8*a^2*d^3*e)*arctan(sqrt(b*x +
a)*d/sqrt(b*c*d - a*d^2))/((b*c^4*d - a*c^3*d^2)*sqrt(b*c*d - a*d^2)) + 2*a*arc
tan(sqrt(b*x + a)/sqrt(-a))*e/(sqrt(-a)*c^3) - 1/4*(sqrt(b*x + a)*b^3*c^3*f - (b
*x + a)^(3/2)*b^2*c^2*d*f - sqrt(b*x + a)*a*b^2*c^2*d*f - 5*sqrt(b*x + a)*b^3*c^
2*d*e - 3*(b*x + a)^(3/2)*b^2*c*d^2*e + 9*sqrt(b*x + a)*a*b^2*c*d^2*e + 4*(b*x +
a)^(3/2)*a*b*d^3*e - 4*sqrt(b*x + a)*a^2*b*d^3*e)/((b*c^3*d - a*c^2*d^2)*(b*c +
(b*x + a)*d - a*d)^2)