3.454 \(\int \frac{(c+d x)^{3/2}}{x (a+b x)^2} \, dx\)

Optimal. Leaf size=115 \[ \frac{\sqrt{b c-a d} (a d+2 b c) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^2 b^{3/2}}-\frac{2 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^2}+\frac{\sqrt{c+d x} (b c-a d)}{a b (a+b x)} \]

[Out]

((b*c - a*d)*Sqrt[c + d*x])/(a*b*(a + b*x)) - (2*c^(3/2)*ArcTanh[Sqrt[c + d*x]/S
qrt[c]])/a^2 + (Sqrt[b*c - a*d]*(2*b*c + a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sq
rt[b*c - a*d]])/(a^2*b^(3/2))

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Rubi [A]  time = 0.295956, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{\sqrt{b c-a d} (a d+2 b c) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^2 b^{3/2}}-\frac{2 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^2}+\frac{\sqrt{c+d x} (b c-a d)}{a b (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(3/2)/(x*(a + b*x)^2),x]

[Out]

((b*c - a*d)*Sqrt[c + d*x])/(a*b*(a + b*x)) - (2*c^(3/2)*ArcTanh[Sqrt[c + d*x]/S
qrt[c]])/a^2 + (Sqrt[b*c - a*d]*(2*b*c + a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sq
rt[b*c - a*d]])/(a^2*b^(3/2))

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Rubi in Sympy [A]  time = 32.3172, size = 100, normalized size = 0.87 \[ - \frac{\sqrt{c + d x} \left (a d - b c\right )}{a b \left (a + b x\right )} - \frac{2 c^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{a^{2}} + \frac{\sqrt{a d - b c} \left (a d + 2 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{a^{2} b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(c + d*x)*(a*d - b*c)/(a*b*(a + b*x)) - 2*c**(3/2)*atanh(sqrt(c + d*x)/sqrt
(c))/a**2 + sqrt(a*d - b*c)*(a*d + 2*b*c)*atan(sqrt(b)*sqrt(c + d*x)/sqrt(a*d -
b*c))/(a**2*b**(3/2))

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Mathematica [A]  time = 0.383727, size = 111, normalized size = 0.97 \[ \frac{\frac{\sqrt{b c-a d} (a d+2 b c) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{3/2}}+\frac{a \sqrt{c+d x} (b c-a d)}{b (a+b x)}-2 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(3/2)/(x*(a + b*x)^2),x]

[Out]

((a*(b*c - a*d)*Sqrt[c + d*x])/(b*(a + b*x)) - 2*c^(3/2)*ArcTanh[Sqrt[c + d*x]/S
qrt[c]] + (Sqrt[b*c - a*d]*(2*b*c + a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*
c - a*d]])/b^(3/2))/a^2

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Maple [A]  time = 0.02, size = 194, normalized size = 1.7 \[ -2\,{\frac{{c}^{3/2}}{{a}^{2}}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }-{\frac{{d}^{2}}{b \left ( bdx+ad \right ) }\sqrt{dx+c}}+{\frac{dc}{a \left ( bdx+ad \right ) }\sqrt{dx+c}}+{\frac{{d}^{2}}{b}\arctan \left ({b\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}}+{\frac{dc}{a}\arctan \left ({b\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}}-2\,{\frac{{c}^{2}b}{{a}^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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((d*x + c)^(3/2)/((b*x + a)^2*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.282248, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (2 \, a b c + a^{2} d +{\left (2 \, b^{2} c + a b d\right )} x\right )} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x + 2 \, b c - a d + 2 \, \sqrt{d x + c} b \sqrt{\frac{b c - a d}{b}}}{b x + a}\right ) + 2 \,{\left (b^{2} c x + a b c\right )} \sqrt{c} \log \left (\frac{d x - 2 \, \sqrt{d x + c} \sqrt{c} + 2 \, c}{x}\right ) + 2 \,{\left (a b c - a^{2} d\right )} \sqrt{d x + c}}{2 \,{\left (a^{2} b^{2} x + a^{3} b\right )}}, \frac{{\left (2 \, a b c + a^{2} d +{\left (2 \, b^{2} c + a b d\right )} x\right )} \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-\frac{b c - a d}{b}}}\right ) +{\left (b^{2} c x + a b c\right )} \sqrt{c} \log \left (\frac{d x - 2 \, \sqrt{d x + c} \sqrt{c} + 2 \, c}{x}\right ) +{\left (a b c - a^{2} d\right )} \sqrt{d x + c}}{a^{2} b^{2} x + a^{3} b}, -\frac{4 \,{\left (b^{2} c x + a b c\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right ) -{\left (2 \, a b c + a^{2} d +{\left (2 \, b^{2} c + a b d\right )} x\right )} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x + 2 \, b c - a d + 2 \, \sqrt{d x + c} b \sqrt{\frac{b c - a d}{b}}}{b x + a}\right ) - 2 \,{\left (a b c - a^{2} d\right )} \sqrt{d x + c}}{2 \,{\left (a^{2} b^{2} x + a^{3} b\right )}}, -\frac{2 \,{\left (b^{2} c x + a b c\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right ) -{\left (2 \, a b c + a^{2} d +{\left (2 \, b^{2} c + a b d\right )} x\right )} \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-\frac{b c - a d}{b}}}\right ) -{\left (a b c - a^{2} d\right )} \sqrt{d x + c}}{a^{2} b^{2} x + a^{3} b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*((2*a*b*c + a^2*d + (2*b^2*c + a*b*d)*x)*sqrt((b*c - a*d)/b)*log((b*d*x + 2
*b*c - a*d + 2*sqrt(d*x + c)*b*sqrt((b*c - a*d)/b))/(b*x + a)) + 2*(b^2*c*x + a*
b*c)*sqrt(c)*log((d*x - 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) + 2*(a*b*c - a^2*d)*sq
rt(d*x + c))/(a^2*b^2*x + a^3*b), ((2*a*b*c + a^2*d + (2*b^2*c + a*b*d)*x)*sqrt(
-(b*c - a*d)/b)*arctan(sqrt(d*x + c)/sqrt(-(b*c - a*d)/b)) + (b^2*c*x + a*b*c)*s
qrt(c)*log((d*x - 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) + (a*b*c - a^2*d)*sqrt(d*x +
c))/(a^2*b^2*x + a^3*b), -1/2*(4*(b^2*c*x + a*b*c)*sqrt(-c)*arctan(sqrt(d*x + c
)/sqrt(-c)) - (2*a*b*c + a^2*d + (2*b^2*c + a*b*d)*x)*sqrt((b*c - a*d)/b)*log((b
*d*x + 2*b*c - a*d + 2*sqrt(d*x + c)*b*sqrt((b*c - a*d)/b))/(b*x + a)) - 2*(a*b*
c - a^2*d)*sqrt(d*x + c))/(a^2*b^2*x + a^3*b), -(2*(b^2*c*x + a*b*c)*sqrt(-c)*ar
ctan(sqrt(d*x + c)/sqrt(-c)) - (2*a*b*c + a^2*d + (2*b^2*c + a*b*d)*x)*sqrt(-(b*
c - a*d)/b)*arctan(sqrt(d*x + c)/sqrt(-(b*c - a*d)/b)) - (a*b*c - a^2*d)*sqrt(d*
x + c))/(a^2*b^2*x + a^3*b)]

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Sympy [A]  time = 97.4062, size = 1192, normalized size = 10.37 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)**(3/2)/x/(b*x+a)**2,x)

[Out]

-2*a*d**3*sqrt(c + d*x)/(2*a**2*b*d**2 - 2*a*b**2*c*d + 2*a*b**2*d**2*x - 2*b**3
*c*d*x) + a*d**3*sqrt(-1/(b*(a*d - b*c)**3))*log(-a**2*d**2*sqrt(-1/(b*(a*d - b*
c)**3)) + 2*a*b*c*d*sqrt(-1/(b*(a*d - b*c)**3)) - b**2*c**2*sqrt(-1/(b*(a*d - b*
c)**3)) + sqrt(c + d*x))/(2*b) - a*d**3*sqrt(-1/(b*(a*d - b*c)**3))*log(a**2*d**
2*sqrt(-1/(b*(a*d - b*c)**3)) - 2*a*b*c*d*sqrt(-1/(b*(a*d - b*c)**3)) + b**2*c**
2*sqrt(-1/(b*(a*d - b*c)**3)) + sqrt(c + d*x))/(2*b) - 2*b*c**2*d*sqrt(c + d*x)/
(2*a**3*d**2 - 2*a**2*b*c*d + 2*a**2*b*d**2*x - 2*a*b**2*c*d*x) - c*d**2*sqrt(-1
/(b*(a*d - b*c)**3))*log(-a**2*d**2*sqrt(-1/(b*(a*d - b*c)**3)) + 2*a*b*c*d*sqrt
(-1/(b*(a*d - b*c)**3)) - b**2*c**2*sqrt(-1/(b*(a*d - b*c)**3)) + sqrt(c + d*x))
+ c*d**2*sqrt(-1/(b*(a*d - b*c)**3))*log(a**2*d**2*sqrt(-1/(b*(a*d - b*c)**3))
- 2*a*b*c*d*sqrt(-1/(b*(a*d - b*c)**3)) + b**2*c**2*sqrt(-1/(b*(a*d - b*c)**3))
+ sqrt(c + d*x)) + 4*c*d**2*sqrt(c + d*x)/(2*a**2*d**2 - 2*a*b*c*d + 2*a*b*d**2*
x - 2*b**2*c*d*x) + 2*d**2*Piecewise((atan(sqrt(c + d*x)/sqrt(a*d/b - c))/(b*sqr
t(a*d/b - c)), a*d/b - c > 0), (-acoth(sqrt(c + d*x)/sqrt(-a*d/b + c))/(b*sqrt(-
a*d/b + c)), (a*d/b - c < 0) & (c + d*x > -a*d/b + c)), (-atanh(sqrt(c + d*x)/sq
rt(-a*d/b + c))/(b*sqrt(-a*d/b + c)), (a*d/b - c < 0) & (c + d*x < -a*d/b + c)))
/b + b*c**2*d*sqrt(-1/(b*(a*d - b*c)**3))*log(-a**2*d**2*sqrt(-1/(b*(a*d - b*c)*
*3)) + 2*a*b*c*d*sqrt(-1/(b*(a*d - b*c)**3)) - b**2*c**2*sqrt(-1/(b*(a*d - b*c)*
*3)) + sqrt(c + d*x))/(2*a) - b*c**2*d*sqrt(-1/(b*(a*d - b*c)**3))*log(a**2*d**2
*sqrt(-1/(b*(a*d - b*c)**3)) - 2*a*b*c*d*sqrt(-1/(b*(a*d - b*c)**3)) + b**2*c**2
*sqrt(-1/(b*(a*d - b*c)**3)) + sqrt(c + d*x))/(2*a) - 2*b*c**2*Piecewise((atan(s
qrt(c + d*x)/sqrt(a*d/b - c))/(b*sqrt(a*d/b - c)), a*d/b - c > 0), (-acoth(sqrt(
c + d*x)/sqrt(-a*d/b + c))/(b*sqrt(-a*d/b + c)), (a*d/b - c < 0) & (c + d*x > -a
*d/b + c)), (-atanh(sqrt(c + d*x)/sqrt(-a*d/b + c))/(b*sqrt(-a*d/b + c)), (a*d/b
- c < 0) & (c + d*x < -a*d/b + c)))/a**2 - 2*c**2*Piecewise((-atan(sqrt(c + d*x
)/sqrt(-c))/sqrt(-c), -c > 0), (acoth(sqrt(c + d*x)/sqrt(c))/sqrt(c), (-c < 0) &
(c < c + d*x)), (atanh(sqrt(c + d*x)/sqrt(c))/sqrt(c), (-c < 0) & (c > c + d*x)
))/a**2

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GIAC/XCAS [A]  time = 0.231175, size = 194, normalized size = 1.69 \[ \frac{2 \, c^{2} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right )}{a^{2} \sqrt{-c}} - \frac{{\left (2 \, b^{2} c^{2} - a b c d - a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a^{2} b} + \frac{\sqrt{d x + c} b c d - \sqrt{d x + c} a d^{2}}{{\left ({\left (d x + c\right )} b - b c + a d\right )} a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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