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

Optimal. Leaf size=142 \[ \frac{(b c-a d)^{3/2} (3 a d+2 b c) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^2 b^{5/2}}-\frac{2 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^2}-\frac{d \sqrt{c+d x} (b c-3 a d)}{a b^2}+\frac{(c+d x)^{3/2} (b c-a d)}{a b (a+b x)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.484099, antiderivative size = 142, 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 \[ \frac{(b c-a d)^{3/2} (3 a d+2 b c) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^2 b^{5/2}}-\frac{2 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^2}-\frac{d \sqrt{c+d x} (b c-3 a d)}{a b^2}+\frac{(c+d x)^{3/2} (b c-a d)}{a b (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 51.2607, size = 126, normalized size = 0.89 \[ - \frac{\left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )}{a b \left (a + b x\right )} + \frac{d \sqrt{c + d x} \left (3 a d - b c\right )}{a b^{2}} - \frac{2 c^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{a^{2}} - \frac{\left (a d - b c\right )^{\frac{3}{2}} \left (3 a d + 2 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{a^{2} b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.219618, size = 125, normalized size = 0.88 \[ \frac{(b c-a d)^{3/2} (3 a d+2 b c) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^2 b^{5/2}}-\frac{2 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^2}+\frac{\sqrt{c+d x} \left (\frac{(b c-a d)^2}{a (a+b x)}+2 d^2\right )}{b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.024, size = 284, normalized size = 2. \[ 2\,{\frac{{d}^{2}\sqrt{dx+c}}{{b}^{2}}}-2\,{\frac{{c}^{5/2}}{{a}^{2}}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }+{\frac{{d}^{3}a}{{b}^{2} \left ( bdx+ad \right ) }\sqrt{dx+c}}-2\,{\frac{{d}^{2}\sqrt{dx+c}c}{b \left ( bdx+ad \right ) }}+{\frac{d{c}^{2}}{a \left ( bdx+ad \right ) }\sqrt{dx+c}}-3\,{\frac{{d}^{3}a}{{b}^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+4\,{\frac{{d}^{2}c}{b\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+{\frac{d{c}^{2}}{a}\arctan \left ({b\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}}-2\,{\frac{b{c}^{3}}{{a}^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*d^2/b^2*(d*x+c)^(1/2)-2*c^(5/2)*arctanh((d*x+c)^(1/2)/c^(1/2))/a^2+d^3/b^2*a*(
d*x+c)^(1/2)/(b*d*x+a*d)-2*d^2/b*(d*x+c)^(1/2)/(b*d*x+a*d)*c+d/a*(d*x+c)^(1/2)/(
b*d*x+a*d)*c^2-3*d^3/b^2*a/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)
*b)^(1/2))+4*d^2/b/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2
))*c+d/a/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*c^2-2*b
/a^2/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*c^3

_______________________________________________________________________________________

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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.408955, 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),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

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/(b*x+a)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.223517, size = 261, normalized size = 1.84 \[ \frac{2 \, c^{3} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right )}{a^{2} \sqrt{-c}} + \frac{2 \, \sqrt{d x + c} d^{2}}{b^{2}} - \frac{{\left (2 \, b^{3} c^{3} - a b^{2} c^{2} d - 4 \, a^{2} b c d^{2} + 3 \, a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a^{2} b^{2}} + \frac{\sqrt{d x + c} b^{2} c^{2} d - 2 \, \sqrt{d x + c} a b c d^{2} + \sqrt{d x + c} a^{2} d^{3}}{{\left ({\left (d x + c\right )} b - b c + a d\right )} a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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