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

Optimal. Leaf size=118 \[ \frac{2 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a b^{5/2}}+\frac{2 d \sqrt{c+d x} (2 b c-a d)}{b^2}-\frac{2 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a}+\frac{2 d (c+d x)^{3/2}}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.442041, antiderivative size = 118, 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 \[ \frac{2 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a b^{5/2}}+\frac{2 d \sqrt{c+d x} (2 b c-a d)}{b^2}-\frac{2 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a}+\frac{2 d (c+d x)^{3/2}}{3 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 45.3331, size = 105, normalized size = 0.89 \[ \frac{2 d \left (c + d x\right )^{\frac{3}{2}}}{3 b} - \frac{2 d \sqrt{c + d x} \left (a d - 2 b c\right )}{b^{2}} - \frac{2 c^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{a} + \frac{2 \left (a d - b c\right )^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{a b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.281958, size = 107, normalized size = 0.91 \[ \frac{2 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a b^{5/2}}+\frac{2 d \sqrt{c+d x} (-3 a d+7 b c+b d x)}{3 b^2}-\frac{2 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a} \]

Antiderivative was successfully verified.

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

[Out]

(2*d*Sqrt[c + d*x]*(7*b*c - 3*a*d + b*d*x))/(3*b^2) - (2*c^(5/2)*ArcTanh[Sqrt[c
+ d*x]/Sqrt[c]])/a + (2*(b*c - a*d)^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b
*c - a*d]])/(a*b^(5/2))

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Maple [B]  time = 0.018, size = 237, normalized size = 2. \[{\frac{2\,d}{3\,b} \left ( dx+c \right ) ^{{\frac{3}{2}}}}-2\,{\frac{{d}^{2}a\sqrt{dx+c}}{{b}^{2}}}+4\,{\frac{d\sqrt{dx+c}c}{b}}-2\,{\frac{{c}^{5/2}}{a}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }+2\,{\frac{{a}^{2}{d}^{3}}{{b}^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-6\,{\frac{ac{d}^{2}}{b\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+6\,{\frac{d{c}^{2}}{\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-2\,{\frac{b{c}^{3}}{a\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(5/2)/x/(b*x+a),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.401278, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b^{2} c^{\frac{5}{2}} \log \left (\frac{d x - 2 \, \sqrt{d x + c} \sqrt{c} + 2 \, c}{x}\right ) + 3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x + 2 \, b c - a d + 2 \, \sqrt{d x + c} b \sqrt{\frac{b c - a d}{b}}}{b x + a}\right ) + 2 \,{\left (a b d^{2} x + 7 \, a b c d - 3 \, a^{2} d^{2}\right )} \sqrt{d x + c}}{3 \, a b^{2}}, \frac{3 \, b^{2} c^{\frac{5}{2}} \log \left (\frac{d x - 2 \, \sqrt{d x + c} \sqrt{c} + 2 \, c}{x}\right ) + 6 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-\frac{b c - a d}{b}}}\right ) + 2 \,{\left (a b d^{2} x + 7 \, a b c d - 3 \, a^{2} d^{2}\right )} \sqrt{d x + c}}{3 \, a b^{2}}, -\frac{6 \, b^{2} \sqrt{-c} c^{2} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right ) - 3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x + 2 \, b c - a d + 2 \, \sqrt{d x + c} b \sqrt{\frac{b c - a d}{b}}}{b x + a}\right ) - 2 \,{\left (a b d^{2} x + 7 \, a b c d - 3 \, a^{2} d^{2}\right )} \sqrt{d x + c}}{3 \, a b^{2}}, -\frac{2 \,{\left (3 \, b^{2} \sqrt{-c} c^{2} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right ) - 3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-\frac{b c - a d}{b}}}\right ) -{\left (a b d^{2} x + 7 \, a b c d - 3 \, a^{2} d^{2}\right )} \sqrt{d x + c}\right )}}{3 \, a b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/3*(3*b^2*c^(5/2)*log((d*x - 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) + 3*(b^2*c^2 -
2*a*b*c*d + a^2*d^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*b*d^2*x + 7*a*b*c*d - 3*a^2*d^2)*sq
rt(d*x + c))/(a*b^2), 1/3*(3*b^2*c^(5/2)*log((d*x - 2*sqrt(d*x + c)*sqrt(c) + 2*
c)/x) + 6*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(-(b*c - a*d)/b)*arctan(sqrt(d*x +
c)/sqrt(-(b*c - a*d)/b)) + 2*(a*b*d^2*x + 7*a*b*c*d - 3*a^2*d^2)*sqrt(d*x + c))
/(a*b^2), -1/3*(6*b^2*sqrt(-c)*c^2*arctan(sqrt(d*x + c)/sqrt(-c)) - 3*(b^2*c^2 -
2*a*b*c*d + a^2*d^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*b*d^2*x + 7*a*b*c*d - 3*a^2*d^2)*s
qrt(d*x + c))/(a*b^2), -2/3*(3*b^2*sqrt(-c)*c^2*arctan(sqrt(d*x + c)/sqrt(-c)) -
3*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(-(b*c - a*d)/b)*arctan(sqrt(d*x + c)/sqr
t(-(b*c - a*d)/b)) - (a*b*d^2*x + 7*a*b*c*d - 3*a^2*d^2)*sqrt(d*x + c))/(a*b^2)]

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Sympy [A]  time = 48.9201, size = 294, normalized size = 2.49 \[ \frac{2 d \left (c + d x\right )^{\frac{3}{2}}}{3 b} + \frac{\sqrt{c + d x} \left (- 2 a d^{2} + 4 b c d\right )}{b^{2}} - \frac{2 c^{3} \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{c + d x}}{\sqrt{- c}} \right )}}{\sqrt{- c}} & \text{for}\: - c > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{\sqrt{c}} & \text{for}\: - c < 0 \wedge c < c + d x \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{\sqrt{c}} & \text{for}\: c > c + d x \wedge - c < 0 \end{cases}\right )}{a} + \frac{2 \left (a d - b c\right )^{3} \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{\sqrt{c + d x}}{\sqrt{\frac{a d - b c}{b}}} \right )}}{b \sqrt{\frac{a d - b c}{b}}} & \text{for}\: \frac{a d - b c}{b} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{\sqrt{c + d x}}{\sqrt{\frac{- a d + b c}{b}}} \right )}}{b \sqrt{\frac{- a d + b c}{b}}} & \text{for}\: c + d x > \frac{- a d + b c}{b} \wedge \frac{a d - b c}{b} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{\frac{- a d + b c}{b}}} \right )}}{b \sqrt{\frac{- a d + b c}{b}}} & \text{for}\: \frac{a d - b c}{b} < 0 \wedge c + d x < \frac{- a d + b c}{b} \end{cases}\right )}{a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*d*(c + d*x)**(3/2)/(3*b) + sqrt(c + d*x)*(-2*a*d**2 + 4*b*c*d)/b**2 - 2*c**3*P
iecewise((-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*(a*d - b*c)**3*Piecewise((atan(sqrt(c + d*x
)/sqrt((a*d - b*c)/b))/(b*sqrt((a*d - b*c)/b)), (a*d - b*c)/b > 0), (-acoth(sqrt
(c + d*x)/sqrt((-a*d + b*c)/b))/(b*sqrt((-a*d + b*c)/b)), ((a*d - b*c)/b < 0) &
(c + d*x > (-a*d + b*c)/b)), (-atanh(sqrt(c + d*x)/sqrt((-a*d + b*c)/b))/(b*sqrt
((-a*d + b*c)/b)), ((a*d - b*c)/b < 0) & (c + d*x < (-a*d + b*c)/b)))/(a*b**2)

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GIAC/XCAS [A]  time = 0.234005, size = 208, normalized size = 1.76 \[ \frac{2 \, c^{3} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right )}{a \sqrt{-c}} - \frac{2 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a b^{2}} + \frac{2 \,{\left ({\left (d x + c\right )}^{\frac{3}{2}} b^{2} d + 6 \, \sqrt{d x + c} b^{2} c d - 3 \, \sqrt{d x + c} a b d^{2}\right )}}{3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*c^3*arctan(sqrt(d*x + c)/sqrt(-c))/(a*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*b^2) + 2/3*((d*x + c)^(3/2)*b^2*d + 6*sqrt(d*x + c)*b^2*c*d - 3*s
qrt(d*x + c)*a*b*d^2)/b^3