3.463 \(\int \frac{(c+d x)^{5/2}}{x^4 (a+b x)^2} \, dx\)

Optimal. Leaf size=284 \[ -\frac{\sqrt{b} (8 b c-3 a d) (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^5}+\frac{c \sqrt{c+d x} (8 b c-9 a d)}{12 a^2 x^2 (a+b x)}-\frac{b \sqrt{c+d x} \left (19 a^2 d^2-52 a b c d+32 b^2 c^2\right )}{8 a^4 (a+b x)}-\frac{\sqrt{c+d x} \left (33 a^2 d^2-82 a b c d+48 b^2 c^2\right )}{24 a^3 x (a+b x)}+\frac{\left (-5 a^3 d^3+60 a^2 b c d^2-120 a b^2 c^2 d+64 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{8 a^5 \sqrt{c}}-\frac{c (c+d x)^{3/2}}{3 a x^3 (a+b x)} \]

[Out]

-(b*(32*b^2*c^2 - 52*a*b*c*d + 19*a^2*d^2)*Sqrt[c + d*x])/(8*a^4*(a + b*x)) + (c
*(8*b*c - 9*a*d)*Sqrt[c + d*x])/(12*a^2*x^2*(a + b*x)) - ((48*b^2*c^2 - 82*a*b*c
*d + 33*a^2*d^2)*Sqrt[c + d*x])/(24*a^3*x*(a + b*x)) - (c*(c + d*x)^(3/2))/(3*a*
x^3*(a + b*x)) + ((64*b^3*c^3 - 120*a*b^2*c^2*d + 60*a^2*b*c*d^2 - 5*a^3*d^3)*Ar
cTanh[Sqrt[c + d*x]/Sqrt[c]])/(8*a^5*Sqrt[c]) - (Sqrt[b]*(8*b*c - 3*a*d)*(b*c -
a*d)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/a^5

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Rubi [A]  time = 1.14759, antiderivative size = 284, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ -\frac{\sqrt{b} (8 b c-3 a d) (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^5}+\frac{c \sqrt{c+d x} (8 b c-9 a d)}{12 a^2 x^2 (a+b x)}-\frac{b \sqrt{c+d x} \left (19 a^2 d^2-52 a b c d+32 b^2 c^2\right )}{8 a^4 (a+b x)}-\frac{\sqrt{c+d x} \left (33 a^2 d^2-82 a b c d+48 b^2 c^2\right )}{24 a^3 x (a+b x)}+\frac{\left (-5 a^3 d^3+60 a^2 b c d^2-120 a b^2 c^2 d+64 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{8 a^5 \sqrt{c}}-\frac{c (c+d x)^{3/2}}{3 a x^3 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(5/2)/(x^4*(a + b*x)^2),x]

[Out]

-(b*(32*b^2*c^2 - 52*a*b*c*d + 19*a^2*d^2)*Sqrt[c + d*x])/(8*a^4*(a + b*x)) + (c
*(8*b*c - 9*a*d)*Sqrt[c + d*x])/(12*a^2*x^2*(a + b*x)) - ((48*b^2*c^2 - 82*a*b*c
*d + 33*a^2*d^2)*Sqrt[c + d*x])/(24*a^3*x*(a + b*x)) - (c*(c + d*x)^(3/2))/(3*a*
x^3*(a + b*x)) + ((64*b^3*c^3 - 120*a*b^2*c^2*d + 60*a^2*b*c*d^2 - 5*a^3*d^3)*Ar
cTanh[Sqrt[c + d*x]/Sqrt[c]])/(8*a^5*Sqrt[c]) - (Sqrt[b]*(8*b*c - 3*a*d)*(b*c -
a*d)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/a^5

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Rubi in Sympy [A]  time = 131.717, size = 267, normalized size = 0.94 \[ - \frac{c \left (c + d x\right )^{\frac{3}{2}}}{3 a x^{3} \left (a + b x\right )} - \frac{\sqrt{c + d x} \left (a d - b c\right ) \left (3 a d - 4 b c\right )}{3 a^{2} b x^{2} \left (a + b x\right )} + \frac{\sqrt{c + d x} \left (12 a^{2} d^{2} - 37 a b c d + 24 b^{2} c^{2}\right )}{12 a^{3} b x^{2}} - \frac{\sqrt{c + d x} \left (19 a^{2} d^{2} - 52 a b c d + 32 b^{2} c^{2}\right )}{8 a^{4} x} - \frac{\sqrt{b} \left (a d - b c\right )^{\frac{3}{2}} \left (3 a d - 8 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{a^{5}} - \frac{\left (5 a^{3} d^{3} - 60 a^{2} b c d^{2} + 120 a b^{2} c^{2} d - 64 b^{3} c^{3}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{8 a^{5} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(5/2)/x**4/(b*x+a)**2,x)

[Out]

-c*(c + d*x)**(3/2)/(3*a*x**3*(a + b*x)) - sqrt(c + d*x)*(a*d - b*c)*(3*a*d - 4*
b*c)/(3*a**2*b*x**2*(a + b*x)) + sqrt(c + d*x)*(12*a**2*d**2 - 37*a*b*c*d + 24*b
**2*c**2)/(12*a**3*b*x**2) - sqrt(c + d*x)*(19*a**2*d**2 - 52*a*b*c*d + 32*b**2*
c**2)/(8*a**4*x) - sqrt(b)*(a*d - b*c)**(3/2)*(3*a*d - 8*b*c)*atan(sqrt(b)*sqrt(
c + d*x)/sqrt(a*d - b*c))/a**5 - (5*a**3*d**3 - 60*a**2*b*c*d**2 + 120*a*b**2*c*
*2*d - 64*b**3*c**3)*atanh(sqrt(c + d*x)/sqrt(c))/(8*a**5*sqrt(c))

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Mathematica [A]  time = 0.483373, size = 223, normalized size = 0.79 \[ -\frac{\frac{a \sqrt{c+d x} \left (a^3 \left (8 c^2+26 c d x+33 d^2 x^2\right )+a^2 b x \left (-16 c^2-82 c d x+57 d^2 x^2\right )+12 a b^2 c x^2 (4 c-13 d x)+96 b^3 c^2 x^3\right )}{x^3 (a+b x)}-\frac{3 \left (-5 a^3 d^3+60 a^2 b c d^2-120 a b^2 c^2 d+64 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{\sqrt{c}}+24 \sqrt{b} (8 b c-3 a d) (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{24 a^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(5/2)/(x^4*(a + b*x)^2),x]

[Out]

-((a*Sqrt[c + d*x]*(96*b^3*c^2*x^3 + 12*a*b^2*c*x^2*(4*c - 13*d*x) + a^3*(8*c^2
+ 26*c*d*x + 33*d^2*x^2) + a^2*b*x*(-16*c^2 - 82*c*d*x + 57*d^2*x^2)))/(x^3*(a +
b*x)) - (3*(64*b^3*c^3 - 120*a*b^2*c^2*d + 60*a^2*b*c*d^2 - 5*a^3*d^3)*ArcTanh[
Sqrt[c + d*x]/Sqrt[c]])/Sqrt[c] + 24*Sqrt[b]*(8*b*c - 3*a*d)*(b*c - a*d)^(3/2)*A
rcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(24*a^5)

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Maple [B]  time = 0.029, size = 545, normalized size = 1.9 \[ -{\frac{11}{8\,{a}^{2}{x}^{3}} \left ( dx+c \right ) ^{{\frac{5}{2}}}}+{\frac{9\,bc}{2\,{a}^{3}d{x}^{3}} \left ( dx+c \right ) ^{{\frac{5}{2}}}}-3\,{\frac{ \left ( dx+c \right ) ^{5/2}{b}^{2}{c}^{2}}{{d}^{2}{a}^{4}{x}^{3}}}+{\frac{5\,c}{3\,{a}^{2}{x}^{3}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}-8\,{\frac{ \left ( dx+c \right ) ^{3/2}b{c}^{2}}{{a}^{3}d{x}^{3}}}+6\,{\frac{ \left ( dx+c \right ) ^{3/2}{b}^{2}{c}^{3}}{{d}^{2}{a}^{4}{x}^{3}}}+{\frac{7\,{c}^{3}b}{2\,{a}^{3}d{x}^{3}}\sqrt{dx+c}}-3\,{\frac{{b}^{2}\sqrt{dx+c}{c}^{4}}{{d}^{2}{a}^{4}{x}^{3}}}-{\frac{5\,{c}^{2}}{8\,{a}^{2}{x}^{3}}\sqrt{dx+c}}-{\frac{5\,{d}^{3}}{8\,{a}^{2}}{\it Artanh} \left ({1\sqrt{dx+c}{\frac{1}{\sqrt{c}}}} \right ){\frac{1}{\sqrt{c}}}}+{\frac{15\,{d}^{2}b}{2\,{a}^{3}}\sqrt{c}{\it Artanh} \left ({1\sqrt{dx+c}{\frac{1}{\sqrt{c}}}} \right ) }-15\,{\frac{d{c}^{3/2}{b}^{2}}{{a}^{4}}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }+8\,{\frac{{c}^{5/2}{b}^{3}}{{a}^{5}}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }-{\frac{{d}^{3}b}{{a}^{2} \left ( bdx+ad \right ) }\sqrt{dx+c}}+2\,{\frac{{d}^{2}{b}^{2}\sqrt{dx+c}c}{{a}^{3} \left ( bdx+ad \right ) }}-{\frac{d{b}^{3}{c}^{2}}{{a}^{4} \left ( bdx+ad \right ) }\sqrt{dx+c}}-3\,{\frac{{d}^{3}b}{{a}^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+14\,{\frac{{d}^{2}{b}^{2}c}{{a}^{3}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-19\,{\frac{d{b}^{3}{c}^{2}}{{a}^{4}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+8\,{\frac{{b}^{4}{c}^{3}}{{a}^{5}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(5/2)/x^4/(b*x+a)^2,x)

[Out]

-11/8/a^2/x^3*(d*x+c)^(5/2)+9/2/d/a^3/x^3*(d*x+c)^(5/2)*c*b-3/d^2/a^4/x^3*(d*x+c
)^(5/2)*b^2*c^2+5/3/a^2/x^3*(d*x+c)^(3/2)*c-8/d/a^3/x^3*(d*x+c)^(3/2)*b*c^2+6/d^
2/a^4/x^3*(d*x+c)^(3/2)*b^2*c^3+7/2/d/a^3/x^3*(d*x+c)^(1/2)*b*c^3-3/d^2/a^4/x^3*
(d*x+c)^(1/2)*b^2*c^4-5/8/a^2/x^3*(d*x+c)^(1/2)*c^2-5/8*d^3/a^2/c^(1/2)*arctanh(
(d*x+c)^(1/2)/c^(1/2))+15/2*d^2/a^3*c^(1/2)*arctanh((d*x+c)^(1/2)/c^(1/2))*b-15*
d/a^4*c^(3/2)*arctanh((d*x+c)^(1/2)/c^(1/2))*b^2+8/a^5*c^(5/2)*arctanh((d*x+c)^(
1/2)/c^(1/2))*b^3-d^3*b/a^2*(d*x+c)^(1/2)/(b*d*x+a*d)+2*d^2*b^2/a^3*(d*x+c)^(1/2
)/(b*d*x+a*d)*c-d*b^3/a^4*(d*x+c)^(1/2)/(b*d*x+a*d)*c^2-3*d^3*b/a^2/((a*d-b*c)*b
)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))+14*d^2*b^2/a^3/((a*d-b*c)*b)
^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*c-19*d*b^3/a^4/((a*d-b*c)*b)^
(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*c^2+8*b^4/a^5/((a*d-b*c)*b)^(1
/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*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^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.565234, 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^4),x, algorithm="fricas")

[Out]

[1/48*(24*((8*b^3*c^2 - 11*a*b^2*c*d + 3*a^2*b*d^2)*x^4 + (8*a*b^2*c^2 - 11*a^2*
b*c*d + 3*a^3*d^2)*x^3)*sqrt(b^2*c - a*b*d)*sqrt(c)*log((b*d*x + 2*b*c - a*d - 2
*sqrt(b^2*c - a*b*d)*sqrt(d*x + c))/(b*x + a)) - 2*(8*a^4*c^2 + 3*(32*a*b^3*c^2
- 52*a^2*b^2*c*d + 19*a^3*b*d^2)*x^3 + (48*a^2*b^2*c^2 - 82*a^3*b*c*d + 33*a^4*d
^2)*x^2 - 2*(8*a^3*b*c^2 - 13*a^4*c*d)*x)*sqrt(d*x + c)*sqrt(c) - 3*((64*b^4*c^3
- 120*a*b^3*c^2*d + 60*a^2*b^2*c*d^2 - 5*a^3*b*d^3)*x^4 + (64*a*b^3*c^3 - 120*a
^2*b^2*c^2*d + 60*a^3*b*c*d^2 - 5*a^4*d^3)*x^3)*log(((d*x + 2*c)*sqrt(c) - 2*sqr
t(d*x + c)*c)/x))/((a^5*b*x^4 + a^6*x^3)*sqrt(c)), 1/48*(48*((8*b^3*c^2 - 11*a*b
^2*c*d + 3*a^2*b*d^2)*x^4 + (8*a*b^2*c^2 - 11*a^2*b*c*d + 3*a^3*d^2)*x^3)*sqrt(-
b^2*c + a*b*d)*sqrt(c)*arctan(sqrt(-b^2*c + a*b*d)/(sqrt(d*x + c)*b)) - 2*(8*a^4
*c^2 + 3*(32*a*b^3*c^2 - 52*a^2*b^2*c*d + 19*a^3*b*d^2)*x^3 + (48*a^2*b^2*c^2 -
82*a^3*b*c*d + 33*a^4*d^2)*x^2 - 2*(8*a^3*b*c^2 - 13*a^4*c*d)*x)*sqrt(d*x + c)*s
qrt(c) - 3*((64*b^4*c^3 - 120*a*b^3*c^2*d + 60*a^2*b^2*c*d^2 - 5*a^3*b*d^3)*x^4
+ (64*a*b^3*c^3 - 120*a^2*b^2*c^2*d + 60*a^3*b*c*d^2 - 5*a^4*d^3)*x^3)*log(((d*x
+ 2*c)*sqrt(c) - 2*sqrt(d*x + c)*c)/x))/((a^5*b*x^4 + a^6*x^3)*sqrt(c)), 1/24*(
12*((8*b^3*c^2 - 11*a*b^2*c*d + 3*a^2*b*d^2)*x^4 + (8*a*b^2*c^2 - 11*a^2*b*c*d +
3*a^3*d^2)*x^3)*sqrt(b^2*c - a*b*d)*sqrt(-c)*log((b*d*x + 2*b*c - a*d - 2*sqrt(
b^2*c - a*b*d)*sqrt(d*x + c))/(b*x + a)) - (8*a^4*c^2 + 3*(32*a*b^3*c^2 - 52*a^2
*b^2*c*d + 19*a^3*b*d^2)*x^3 + (48*a^2*b^2*c^2 - 82*a^3*b*c*d + 33*a^4*d^2)*x^2
- 2*(8*a^3*b*c^2 - 13*a^4*c*d)*x)*sqrt(d*x + c)*sqrt(-c) - 3*((64*b^4*c^3 - 120*
a*b^3*c^2*d + 60*a^2*b^2*c*d^2 - 5*a^3*b*d^3)*x^4 + (64*a*b^3*c^3 - 120*a^2*b^2*
c^2*d + 60*a^3*b*c*d^2 - 5*a^4*d^3)*x^3)*arctan(c/(sqrt(d*x + c)*sqrt(-c))))/((a
^5*b*x^4 + a^6*x^3)*sqrt(-c)), 1/24*(24*((8*b^3*c^2 - 11*a*b^2*c*d + 3*a^2*b*d^2
)*x^4 + (8*a*b^2*c^2 - 11*a^2*b*c*d + 3*a^3*d^2)*x^3)*sqrt(-b^2*c + a*b*d)*sqrt(
-c)*arctan(sqrt(-b^2*c + a*b*d)/(sqrt(d*x + c)*b)) - (8*a^4*c^2 + 3*(32*a*b^3*c^
2 - 52*a^2*b^2*c*d + 19*a^3*b*d^2)*x^3 + (48*a^2*b^2*c^2 - 82*a^3*b*c*d + 33*a^4
*d^2)*x^2 - 2*(8*a^3*b*c^2 - 13*a^4*c*d)*x)*sqrt(d*x + c)*sqrt(-c) - 3*((64*b^4*
c^3 - 120*a*b^3*c^2*d + 60*a^2*b^2*c*d^2 - 5*a^3*b*d^3)*x^4 + (64*a*b^3*c^3 - 12
0*a^2*b^2*c^2*d + 60*a^3*b*c*d^2 - 5*a^4*d^3)*x^3)*arctan(c/(sqrt(d*x + c)*sqrt(
-c))))/((a^5*b*x^4 + a^6*x^3)*sqrt(-c))]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.231037, size = 500, normalized size = 1.76 \[ \frac{{\left (8 \, b^{4} c^{3} - 19 \, a b^{3} c^{2} d + 14 \, a^{2} b^{2} c d^{2} - 3 \, a^{3} b d^{3}\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a^{5}} - \frac{{\left (64 \, b^{3} c^{3} - 120 \, a b^{2} c^{2} d + 60 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right )}{8 \, a^{5} \sqrt{-c}} - \frac{\sqrt{d x + c} b^{3} c^{2} d - 2 \, \sqrt{d x + c} a b^{2} c d^{2} + \sqrt{d x + c} a^{2} b d^{3}}{{\left ({\left (d x + c\right )} b - b c + a d\right )} a^{4}} - \frac{72 \,{\left (d x + c\right )}^{\frac{5}{2}} b^{2} c^{2} d - 144 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{2} c^{3} d + 72 \, \sqrt{d x + c} b^{2} c^{4} d - 108 \,{\left (d x + c\right )}^{\frac{5}{2}} a b c d^{2} + 192 \,{\left (d x + c\right )}^{\frac{3}{2}} a b c^{2} d^{2} - 84 \, \sqrt{d x + c} a b c^{3} d^{2} + 33 \,{\left (d x + c\right )}^{\frac{5}{2}} a^{2} d^{3} - 40 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{2} c d^{3} + 15 \, \sqrt{d x + c} a^{2} c^{2} d^{3}}{24 \, a^{4} d^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/2)/((b*x + a)^2*x^4),x, algorithm="giac")

[Out]

(8*b^4*c^3 - 19*a*b^3*c^2*d + 14*a^2*b^2*c*d^2 - 3*a^3*b*d^3)*arctan(sqrt(d*x +
c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*a^5) - 1/8*(64*b^3*c^3 - 120*a*
b^2*c^2*d + 60*a^2*b*c*d^2 - 5*a^3*d^3)*arctan(sqrt(d*x + c)/sqrt(-c))/(a^5*sqrt
(-c)) - (sqrt(d*x + c)*b^3*c^2*d - 2*sqrt(d*x + c)*a*b^2*c*d^2 + sqrt(d*x + c)*a
^2*b*d^3)/(((d*x + c)*b - b*c + a*d)*a^4) - 1/24*(72*(d*x + c)^(5/2)*b^2*c^2*d -
144*(d*x + c)^(3/2)*b^2*c^3*d + 72*sqrt(d*x + c)*b^2*c^4*d - 108*(d*x + c)^(5/2
)*a*b*c*d^2 + 192*(d*x + c)^(3/2)*a*b*c^2*d^2 - 84*sqrt(d*x + c)*a*b*c^3*d^2 + 3
3*(d*x + c)^(5/2)*a^2*d^3 - 40*(d*x + c)^(3/2)*a^2*c*d^3 + 15*sqrt(d*x + c)*a^2*
c^2*d^3)/(a^4*d^3*x^3)