3.461 \(\int \frac{(c+d x)^{5/2}}{x^2 (a+b x)^2} \, dx\)

Optimal. Leaf size=147 \[ -\frac{(b c-a d)^{3/2} (a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^3 b^{3/2}}+\frac{c^{3/2} (4 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^3}-\frac{\sqrt{c+d x} (b c-a d)^2}{a^2 b (a+b x)}-\frac{c^2 \sqrt{c+d x}}{a^2 x} \]

[Out]

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

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Rubi [A]  time = 0.566549, antiderivative size = 159, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ -\frac{(b c-a d)^{3/2} (a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^3 b^{3/2}}+\frac{c^{3/2} (4 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^3}-\frac{\sqrt{c+d x} (b c-a d) (2 b c-a d)}{a^2 b (a+b x)}-\frac{c (c+d x)^{3/2}}{a x (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(5/2)/(x^2*(a + b*x)^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 56.9539, size = 138, normalized size = 0.94 \[ - \frac{c \left (c + d x\right )^{\frac{3}{2}}}{a x \left (a + b x\right )} - \frac{\sqrt{c + d x} \left (a d - 2 b c\right ) \left (a d - b c\right )}{a^{2} b \left (a + b x\right )} - \frac{c^{\frac{3}{2}} \left (5 a d - 4 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{a^{3}} + \frac{\left (a d - b c\right )^{\frac{3}{2}} \left (a d + 4 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{a^{3} b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(5/2)/x**2/(b*x+a)**2,x)

[Out]

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

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Mathematica [A]  time = 0.29412, size = 132, normalized size = 0.9 \[ \frac{-\frac{(b c-a d)^{3/2} (a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{3/2}}+c^{3/2} (4 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )-a \sqrt{c+d x} \left (\frac{(b c-a d)^2}{b (a+b x)}+\frac{c^2}{x}\right )}{a^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(5/2)/(x^2*(a + b*x)^2),x]

[Out]

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

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Maple [B]  time = 0.027, size = 313, normalized size = 2.1 \[ -{\frac{{c}^{2}}{{a}^{2}x}\sqrt{dx+c}}-5\,{\frac{d{c}^{3/2}}{{a}^{2}}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }+4\,{\frac{{c}^{5/2}b}{{a}^{3}}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }-{\frac{{d}^{3}}{b \left ( bdx+ad \right ) }\sqrt{dx+c}}+2\,{\frac{{d}^{2}\sqrt{dx+c}c}{a \left ( bdx+ad \right ) }}-{\frac{bd{c}^{2}}{{a}^{2} \left ( bdx+ad \right ) }\sqrt{dx+c}}+{\frac{{d}^{3}}{b}\arctan \left ({b\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}}+2\,{\frac{{d}^{2}c}{a\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-7\,{\frac{bd{c}^{2}}{{a}^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+4\,{\frac{{c}^{3}{b}^{2}}{{a}^{3}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(5/2)/x^2/(b*x+a)^2,x)

[Out]

-c^2*(d*x+c)^(1/2)/a^2/x-5*d/a^2*c^(3/2)*arctanh((d*x+c)^(1/2)/c^(1/2))+4*c^(5/2
)/a^3*arctanh((d*x+c)^(1/2)/c^(1/2))*b-d^3/b*(d*x+c)^(1/2)/(b*d*x+a*d)+2*d^2/a*(
d*x+c)^(1/2)/(b*d*x+a*d)*c-d/a^2*b*(d*x+c)^(1/2)/(b*d*x+a*d)*c^2+d^3/b/((a*d-b*c
)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))+2*d^2/a/((a*d-b*c)*b)^(1/
2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*c-7*d/a^2*b/((a*d-b*c)*b)^(1/2)*a
rctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*c^2+4/a^3/((a*d-b*c)*b)^(1/2)*arctan(
(d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*c^3*b^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)^(5/2)/((b*x + a)^2*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.406501, 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)^(5/2)/((b*x + a)^2*x^2),x, algorithm="fricas")

[Out]

[-1/2*(((4*b^3*c^2 - 3*a*b^2*c*d - a^2*b*d^2)*x^2 + (4*a*b^2*c^2 - 3*a^2*b*c*d -
a^3*d^2)*x)*sqrt((b*c - a*d)/b)*log((b*d*x + 2*b*c - a*d + 2*sqrt(d*x + c)*b*sq
rt((b*c - a*d)/b))/(b*x + a)) + ((4*b^3*c^2 - 5*a*b^2*c*d)*x^2 + (4*a*b^2*c^2 -
5*a^2*b*c*d)*x)*sqrt(c)*log((d*x - 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) + 2*(a^2*b*
c^2 + (2*a*b^2*c^2 - 2*a^2*b*c*d + a^3*d^2)*x)*sqrt(d*x + c))/(a^3*b^2*x^2 + a^4
*b*x), -1/2*(2*((4*b^3*c^2 - 3*a*b^2*c*d - a^2*b*d^2)*x^2 + (4*a*b^2*c^2 - 3*a^2
*b*c*d - a^3*d^2)*x)*sqrt(-(b*c - a*d)/b)*arctan(sqrt(d*x + c)/sqrt(-(b*c - a*d)
/b)) + ((4*b^3*c^2 - 5*a*b^2*c*d)*x^2 + (4*a*b^2*c^2 - 5*a^2*b*c*d)*x)*sqrt(c)*l
og((d*x - 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) + 2*(a^2*b*c^2 + (2*a*b^2*c^2 - 2*a^
2*b*c*d + a^3*d^2)*x)*sqrt(d*x + c))/(a^3*b^2*x^2 + a^4*b*x), 1/2*(2*((4*b^3*c^2
- 5*a*b^2*c*d)*x^2 + (4*a*b^2*c^2 - 5*a^2*b*c*d)*x)*sqrt(-c)*arctan(sqrt(d*x +
c)/sqrt(-c)) - ((4*b^3*c^2 - 3*a*b^2*c*d - a^2*b*d^2)*x^2 + (4*a*b^2*c^2 - 3*a^2
*b*c*d - a^3*d^2)*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*b*c^2 + (2*a*b^2*c^2 - 2*a^2*b*c*
d + a^3*d^2)*x)*sqrt(d*x + c))/(a^3*b^2*x^2 + a^4*b*x), (((4*b^3*c^2 - 5*a*b^2*c
*d)*x^2 + (4*a*b^2*c^2 - 5*a^2*b*c*d)*x)*sqrt(-c)*arctan(sqrt(d*x + c)/sqrt(-c))
- ((4*b^3*c^2 - 3*a*b^2*c*d - a^2*b*d^2)*x^2 + (4*a*b^2*c^2 - 3*a^2*b*c*d - a^3
*d^2)*x)*sqrt(-(b*c - a*d)/b)*arctan(sqrt(d*x + c)/sqrt(-(b*c - a*d)/b)) - (a^2*
b*c^2 + (2*a*b^2*c^2 - 2*a^2*b*c*d + a^3*d^2)*x)*sqrt(d*x + c))/(a^3*b^2*x^2 + a
^4*b*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)**(5/2)/x**2/(b*x+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.230165, size = 351, normalized size = 2.39 \[ -\frac{{\left (4 \, b c^{3} - 5 \, a c^{2} d\right )} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right )}{a^{3} \sqrt{-c}} + \frac{{\left (4 \, b^{3} c^{3} - 7 \, a b^{2} c^{2} d + 2 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a^{3} b} - \frac{2 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{2} c^{2} d - 2 \, \sqrt{d x + c} b^{2} c^{3} d - 2 \,{\left (d x + c\right )}^{\frac{3}{2}} a b c d^{2} + 3 \, \sqrt{d x + c} a b c^{2} d^{2} +{\left (d x + c\right )}^{\frac{3}{2}} a^{2} d^{3} - \sqrt{d x + c} a^{2} c d^{3}}{{\left ({\left (d x + c\right )}^{2} b - 2 \,{\left (d x + c\right )} b c + b c^{2} +{\left (d x + c\right )} a d - a c d\right )} a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/2)/((b*x + a)^2*x^2),x, algorithm="giac")

[Out]

-(4*b*c^3 - 5*a*c^2*d)*arctan(sqrt(d*x + c)/sqrt(-c))/(a^3*sqrt(-c)) + (4*b^3*c^
3 - 7*a*b^2*c^2*d + 2*a^2*b*c*d^2 + a^3*d^3)*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c
+ a*b*d))/(sqrt(-b^2*c + a*b*d)*a^3*b) - (2*(d*x + c)^(3/2)*b^2*c^2*d - 2*sqrt(d
*x + c)*b^2*c^3*d - 2*(d*x + c)^(3/2)*a*b*c*d^2 + 3*sqrt(d*x + c)*a*b*c^2*d^2 +
(d*x + c)^(3/2)*a^2*d^3 - sqrt(d*x + c)*a^2*c*d^3)/(((d*x + c)^2*b - 2*(d*x + c)
*b*c + b*c^2 + (d*x + c)*a*d - a*c*d)*a^2*b)