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

Optimal. Leaf size=151 \[ \frac{2 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^3 \sqrt{b}}+\frac{c \sqrt{c+d x} (4 b c-7 a d)}{4 a^2 x}-\frac{\sqrt{c} \left (15 a^2 d^2-20 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{4 a^3}-\frac{c (c+d x)^{3/2}}{2 a x^2} \]

[Out]

(c*(4*b*c - 7*a*d)*Sqrt[c + d*x])/(4*a^2*x) - (c*(c + d*x)^(3/2))/(2*a*x^2) - (S
qrt[c]*(8*b^2*c^2 - 20*a*b*c*d + 15*a^2*d^2)*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/(4*
a^3) + (2*(b*c - a*d)^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(a
^3*Sqrt[b])

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Rubi [A]  time = 0.474951, antiderivative size = 151, 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 \[ \frac{2 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^3 \sqrt{b}}+\frac{c \sqrt{c+d x} (4 b c-7 a d)}{4 a^2 x}-\frac{\sqrt{c} \left (15 a^2 d^2-20 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{4 a^3}-\frac{c (c+d x)^{3/2}}{2 a x^2} \]

Antiderivative was successfully verified.

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

[Out]

(c*(4*b*c - 7*a*d)*Sqrt[c + d*x])/(4*a^2*x) - (c*(c + d*x)^(3/2))/(2*a*x^2) - (S
qrt[c]*(8*b^2*c^2 - 20*a*b*c*d + 15*a^2*d^2)*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/(4*
a^3) + (2*(b*c - a*d)^(5/2)*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 = 51.2073, size = 139, normalized size = 0.92 \[ - \frac{c \left (c + d x\right )^{\frac{3}{2}}}{2 a x^{2}} - \frac{c \sqrt{c + d x} \left (7 a d - 4 b c\right )}{4 a^{2} x} - \frac{\sqrt{c} \left (15 a^{2} d^{2} - 20 a b c d + 8 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{4 a^{3}} + \frac{2 \left (a d - b c\right )^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{a^{3} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-c*(c + d*x)**(3/2)/(2*a*x**2) - c*sqrt(c + d*x)*(7*a*d - 4*b*c)/(4*a**2*x) - sq
rt(c)*(15*a**2*d**2 - 20*a*b*c*d + 8*b**2*c**2)*atanh(sqrt(c + d*x)/sqrt(c))/(4*
a**3) + 2*(a*d - b*c)**(5/2)*atan(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c))/(a**3*s
qrt(b))

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Mathematica [A]  time = 0.184538, size = 131, normalized size = 0.87 \[ \frac{-\sqrt{c} \left (15 a^2 d^2-20 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )+\frac{a c \sqrt{c+d x} (-2 a c-9 a d x+4 b c x)}{x^2}+\frac{8 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{\sqrt{b}}}{4 a^3} \]

Antiderivative was successfully verified.

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

[Out]

((a*c*Sqrt[c + d*x]*(-2*a*c + 4*b*c*x - 9*a*d*x))/x^2 - Sqrt[c]*(8*b^2*c^2 - 20*
a*b*c*d + 15*a^2*d^2)*ArcTanh[Sqrt[c + d*x]/Sqrt[c]] + (8*(b*c - a*d)^(5/2)*ArcT
anh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/Sqrt[b])/(4*a^3)

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Maple [B]  time = 0.023, size = 321, normalized size = 2.1 \[ -{\frac{9\,c}{4\,a{x}^{2}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}+{\frac{{c}^{2}b}{d{a}^{2}{x}^{2}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}+{\frac{7\,{c}^{2}}{4\,a{x}^{2}}\sqrt{dx+c}}-{\frac{{c}^{3}b}{d{a}^{2}{x}^{2}}\sqrt{dx+c}}-{\frac{15\,{d}^{2}}{4\,a}\sqrt{c}{\it Artanh} \left ({1\sqrt{dx+c}{\frac{1}{\sqrt{c}}}} \right ) }+5\,{\frac{d{c}^{3/2}b}{{a}^{2}}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }-2\,{\frac{{c}^{5/2}{b}^{2}}{{a}^{3}}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }+2\,{\frac{{d}^{3}}{\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-6\,{\frac{{d}^{2}cb}{a\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+6\,{\frac{{b}^{2}d{c}^{2}}{{a}^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-2\,{\frac{{b}^{3}{c}^{3}}{{a}^{3}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.403419, size = 1, normalized size = 0.01 \[ \left [\frac{8 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x + 2 \, b c - a d + 2 \, \sqrt{d x + c} b \sqrt{\frac{b c - a d}{b}}}{b x + a}\right ) +{\left (8 \, b^{2} c^{2} - 20 \, a b c d + 15 \, a^{2} d^{2}\right )} \sqrt{c} x^{2} \log \left (\frac{d x - 2 \, \sqrt{d x + c} \sqrt{c} + 2 \, c}{x}\right ) - 2 \,{\left (2 \, a^{2} c^{2} -{\left (4 \, a b c^{2} - 9 \, a^{2} c d\right )} x\right )} \sqrt{d x + c}}{8 \, a^{3} x^{2}}, \frac{16 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2} \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-\frac{b c - a d}{b}}}\right ) +{\left (8 \, b^{2} c^{2} - 20 \, a b c d + 15 \, a^{2} d^{2}\right )} \sqrt{c} x^{2} \log \left (\frac{d x - 2 \, \sqrt{d x + c} \sqrt{c} + 2 \, c}{x}\right ) - 2 \,{\left (2 \, a^{2} c^{2} -{\left (4 \, a b c^{2} - 9 \, a^{2} c d\right )} x\right )} \sqrt{d x + c}}{8 \, a^{3} x^{2}}, -\frac{{\left (8 \, b^{2} c^{2} - 20 \, a b c d + 15 \, a^{2} d^{2}\right )} \sqrt{-c} x^{2} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right ) - 4 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x + 2 \, b c - a d + 2 \, \sqrt{d x + c} b \sqrt{\frac{b c - a d}{b}}}{b x + a}\right ) +{\left (2 \, a^{2} c^{2} -{\left (4 \, a b c^{2} - 9 \, a^{2} c d\right )} x\right )} \sqrt{d x + c}}{4 \, a^{3} x^{2}}, -\frac{{\left (8 \, b^{2} c^{2} - 20 \, a b c d + 15 \, a^{2} d^{2}\right )} \sqrt{-c} x^{2} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right ) - 8 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2} \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-\frac{b c - a d}{b}}}\right ) +{\left (2 \, a^{2} c^{2} -{\left (4 \, a b c^{2} - 9 \, a^{2} c d\right )} x\right )} \sqrt{d x + c}}{4 \, a^{3} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(8*(b^2*c^2 - 2*a*b*c*d + a^2*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)) + (8*b^2*c^2 - 20*a
*b*c*d + 15*a^2*d^2)*sqrt(c)*x^2*log((d*x - 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) -
2*(2*a^2*c^2 - (4*a*b*c^2 - 9*a^2*c*d)*x)*sqrt(d*x + c))/(a^3*x^2), 1/8*(16*(b^2
*c^2 - 2*a*b*c*d + a^2*d^2)*x^2*sqrt(-(b*c - a*d)/b)*arctan(sqrt(d*x + c)/sqrt(-
(b*c - a*d)/b)) + (8*b^2*c^2 - 20*a*b*c*d + 15*a^2*d^2)*sqrt(c)*x^2*log((d*x - 2
*sqrt(d*x + c)*sqrt(c) + 2*c)/x) - 2*(2*a^2*c^2 - (4*a*b*c^2 - 9*a^2*c*d)*x)*sqr
t(d*x + c))/(a^3*x^2), -1/4*((8*b^2*c^2 - 20*a*b*c*d + 15*a^2*d^2)*sqrt(-c)*x^2*
arctan(sqrt(d*x + c)/sqrt(-c)) - 4*(b^2*c^2 - 2*a*b*c*d + a^2*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^2*c^2 - (4*a*b*c^2 - 9*a^2*c*d)*x)*sqrt(d*x + c))/(a^3*x^2), -1/4
*((8*b^2*c^2 - 20*a*b*c*d + 15*a^2*d^2)*sqrt(-c)*x^2*arctan(sqrt(d*x + c)/sqrt(-
c)) - 8*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x^2*sqrt(-(b*c - a*d)/b)*arctan(sqrt(d*x
+ c)/sqrt(-(b*c - a*d)/b)) + (2*a^2*c^2 - (4*a*b*c^2 - 9*a^2*c*d)*x)*sqrt(d*x +
c))/(a^3*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),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.223255, size = 267, normalized size = 1.77 \[ -\frac{2 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a^{3}} + \frac{{\left (8 \, b^{2} c^{3} - 20 \, a b c^{2} d + 15 \, a^{2} c d^{2}\right )} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right )}{4 \, a^{3} \sqrt{-c}} + \frac{4 \,{\left (d x + c\right )}^{\frac{3}{2}} b c^{2} d - 4 \, \sqrt{d x + c} b c^{3} d - 9 \,{\left (d x + c\right )}^{\frac{3}{2}} a c d^{2} + 7 \, \sqrt{d x + c} a c^{2} d^{2}}{4 \, a^{2} d^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*arctan(sqrt(d*x + c)*b/sq
rt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*a^3) + 1/4*(8*b^2*c^3 - 20*a*b*c^2*d +
15*a^2*c*d^2)*arctan(sqrt(d*x + c)/sqrt(-c))/(a^3*sqrt(-c)) + 1/4*(4*(d*x + c)^
(3/2)*b*c^2*d - 4*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^2*d^2*x^2)