3.1394 \(\int \frac{(c+d x)^{3/2}}{(a+b x)^2} \, dx\)

Optimal. Leaf size=85 \[ -\frac{3 d \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{5/2}}-\frac{(c+d x)^{3/2}}{b (a+b x)}+\frac{3 d \sqrt{c+d x}}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.101582, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ -\frac{3 d \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{5/2}}-\frac{(c+d x)^{3/2}}{b (a+b x)}+\frac{3 d \sqrt{c+d x}}{b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 18.0712, size = 73, normalized size = 0.86 \[ - \frac{\left (c + d x\right )^{\frac{3}{2}}}{b \left (a + b x\right )} + \frac{3 d \sqrt{c + d x}}{b^{2}} - \frac{3 d \sqrt{a d - b c} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.144958, size = 85, normalized size = 1. \[ \sqrt{c+d x} \left (\frac{a d-b c}{b^2 (a+b x)}+\frac{2 d}{b^2}\right )-\frac{3 d \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.019, size = 148, normalized size = 1.7 \[ 2\,{\frac{d\sqrt{dx+c}}{{b}^{2}}}+{\frac{a{d}^{2}}{{b}^{2} \left ( bdx+ad \right ) }\sqrt{dx+c}}-{\frac{dc}{b \left ( bdx+ad \right ) }\sqrt{dx+c}}-3\,{\frac{a{d}^{2}}{{b}^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+3\,{\frac{dc}{b\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(3/2)/(b*x+a)^2,x)

[Out]

2*d*(d*x+c)^(1/2)/b^2+1/b^2*(d*x+c)^(1/2)/(b*d*x+a*d)*a*d^2-d/b*(d*x+c)^(1/2)/(b
*d*x+a*d)*c-3/b^2/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2)
)*a*d^2+3*d/b/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*c

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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)^(3/2)/(b*x + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224094, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (b d x + a d\right )} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x + 2 \, b c - a d - 2 \, \sqrt{d x + c} b \sqrt{\frac{b c - a d}{b}}}{b x + a}\right ) + 2 \,{\left (2 \, b d x - b c + 3 \, a d\right )} \sqrt{d x + c}}{2 \,{\left (b^{3} x + a b^{2}\right )}}, -\frac{3 \,{\left (b d x + a d\right )} \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-\frac{b c - a d}{b}}}\right ) -{\left (2 \, b d x - b c + 3 \, a d\right )} \sqrt{d x + c}}{b^{3} x + a b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(3*(b*d*x + a*d)*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*(2*b*d*x - b*c + 3*a*d)*sqrt(d*x + c)
)/(b^3*x + a*b^2), -(3*(b*d*x + a*d)*sqrt(-(b*c - a*d)/b)*arctan(sqrt(d*x + c)/s
qrt(-(b*c - a*d)/b)) - (2*b*d*x - b*c + 3*a*d)*sqrt(d*x + c))/(b^3*x + a*b^2)]

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Sympy [A]  time = 29.854, size = 1129, normalized size = 13.28 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**2*d**3*sqrt(c + d*x)/(2*a**2*b**2*d**2 - 2*a*b**3*c*d + 2*a*b**3*d**2*x - 2
*b**4*c*d*x) - a**2*d**3*sqrt(-1/(b*(a*d - b*c)**3))*log(-a**2*d**2*sqrt(-1/(b*(
a*d - b*c)**3)) + 2*a*b*c*d*sqrt(-1/(b*(a*d - b*c)**3)) - b**2*c**2*sqrt(-1/(b*(
a*d - b*c)**3)) + sqrt(c + d*x))/(2*b**2) + a**2*d**3*sqrt(-1/(b*(a*d - b*c)**3)
)*log(a**2*d**2*sqrt(-1/(b*(a*d - b*c)**3)) - 2*a*b*c*d*sqrt(-1/(b*(a*d - b*c)**
3)) + b**2*c**2*sqrt(-1/(b*(a*d - b*c)**3)) + sqrt(c + d*x))/(2*b**2) - 4*a*c*d*
*2*sqrt(c + d*x)/(2*a**2*b*d**2 - 2*a*b**2*c*d + 2*a*b**2*d**2*x - 2*b**3*c*d*x)
+ a*c*d**2*sqrt(-1/(b*(a*d - b*c)**3))*log(-a**2*d**2*sqrt(-1/(b*(a*d - b*c)**3
)) + 2*a*b*c*d*sqrt(-1/(b*(a*d - b*c)**3)) - b**2*c**2*sqrt(-1/(b*(a*d - b*c)**3
)) + sqrt(c + d*x))/b - a*c*d**2*sqrt(-1/(b*(a*d - b*c)**3))*log(a**2*d**2*sqrt(
-1/(b*(a*d - b*c)**3)) - 2*a*b*c*d*sqrt(-1/(b*(a*d - b*c)**3)) + b**2*c**2*sqrt(
-1/(b*(a*d - b*c)**3)) + sqrt(c + d*x))/b - 4*a*d**2*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)/s
qrt(-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**2 - c**2*d*sqrt(-1/(b*(a*d - b*c)**3))*log(-a**2*d
**2*sqrt(-1/(b*(a*d - b*c)**3)) + 2*a*b*c*d*sqrt(-1/(b*(a*d - b*c)**3)) - b**2*c
**2*sqrt(-1/(b*(a*d - b*c)**3)) + sqrt(c + d*x))/2 + c**2*d*sqrt(-1/(b*(a*d - b*
c)**3))*log(a**2*d**2*sqrt(-1/(b*(a*d - b*c)**3)) - 2*a*b*c*d*sqrt(-1/(b*(a*d -
b*c)**3)) + b**2*c**2*sqrt(-1/(b*(a*d - b*c)**3)) + sqrt(c + d*x))/2 + 2*c**2*d*
sqrt(c + d*x)/(2*a**2*d**2 - 2*a*b*c*d + 2*a*b*d**2*x - 2*b**2*c*d*x) + 4*c*d*Pi
ecewise((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 + 2*d*sqrt(c + d*x)/b**2

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GIAC/XCAS [A]  time = 0.223665, size = 153, normalized size = 1.8 \[ \frac{2 \, \sqrt{d x + c} d}{b^{2}} + \frac{3 \,{\left (b c d - a d^{2}\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} b^{2}} - \frac{\sqrt{d x + c} b c d - \sqrt{d x + c} a d^{2}}{{\left ({\left (d x + c\right )} b - b c + a d\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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