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

Optimal. Leaf size=128 \[ -\frac{2 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^2 b^{3/2}}+\frac{c^{3/2} (2 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^2}+\frac{d \sqrt{c+d x} (2 a d+b c)}{a b}-\frac{c (c+d x)^{3/2}}{a x} \]

[Out]

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

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Rubi [A]  time = 0.546668, antiderivative size = 128, 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 \[ -\frac{2 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^2 b^{3/2}}+\frac{c^{3/2} (2 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^2}+\frac{d \sqrt{c+d x} (2 a d+b c)}{a b}-\frac{c (c+d x)^{3/2}}{a x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 52.2457, size = 114, normalized size = 0.89 \[ - \frac{c \left (c + d x\right )^{\frac{3}{2}}}{a x} + \frac{d \sqrt{c + d x} \left (2 a d + b c\right )}{a b} - \frac{c^{\frac{3}{2}} \left (5 a d - 2 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{a^{2}} - \frac{2 \left (a d - b c\right )^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{a^{2} b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.142932, size = 115, normalized size = 0.9 \[ -\frac{2 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^2 b^{3/2}}+\frac{c^{3/2} (2 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^2}+\sqrt{c+d x} \left (\frac{2 d^2}{b}-\frac{c^2}{a x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.027, size = 249, normalized size = 2. \[ 2\,{\frac{{d}^{2}\sqrt{dx+c}}{b}}-{\frac{{c}^{2}}{ax}\sqrt{dx+c}}-5\,{\frac{d{c}^{3/2}}{a}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }+2\,{\frac{{c}^{5/2}b}{{a}^{2}}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }-2\,{\frac{{d}^{3}a}{b\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+6\,{\frac{{d}^{2}c}{\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-6\,{\frac{bd{c}^{2}}{a\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+2\,{\frac{{b}^{2}{c}^{3}}{{a}^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*d^2/b*(d*x+c)^(1/2)-c^2/a*(d*x+c)^(1/2)/x-5*d*c^(3/2)/a*arctanh((d*x+c)^(1/2)/
c^(1/2))+2*c^(5/2)/a^2*arctanh((d*x+c)^(1/2)/c^(1/2))*b-2*d^3/b*a/((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*d-b*c)*b)^(1/2)*arct
an((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*c-6*d*b/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^2/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

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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^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.397042, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x + 2 \, b c - a d - 2 \, \sqrt{d x + c} b \sqrt{\frac{b c - a d}{b}}}{b x + a}\right ) -{\left (2 \, b^{2} c^{2} - 5 \, a b c d\right )} \sqrt{c} x \log \left (\frac{d x - 2 \, \sqrt{d x + c} \sqrt{c} + 2 \, c}{x}\right ) + 2 \,{\left (2 \, a^{2} d^{2} x - a b c^{2}\right )} \sqrt{d x + c}}{2 \, a^{2} b x}, -\frac{4 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-\frac{b c - a d}{b}}}\right ) +{\left (2 \, b^{2} c^{2} - 5 \, a b c d\right )} \sqrt{c} x \log \left (\frac{d x - 2 \, \sqrt{d x + c} \sqrt{c} + 2 \, c}{x}\right ) - 2 \,{\left (2 \, a^{2} d^{2} x - a b c^{2}\right )} \sqrt{d x + c}}{2 \, a^{2} b x}, \frac{{\left (2 \, b^{2} c^{2} - 5 \, a b c d\right )} \sqrt{-c} x \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right ) +{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x + 2 \, b c - a d - 2 \, \sqrt{d x + c} b \sqrt{\frac{b c - a d}{b}}}{b x + a}\right ) +{\left (2 \, a^{2} d^{2} x - a b c^{2}\right )} \sqrt{d x + c}}{a^{2} b x}, \frac{{\left (2 \, b^{2} c^{2} - 5 \, a b c d\right )} \sqrt{-c} x \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right ) - 2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-\frac{b c - a d}{b}}}\right ) +{\left (2 \, a^{2} d^{2} x - a b c^{2}\right )} \sqrt{d x + c}}{a^{2} b x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(2*(b^2*c^2 - 2*a*b*c*d + a^2*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^2*c^2 - 5*a*b*
c*d)*sqrt(c)*x*log((d*x - 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) + 2*(2*a^2*d^2*x - a
*b*c^2)*sqrt(d*x + c))/(a^2*b*x), -1/2*(4*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x*sqrt
(-(b*c - a*d)/b)*arctan(sqrt(d*x + c)/sqrt(-(b*c - a*d)/b)) + (2*b^2*c^2 - 5*a*b
*c*d)*sqrt(c)*x*log((d*x - 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) - 2*(2*a^2*d^2*x -
a*b*c^2)*sqrt(d*x + c))/(a^2*b*x), ((2*b^2*c^2 - 5*a*b*c*d)*sqrt(-c)*x*arctan(sq
rt(d*x + c)/sqrt(-c)) + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x*sqrt((b*c - a*d)/b)*lo
g((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*d^2*x - a*b*c^2)*sqrt(d*x + c))/(a^2*b*x), ((2*b^2*c^2 - 5*a*b*c*d)*sqrt(-c)
*x*arctan(sqrt(d*x + c)/sqrt(-c)) - 2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x*sqrt(-(b
*c - a*d)/b)*arctan(sqrt(d*x + c)/sqrt(-(b*c - a*d)/b)) + (2*a^2*d^2*x - a*b*c^2
)*sqrt(d*x + c))/(a^2*b*x)]

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Sympy [A]  time = 113.063, size = 860, normalized size = 6.72 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a*d**3*Piecewise((atan(sqrt(c + d*x)/sqrt(a*d/b - c))/(b*sqrt(a*d/b - c)), a*
d/b - c > 0), (-acoth(sqrt(c + d*x)/sqrt(-a*d/b + c))/(b*sqrt(-a*d/b + c)), (a*d
/b - c < 0) & (c + d*x > -a*d/b + c)), (-atanh(sqrt(c + d*x)/sqrt(-a*d/b + c))/(
b*sqrt(-a*d/b + c)), (a*d/b - c < 0) & (c + d*x < -a*d/b + c)))/b + 6*c*d**2*Pie
cewise((atan(sqrt(c + d*x)/sqrt(a*d/b - c))/(b*sqrt(a*d/b - c)), a*d/b - c > 0),
(-acoth(sqrt(c + d*x)/sqrt(-a*d/b + c))/(b*sqrt(-a*d/b + c)), (a*d/b - c < 0) &
(c + d*x > -a*d/b + c)), (-atanh(sqrt(c + d*x)/sqrt(-a*d/b + c))/(b*sqrt(-a*d/b
+ c)), (a*d/b - c < 0) & (c + d*x < -a*d/b + c))) + 2*d**2*sqrt(c + d*x)/b - 6*
b*c**2*d*Piecewise((atan(sqrt(c + d*x)/sqrt(a*d/b - c))/(b*sqrt(a*d/b - c)), a*d
/b - c > 0), (-acoth(sqrt(c + d*x)/sqrt(-a*d/b + c))/(b*sqrt(-a*d/b + c)), (a*d/
b - c < 0) & (c + d*x > -a*d/b + c)), (-atanh(sqrt(c + d*x)/sqrt(-a*d/b + c))/(b
*sqrt(-a*d/b + c)), (a*d/b - c < 0) & (c + d*x < -a*d/b + c)))/a - c**3*d*sqrt(c
**(-3))*log(-c**2*sqrt(c**(-3)) + sqrt(c + d*x))/(2*a) + c**3*d*sqrt(c**(-3))*lo
g(c**2*sqrt(c**(-3)) + sqrt(c + d*x))/(2*a) - 6*c**2*d*Piecewise((-atan(sqrt(c +
d*x)/sqrt(-c))/sqrt(-c), -c > 0), (acoth(sqrt(c + d*x)/sqrt(c))/sqrt(c), (-c <
0) & (c < c + d*x)), (atanh(sqrt(c + d*x)/sqrt(c))/sqrt(c), (-c < 0) & (c > c +
d*x)))/a - c**2*sqrt(c + d*x)/(a*x) + 2*b**2*c**3*Piecewise((atan(sqrt(c + d*x)/
sqrt(a*d/b - c))/(b*sqrt(a*d/b - c)), a*d/b - c > 0), (-acoth(sqrt(c + d*x)/sqrt
(-a*d/b + c))/(b*sqrt(-a*d/b + c)), (a*d/b - c < 0) & (c + d*x > -a*d/b + c)), (
-atanh(sqrt(c + d*x)/sqrt(-a*d/b + c))/(b*sqrt(-a*d/b + c)), (a*d/b - c < 0) & (
c + d*x < -a*d/b + c)))/a**2 + 2*b*c**3*Piecewise((-atan(sqrt(c + d*x)/sqrt(-c))
/sqrt(-c), -c > 0), (acoth(sqrt(c + d*x)/sqrt(c))/sqrt(c), (-c < 0) & (c < c + d
*x)), (atanh(sqrt(c + d*x)/sqrt(c))/sqrt(c), (-c < 0) & (c > c + d*x)))/a**2

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GIAC/XCAS [A]  time = 0.217624, size = 205, normalized size = 1.6 \[ \frac{2 \, \sqrt{d x + c} d^{2}}{b} - \frac{\sqrt{d x + c} c^{2}}{a x} - \frac{{\left (2 \, b c^{3} - 5 \, a c^{2} d\right )} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right )}{a^{2} \sqrt{-c}} + \frac{2 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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