3.462 \(\int \frac{(c+d x)^{5/2}}{x^3 (a+b x)^2} \, dx\)

Optimal. Leaf size=219 \[ \frac{(6 b c-a d) (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^4 \sqrt{b}}+\frac{c \sqrt{c+d x} (6 b c-7 a d)}{4 a^2 x (a+b x)}-\frac{\sqrt{c} \left (15 a^2 d^2-40 a b c d+24 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{4 a^4}+\frac{\sqrt{c+d x} \left (4 a^2 d^2-17 a b c d+12 b^2 c^2\right )}{4 a^3 (a+b x)}-\frac{c (c+d x)^{3/2}}{2 a x^2 (a+b x)} \]

[Out]

((12*b^2*c^2 - 17*a*b*c*d + 4*a^2*d^2)*Sqrt[c + d*x])/(4*a^3*(a + b*x)) + (c*(6*
b*c - 7*a*d)*Sqrt[c + d*x])/(4*a^2*x*(a + b*x)) - (c*(c + d*x)^(3/2))/(2*a*x^2*(
a + b*x)) - (Sqrt[c]*(24*b^2*c^2 - 40*a*b*c*d + 15*a^2*d^2)*ArcTanh[Sqrt[c + d*x
]/Sqrt[c]])/(4*a^4) + ((b*c - a*d)^(3/2)*(6*b*c - a*d)*ArcTanh[(Sqrt[b]*Sqrt[c +
d*x])/Sqrt[b*c - a*d]])/(a^4*Sqrt[b])

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Rubi [A]  time = 0.75348, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ \frac{(6 b c-a d) (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^4 \sqrt{b}}+\frac{c \sqrt{c+d x} (6 b c-7 a d)}{4 a^2 x (a+b x)}-\frac{\sqrt{c} \left (15 a^2 d^2-40 a b c d+24 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{4 a^4}+\frac{\sqrt{c+d x} \left (4 a^2 d^2-17 a b c d+12 b^2 c^2\right )}{4 a^3 (a+b x)}-\frac{c (c+d x)^{3/2}}{2 a x^2 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(5/2)/(x^3*(a + b*x)^2),x]

[Out]

((12*b^2*c^2 - 17*a*b*c*d + 4*a^2*d^2)*Sqrt[c + d*x])/(4*a^3*(a + b*x)) + (c*(6*
b*c - 7*a*d)*Sqrt[c + d*x])/(4*a^2*x*(a + b*x)) - (c*(c + d*x)^(3/2))/(2*a*x^2*(
a + b*x)) - (Sqrt[c]*(24*b^2*c^2 - 40*a*b*c*d + 15*a^2*d^2)*ArcTanh[Sqrt[c + d*x
]/Sqrt[c]])/(4*a^4) + ((b*c - a*d)^(3/2)*(6*b*c - a*d)*ArcTanh[(Sqrt[b]*Sqrt[c +
d*x])/Sqrt[b*c - a*d]])/(a^4*Sqrt[b])

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Rubi in Sympy [A]  time = 93.0765, size = 206, normalized size = 0.94 \[ - \frac{c \left (c + d x\right )^{\frac{3}{2}}}{2 a x^{2} \left (a + b x\right )} - \frac{\sqrt{c + d x} \left (a d - b c\right ) \left (2 a d - 3 b c\right )}{2 a^{2} b x \left (a + b x\right )} + \frac{\sqrt{c + d x} \left (4 a^{2} d^{2} - 17 a b c d + 12 b^{2} c^{2}\right )}{4 a^{3} b x} - \frac{\sqrt{c} \left (15 a^{2} d^{2} - 40 a b c d + 24 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{4 a^{4}} + \frac{\left (a d - 6 b c\right ) \left (a d - b c\right )^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{a^{4} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(5/2)/x**3/(b*x+a)**2,x)

[Out]

-c*(c + d*x)**(3/2)/(2*a*x**2*(a + b*x)) - sqrt(c + d*x)*(a*d - b*c)*(2*a*d - 3*
b*c)/(2*a**2*b*x*(a + b*x)) + sqrt(c + d*x)*(4*a**2*d**2 - 17*a*b*c*d + 12*b**2*
c**2)/(4*a**3*b*x) - sqrt(c)*(15*a**2*d**2 - 40*a*b*c*d + 24*b**2*c**2)*atanh(sq
rt(c + d*x)/sqrt(c))/(4*a**4) + (a*d - 6*b*c)*(a*d - b*c)**(3/2)*atan(sqrt(b)*sq
rt(c + d*x)/sqrt(a*d - b*c))/(a**4*sqrt(b))

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Mathematica [A]  time = 0.380246, size = 164, normalized size = 0.75 \[ \frac{-\sqrt{c} \left (15 a^2 d^2-40 a b c d+24 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )+a \sqrt{c+d x} \left (\frac{c (8 b c-9 a d)}{x}+\frac{4 (b c-a d)^2}{a+b x}-\frac{2 a c^2}{x^2}\right )+\frac{4 (6 b c-a d) (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{\sqrt{b}}}{4 a^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(5/2)/(x^3*(a + b*x)^2),x]

[Out]

(a*Sqrt[c + d*x]*((-2*a*c^2)/x^2 + (c*(8*b*c - 9*a*d))/x + (4*(b*c - a*d)^2)/(a
+ b*x)) - Sqrt[c]*(24*b^2*c^2 - 40*a*b*c*d + 15*a^2*d^2)*ArcTanh[Sqrt[c + d*x]/S
qrt[c]] + (4*(b*c - a*d)^(3/2)*(6*b*c - a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqr
t[b*c - a*d]])/Sqrt[b])/(4*a^4)

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Maple [B]  time = 0.028, size = 403, normalized size = 1.8 \[ -{\frac{9\,c}{4\,{a}^{2}{x}^{2}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}+2\,{\frac{{c}^{2} \left ( dx+c \right ) ^{3/2}b}{{a}^{3}d{x}^{2}}}+{\frac{7\,{c}^{2}}{4\,{a}^{2}{x}^{2}}\sqrt{dx+c}}-2\,{\frac{{c}^{3}\sqrt{dx+c}b}{{a}^{3}d{x}^{2}}}-{\frac{15\,{d}^{2}}{4\,{a}^{2}}\sqrt{c}{\it Artanh} \left ({1\sqrt{dx+c}{\frac{1}{\sqrt{c}}}} \right ) }+10\,{\frac{d{c}^{3/2}b}{{a}^{3}}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }-6\,{\frac{{c}^{5/2}{b}^{2}}{{a}^{4}}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }+{\frac{{d}^{3}}{a \left ( bdx+ad \right ) }\sqrt{dx+c}}-2\,{\frac{{d}^{2}\sqrt{dx+c}bc}{{a}^{2} \left ( bdx+ad \right ) }}+{\frac{{b}^{2}d{c}^{2}}{{a}^{3} \left ( bdx+ad \right ) }\sqrt{dx+c}}+{\frac{{d}^{3}}{a}\arctan \left ({b\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}}-8\,{\frac{{d}^{2}bc}{{a}^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+13\,{\frac{{b}^{2}d{c}^{2}}{{a}^{3}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-6\,{\frac{{b}^{3}{c}^{3}}{{a}^{4}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(5/2)/x^3/(b*x+a)^2,x)

[Out]

-9/4*c/a^2/x^2*(d*x+c)^(3/2)+2/d*c^2/a^3/x^2*(d*x+c)^(3/2)*b+7/4*c^2/a^2/x^2*(d*
x+c)^(1/2)-2/d*c^3/a^3/x^2*(d*x+c)^(1/2)*b-15/4*d^2*c^(1/2)/a^2*arctanh((d*x+c)^
(1/2)/c^(1/2))+10*d*c^(3/2)/a^3*arctanh((d*x+c)^(1/2)/c^(1/2))*b-6*c^(5/2)/a^4*a
rctanh((d*x+c)^(1/2)/c^(1/2))*b^2+d^3/a*(d*x+c)^(1/2)/(b*d*x+a*d)-2*d^2/a^2*(d*x
+c)^(1/2)/(b*d*x+a*d)*b*c+d/a^3*(d*x+c)^(1/2)/(b*d*x+a*d)*b^2*c^2+d^3/a/((a*d-b*
c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))-8*d^2/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+13*d/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^2-6/a^4/((a*d-b*c)*b)^(1/2)
*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*b^3*c^3

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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^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.434999, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/2)/((b*x + a)^2*x^3),x, algorithm="fricas")

[Out]

[1/8*(4*((6*b^3*c^2 - 7*a*b^2*c*d + a^2*b*d^2)*x^3 + (6*a*b^2*c^2 - 7*a^2*b*c*d
+ a^3*d^2)*x^2)*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)) + ((24*b^3*c^2 - 40*a*b^2*c*d + 15*a^2*b*d^2)*x
^3 + (24*a*b^2*c^2 - 40*a^2*b*c*d + 15*a^3*d^2)*x^2)*sqrt(c)*log((d*x - 2*sqrt(d
*x + c)*sqrt(c) + 2*c)/x) - 2*(2*a^3*c^2 - (12*a*b^2*c^2 - 17*a^2*b*c*d + 4*a^3*
d^2)*x^2 - 3*(2*a^2*b*c^2 - 3*a^3*c*d)*x)*sqrt(d*x + c))/(a^4*b*x^3 + a^5*x^2),
1/8*(8*((6*b^3*c^2 - 7*a*b^2*c*d + a^2*b*d^2)*x^3 + (6*a*b^2*c^2 - 7*a^2*b*c*d +
a^3*d^2)*x^2)*sqrt(-(b*c - a*d)/b)*arctan(sqrt(d*x + c)/sqrt(-(b*c - a*d)/b)) +
((24*b^3*c^2 - 40*a*b^2*c*d + 15*a^2*b*d^2)*x^3 + (24*a*b^2*c^2 - 40*a^2*b*c*d
+ 15*a^3*d^2)*x^2)*sqrt(c)*log((d*x - 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) - 2*(2*a
^3*c^2 - (12*a*b^2*c^2 - 17*a^2*b*c*d + 4*a^3*d^2)*x^2 - 3*(2*a^2*b*c^2 - 3*a^3*
c*d)*x)*sqrt(d*x + c))/(a^4*b*x^3 + a^5*x^2), -1/4*(((24*b^3*c^2 - 40*a*b^2*c*d
+ 15*a^2*b*d^2)*x^3 + (24*a*b^2*c^2 - 40*a^2*b*c*d + 15*a^3*d^2)*x^2)*sqrt(-c)*a
rctan(sqrt(d*x + c)/sqrt(-c)) - 2*((6*b^3*c^2 - 7*a*b^2*c*d + a^2*b*d^2)*x^3 + (
6*a*b^2*c^2 - 7*a^2*b*c*d + a^3*d^2)*x^2)*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^3*c^2 - (12*a*
b^2*c^2 - 17*a^2*b*c*d + 4*a^3*d^2)*x^2 - 3*(2*a^2*b*c^2 - 3*a^3*c*d)*x)*sqrt(d*
x + c))/(a^4*b*x^3 + a^5*x^2), -1/4*(((24*b^3*c^2 - 40*a*b^2*c*d + 15*a^2*b*d^2)
*x^3 + (24*a*b^2*c^2 - 40*a^2*b*c*d + 15*a^3*d^2)*x^2)*sqrt(-c)*arctan(sqrt(d*x
+ c)/sqrt(-c)) - 4*((6*b^3*c^2 - 7*a*b^2*c*d + a^2*b*d^2)*x^3 + (6*a*b^2*c^2 - 7
*a^2*b*c*d + a^3*d^2)*x^2)*sqrt(-(b*c - a*d)/b)*arctan(sqrt(d*x + c)/sqrt(-(b*c
- a*d)/b)) + (2*a^3*c^2 - (12*a*b^2*c^2 - 17*a^2*b*c*d + 4*a^3*d^2)*x^2 - 3*(2*a
^2*b*c^2 - 3*a^3*c*d)*x)*sqrt(d*x + c))/(a^4*b*x^3 + a^5*x^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((d*x+c)**(5/2)/x**3/(b*x+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.226779, size = 358, normalized size = 1.63 \[ -\frac{{\left (6 \, b^{3} c^{3} - 13 \, a b^{2} c^{2} d + 8 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a^{4}} + \frac{{\left (24 \, b^{2} c^{3} - 40 \, a b c^{2} d + 15 \, a^{2} c d^{2}\right )} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right )}{4 \, a^{4} \sqrt{-c}} + \frac{\sqrt{d x + c} b^{2} c^{2} d - 2 \, \sqrt{d x + c} a b c d^{2} + \sqrt{d x + c} a^{2} d^{3}}{{\left ({\left (d x + c\right )} b - b c + a d\right )} a^{3}} + \frac{8 \,{\left (d x + c\right )}^{\frac{3}{2}} b c^{2} d - 8 \, \sqrt{d x + c} b c^{3} d - 9 \,{\left (d x + c\right )}^{\frac{3}{2}} a c d^{2} + 7 \, \sqrt{d x + c} a c^{2} d^{2}}{4 \, a^{3} d^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/2)/((b*x + a)^2*x^3),x, algorithm="giac")

[Out]

-(6*b^3*c^3 - 13*a*b^2*c^2*d + 8*a^2*b*c*d^2 - a^3*d^3)*arctan(sqrt(d*x + c)*b/s
qrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*a^4) + 1/4*(24*b^2*c^3 - 40*a*b*c^2*d
+ 15*a^2*c*d^2)*arctan(sqrt(d*x + c)/sqrt(-c))/(a^4*sqrt(-c)) + (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^3) + 1/4*(8*(d*x + c)^(3/2)*b*c^2*d - 8*sqrt(d*x + c)*b*c^3*d - 9*(d
*x + c)^(3/2)*a*c*d^2 + 7*sqrt(d*x + c)*a*c^2*d^2)/(a^3*d^2*x^2)