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

Optimal. Leaf size=149 \[ \frac{\sqrt{c} (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^3}-\frac{\sqrt{b c-a d} (4 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^3 \sqrt{b}}-\frac{\sqrt{c+d x} (2 b c-a d)}{a^2 (a+b x)}-\frac{c \sqrt{c+d x}}{a x (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.562979, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{\sqrt{c} (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^3}-\frac{\sqrt{b c-a d} (4 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^3 \sqrt{b}}-\frac{\sqrt{c+d x} (2 b c-a d)}{a^2 (a+b x)}-\frac{c \sqrt{c+d x}}{a x (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 57.6269, size = 129, normalized size = 0.87 \[ - \frac{c \sqrt{c + d x}}{a x \left (a + b x\right )} + \frac{\sqrt{c + d x} \left (a d - 2 b c\right )}{a^{2} \left (a + b x\right )} - \frac{\sqrt{c} \left (3 a d - 4 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{a^{3}} + \frac{\left (a d - 4 b c\right ) \sqrt{a d - b c} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{a^{3} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-c*sqrt(c + d*x)/(a*x*(a + b*x)) + sqrt(c + d*x)*(a*d - 2*b*c)/(a**2*(a + b*x))
- sqrt(c)*(3*a*d - 4*b*c)*atanh(sqrt(c + d*x)/sqrt(c))/a**3 + (a*d - 4*b*c)*sqrt
(a*d - b*c)*atan(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c))/(a**3*sqrt(b))

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Mathematica [A]  time = 0.413918, size = 140, normalized size = 0.94 \[ \frac{-\frac{\left (a^2 d^2-5 a b c d+4 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{\sqrt{b} \sqrt{b c-a d}}+\frac{a \sqrt{c+d x} (-a c+a d x-2 b c x)}{x (a+b x)}+\sqrt{c} (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^3} \]

Antiderivative was successfully verified.

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

[Out]

((a*Sqrt[c + d*x]*(-(a*c) - 2*b*c*x + a*d*x))/(x*(a + b*x)) + Sqrt[c]*(4*b*c - 3
*a*d)*ArcTanh[Sqrt[c + d*x]/Sqrt[c]] - ((4*b^2*c^2 - 5*a*b*c*d + a^2*d^2)*ArcTan
h[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(Sqrt[b]*Sqrt[b*c - a*d]))/a^3

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Maple [A]  time = 0.026, size = 237, normalized size = 1.6 \[ -{\frac{c}{{a}^{2}x}\sqrt{dx+c}}-3\,{\frac{d\sqrt{c}}{{a}^{2}}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }+4\,{\frac{{c}^{3/2}b}{{a}^{3}}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }+{\frac{{d}^{2}}{a \left ( bdx+ad \right ) }\sqrt{dx+c}}-{\frac{bdc}{{a}^{2} \left ( bdx+ad \right ) }\sqrt{dx+c}}+{\frac{{d}^{2}}{a}\arctan \left ({b\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}}-5\,{\frac{bdc}{{a}^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+4\,{\frac{{b}^{2}{c}^{2}}{{a}^{3}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-c/a^2*(d*x+c)^(1/2)/x-3*d*c^(1/2)/a^2*arctanh((d*x+c)^(1/2)/c^(1/2))+4*c^(3/2)/
a^3*arctanh((d*x+c)^(1/2)/c^(1/2))*b+d^2/a*(d*x+c)^(1/2)/(b*d*x+a*d)-d/a^2*(d*x+
c)^(1/2)/(b*d*x+a*d)*b*c+d^2/a/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-
b*c)*b)^(1/2))-5*d/a^2/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^
(1/2))*b*c+4/a^3/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))
*b^2*c^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((d*x + c)^(3/2)/((b*x + a)^2*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.285522, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/2*(((4*b^2*c - a*b*d)*x^2 + (4*a*b*c - a^2*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)) + ((4*b^2*
c - 3*a*b*d)*x^2 + (4*a*b*c - 3*a^2*d)*x)*sqrt(c)*log((d*x - 2*sqrt(d*x + c)*sqr
t(c) + 2*c)/x) + 2*(a^2*c + (2*a*b*c - a^2*d)*x)*sqrt(d*x + c))/(a^3*b*x^2 + a^4
*x), -1/2*(2*((4*b^2*c - a*b*d)*x^2 + (4*a*b*c - a^2*d)*x)*sqrt(-(b*c - a*d)/b)*
arctan(sqrt(d*x + c)/sqrt(-(b*c - a*d)/b)) + ((4*b^2*c - 3*a*b*d)*x^2 + (4*a*b*c
- 3*a^2*d)*x)*sqrt(c)*log((d*x - 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) + 2*(a^2*c +
(2*a*b*c - a^2*d)*x)*sqrt(d*x + c))/(a^3*b*x^2 + a^4*x), 1/2*(2*((4*b^2*c - 3*a
*b*d)*x^2 + (4*a*b*c - 3*a^2*d)*x)*sqrt(-c)*arctan(sqrt(d*x + c)/sqrt(-c)) - ((4
*b^2*c - a*b*d)*x^2 + (4*a*b*c - a^2*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^2*c + (2*a*b*
c - a^2*d)*x)*sqrt(d*x + c))/(a^3*b*x^2 + a^4*x), (((4*b^2*c - 3*a*b*d)*x^2 + (4
*a*b*c - 3*a^2*d)*x)*sqrt(-c)*arctan(sqrt(d*x + c)/sqrt(-c)) - ((4*b^2*c - a*b*d
)*x^2 + (4*a*b*c - a^2*d)*x)*sqrt(-(b*c - a*d)/b)*arctan(sqrt(d*x + c)/sqrt(-(b*
c - a*d)/b)) - (a^2*c + (2*a*b*c - a^2*d)*x)*sqrt(d*x + c))/(a^3*b*x^2 + a^4*x)]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.225548, size = 266, normalized size = 1.79 \[ \frac{{\left (4 \, b^{2} c^{2} - 5 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a^{3}} - \frac{{\left (4 \, b c^{2} - 3 \, a c d\right )} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right )}{a^{3} \sqrt{-c}} - \frac{2 \,{\left (d x + c\right )}^{\frac{3}{2}} b c d - 2 \, \sqrt{d x + c} b c^{2} d -{\left (d x + c\right )}^{\frac{3}{2}} a d^{2} + 2 \, \sqrt{d x + c} a c d^{2}}{{\left ({\left (d x + c\right )}^{2} b - 2 \,{\left (d x + c\right )} b c + b c^{2} +{\left (d x + c\right )} a d - a c d\right )} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(4*b^2*c^2 - 5*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^3) - (4*b*c^2 - 3*a*c*d)*arctan(sqrt(d*x + c)/sqrt(-c))/(
a^3*sqrt(-c)) - (2*(d*x + c)^(3/2)*b*c*d - 2*sqrt(d*x + c)*b*c^2*d - (d*x + c)^(
3/2)*a*d^2 + 2*sqrt(d*x + c)*a*c*d^2)/(((d*x + c)^2*b - 2*(d*x + c)*b*c + b*c^2
+ (d*x + c)*a*d - a*c*d)*a^2)