3.1395 \(\int \frac{(c+d x)^{3/2}}{(a+b x)^3} \, dx\)

Optimal. Leaf size=100 \[ -\frac{3 d^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{4 b^{5/2} \sqrt{b c-a d}}-\frac{3 d \sqrt{c+d x}}{4 b^2 (a+b x)}-\frac{(c+d x)^{3/2}}{2 b (a+b x)^2} \]

[Out]

(-3*d*Sqrt[c + d*x])/(4*b^2*(a + b*x)) - (c + d*x)^(3/2)/(2*b*(a + b*x)^2) - (3*
d^2*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(4*b^(5/2)*Sqrt[b*c - a*d]
)

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Rubi [A]  time = 0.120321, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\frac{3 d^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{4 b^{5/2} \sqrt{b c-a d}}-\frac{3 d \sqrt{c+d x}}{4 b^2 (a+b x)}-\frac{(c+d x)^{3/2}}{2 b (a+b x)^2} \]

Antiderivative was successfully verified.

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

[Out]

(-3*d*Sqrt[c + d*x])/(4*b^2*(a + b*x)) - (c + d*x)^(3/2)/(2*b*(a + b*x)^2) - (3*
d^2*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(4*b^(5/2)*Sqrt[b*c - a*d]
)

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Rubi in Sympy [A]  time = 19.0541, size = 87, normalized size = 0.87 \[ - \frac{\left (c + d x\right )^{\frac{3}{2}}}{2 b \left (a + b x\right )^{2}} - \frac{3 d \sqrt{c + d x}}{4 b^{2} \left (a + b x\right )} + \frac{3 d^{2} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{4 b^{\frac{5}{2}} \sqrt{a d - b c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(c + d*x)**(3/2)/(2*b*(a + b*x)**2) - 3*d*sqrt(c + d*x)/(4*b**2*(a + b*x)) + 3*
d**2*atan(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c))/(4*b**(5/2)*sqrt(a*d - b*c))

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Mathematica [A]  time = 0.110736, size = 90, normalized size = 0.9 \[ -\frac{3 d^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{4 b^{5/2} \sqrt{b c-a d}}-\frac{\sqrt{c+d x} (3 a d+2 b c+5 b d x)}{4 b^2 (a+b x)^2} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[c + d*x]*(2*b*c + 3*a*d + 5*b*d*x))/(4*b^2*(a + b*x)^2) - (3*d^2*ArcTanh[
(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(4*b^(5/2)*Sqrt[b*c - a*d])

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Maple [A]  time = 0.017, size = 121, normalized size = 1.2 \[ -{\frac{5\,{d}^{2}}{4\, \left ( bdx+ad \right ) ^{2}b} \left ( dx+c \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{d}^{3}a}{4\, \left ( bdx+ad \right ) ^{2}{b}^{2}}\sqrt{dx+c}}+{\frac{3\,{d}^{2}c}{4\, \left ( bdx+ad \right ) ^{2}b}\sqrt{dx+c}}+{\frac{3\,{d}^{2}}{4\,{b}^{2}}\arctan \left ({b\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(3/2)/(b*x+a)^3,x)

[Out]

-5/4*d^2/(b*d*x+a*d)^2/b*(d*x+c)^(3/2)-3/4*d^3/(b*d*x+a*d)^2/b^2*(d*x+c)^(1/2)*a
+3/4*d^2/(b*d*x+a*d)^2/b*(d*x+c)^(1/2)*c+3/4*d^2/b^2/((a*d-b*c)*b)^(1/2)*arctan(
(d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))

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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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.234609, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \, \sqrt{b^{2} c - a b d}{\left (5 \, b d x + 2 \, b c + 3 \, a d\right )} \sqrt{d x + c} - 3 \,{\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (\frac{\sqrt{b^{2} c - a b d}{\left (b d x + 2 \, b c - a d\right )} - 2 \,{\left (b^{2} c - a b d\right )} \sqrt{d x + c}}{b x + a}\right )}{8 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )} \sqrt{b^{2} c - a b d}}, -\frac{\sqrt{-b^{2} c + a b d}{\left (5 \, b d x + 2 \, b c + 3 \, a d\right )} \sqrt{d x + c} + 3 \,{\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \arctan \left (-\frac{b c - a d}{\sqrt{-b^{2} c + a b d} \sqrt{d x + c}}\right )}{4 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )} \sqrt{-b^{2} c + a b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/8*(2*sqrt(b^2*c - a*b*d)*(5*b*d*x + 2*b*c + 3*a*d)*sqrt(d*x + c) - 3*(b^2*d^
2*x^2 + 2*a*b*d^2*x + a^2*d^2)*log((sqrt(b^2*c - a*b*d)*(b*d*x + 2*b*c - a*d) -
2*(b^2*c - a*b*d)*sqrt(d*x + c))/(b*x + a)))/((b^4*x^2 + 2*a*b^3*x + a^2*b^2)*sq
rt(b^2*c - a*b*d)), -1/4*(sqrt(-b^2*c + a*b*d)*(5*b*d*x + 2*b*c + 3*a*d)*sqrt(d*
x + c) + 3*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*arctan(-(b*c - a*d)/(sqrt(-b^2*
c + a*b*d)*sqrt(d*x + c))))/((b^4*x^2 + 2*a*b^3*x + a^2*b^2)*sqrt(-b^2*c + a*b*d
))]

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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)**(3/2)/(b*x+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.233498, size = 146, normalized size = 1.46 \[ \frac{3 \, d^{2} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{4 \, \sqrt{-b^{2} c + a b d} b^{2}} - \frac{5 \,{\left (d x + c\right )}^{\frac{3}{2}} b d^{2} - 3 \, \sqrt{d x + c} b c d^{2} + 3 \, \sqrt{d x + c} a d^{3}}{4 \,{\left ({\left (d x + c\right )} b - b c + a d\right )}^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/4*d^2*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*b^2)
- 1/4*(5*(d*x + c)^(3/2)*b*d^2 - 3*sqrt(d*x + c)*b*c*d^2 + 3*sqrt(d*x + c)*a*d^3
)/(((d*x + c)*b - b*c + a*d)^2*b^2)