3.1396 \(\int \frac{(c+d x)^{3/2}}{(a+b x)^4} \, dx\)

Optimal. Leaf size=136 \[ \frac{d^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{8 b^{5/2} (b c-a d)^{3/2}}-\frac{d^2 \sqrt{c+d x}}{8 b^2 (a+b x) (b c-a d)}-\frac{d \sqrt{c+d x}}{4 b^2 (a+b x)^2}-\frac{(c+d x)^{3/2}}{3 b (a+b x)^3} \]

[Out]

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

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Rubi [A]  time = 0.167535, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{d^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{8 b^{5/2} (b c-a d)^{3/2}}-\frac{d^2 \sqrt{c+d x}}{8 b^2 (a+b x) (b c-a d)}-\frac{d \sqrt{c+d x}}{4 b^2 (a+b x)^2}-\frac{(c+d x)^{3/2}}{3 b (a+b x)^3} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(3/2)/(a + b*x)^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 27.9521, size = 114, normalized size = 0.84 \[ - \frac{\left (c + d x\right )^{\frac{3}{2}}}{3 b \left (a + b x\right )^{3}} + \frac{d^{2} \sqrt{c + d x}}{8 b^{2} \left (a + b x\right ) \left (a d - b c\right )} - \frac{d \sqrt{c + d x}}{4 b^{2} \left (a + b x\right )^{2}} + \frac{d^{3} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{8 b^{\frac{5}{2}} \left (a d - b c\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(3/2)/(b*x+a)**4,x)

[Out]

-(c + d*x)**(3/2)/(3*b*(a + b*x)**3) + d**2*sqrt(c + d*x)/(8*b**2*(a + b*x)*(a*d
- b*c)) - d*sqrt(c + d*x)/(4*b**2*(a + b*x)**2) + d**3*atan(sqrt(b)*sqrt(c + d*
x)/sqrt(a*d - b*c))/(8*b**(5/2)*(a*d - b*c)**(3/2))

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Mathematica [A]  time = 0.178702, size = 128, normalized size = 0.94 \[ \frac{d^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{8 b^{5/2} (b c-a d)^{3/2}}+\sqrt{c+d x} \left (-\frac{d^2}{8 b^2 (a+b x) (b c-a d)}+\frac{a d-b c}{3 b^2 (a+b x)^3}-\frac{7 d}{12 b^2 (a+b x)^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(3/2)/(a + b*x)^4,x]

[Out]

Sqrt[c + d*x]*((-(b*c) + a*d)/(3*b^2*(a + b*x)^3) - (7*d)/(12*b^2*(a + b*x)^2) -
d^2/(8*b^2*(b*c - a*d)*(a + b*x))) + (d^3*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[
b*c - a*d]])/(8*b^(5/2)*(b*c - a*d)^(3/2))

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Maple [A]  time = 0.02, size = 163, normalized size = 1.2 \[{\frac{{d}^{3}}{8\, \left ( bdx+ad \right ) ^{3} \left ( ad-bc \right ) } \left ( dx+c \right ) ^{{\frac{5}{2}}}}-{\frac{{d}^{3}}{3\, \left ( bdx+ad \right ) ^{3}b} \left ( dx+c \right ) ^{{\frac{3}{2}}}}-{\frac{{d}^{4}a}{8\, \left ( bdx+ad \right ) ^{3}{b}^{2}}\sqrt{dx+c}}+{\frac{{d}^{3}c}{8\, \left ( bdx+ad \right ) ^{3}b}\sqrt{dx+c}}+{\frac{{d}^{3}}{ \left ( 8\,ad-8\,bc \right ){b}^{2}}\arctan \left ({b\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(3/2)/(b*x+a)^4,x)

[Out]

1/8*d^3/(b*d*x+a*d)^3/(a*d-b*c)*(d*x+c)^(5/2)-1/3*d^3/(b*d*x+a*d)^3/b*(d*x+c)^(3
/2)-1/8*d^4/(b*d*x+a*d)^3/b^2*(d*x+c)^(1/2)*a+1/8*d^3/(b*d*x+a*d)^3/b*(d*x+c)^(1
/2)*c+1/8*d^3/(a*d-b*c)/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)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.235603, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (3 \, b^{2} d^{2} x^{2} + 8 \, b^{2} c^{2} - 2 \, a b c d - 3 \, a^{2} d^{2} + 2 \,{\left (7 \, b^{2} c d - 4 \, a b d^{2}\right )} x\right )} \sqrt{b^{2} c - a b d} \sqrt{d x + c} + 3 \,{\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )} \log \left (\frac{\sqrt{b^{2} c - a b d}{\left (b d x + 2 \, b c - a d\right )} - 2 \,{\left (b^{2} c - a b d\right )} \sqrt{d x + c}}{b x + a}\right )}{48 \,{\left (a^{3} b^{3} c - a^{4} b^{2} d +{\left (b^{6} c - a b^{5} d\right )} x^{3} + 3 \,{\left (a b^{5} c - a^{2} b^{4} d\right )} x^{2} + 3 \,{\left (a^{2} b^{4} c - a^{3} b^{3} d\right )} x\right )} \sqrt{b^{2} c - a b d}}, -\frac{{\left (3 \, b^{2} d^{2} x^{2} + 8 \, b^{2} c^{2} - 2 \, a b c d - 3 \, a^{2} d^{2} + 2 \,{\left (7 \, b^{2} c d - 4 \, a b d^{2}\right )} x\right )} \sqrt{-b^{2} c + a b d} \sqrt{d x + c} - 3 \,{\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )} \arctan \left (-\frac{b c - a d}{\sqrt{-b^{2} c + a b d} \sqrt{d x + c}}\right )}{24 \,{\left (a^{3} b^{3} c - a^{4} b^{2} d +{\left (b^{6} c - a b^{5} d\right )} x^{3} + 3 \,{\left (a b^{5} c - a^{2} b^{4} d\right )} x^{2} + 3 \,{\left (a^{2} b^{4} c - a^{3} b^{3} d\right )} x\right )} \sqrt{-b^{2} c + a b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(3/2)/(b*x + a)^4,x, algorithm="fricas")

[Out]

[-1/48*(2*(3*b^2*d^2*x^2 + 8*b^2*c^2 - 2*a*b*c*d - 3*a^2*d^2 + 2*(7*b^2*c*d - 4*
a*b*d^2)*x)*sqrt(b^2*c - a*b*d)*sqrt(d*x + c) + 3*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2
+ 3*a^2*b*d^3*x + a^3*d^3)*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)))/((a^3*b^3*c - a^4*b^2*d + (b^6*c - a*b
^5*d)*x^3 + 3*(a*b^5*c - a^2*b^4*d)*x^2 + 3*(a^2*b^4*c - a^3*b^3*d)*x)*sqrt(b^2*
c - a*b*d)), -1/24*((3*b^2*d^2*x^2 + 8*b^2*c^2 - 2*a*b*c*d - 3*a^2*d^2 + 2*(7*b^
2*c*d - 4*a*b*d^2)*x)*sqrt(-b^2*c + a*b*d)*sqrt(d*x + c) - 3*(b^3*d^3*x^3 + 3*a*
b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*arctan(-(b*c - a*d)/(sqrt(-b^2*c + a*b*d)
*sqrt(d*x + c))))/((a^3*b^3*c - a^4*b^2*d + (b^6*c - a*b^5*d)*x^3 + 3*(a*b^5*c -
a^2*b^4*d)*x^2 + 3*(a^2*b^4*c - a^3*b^3*d)*x)*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)**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.230435, size = 250, normalized size = 1.84 \[ -\frac{d^{3} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{8 \,{\left (b^{3} c - a b^{2} d\right )} \sqrt{-b^{2} c + a b d}} - \frac{3 \,{\left (d x + c\right )}^{\frac{5}{2}} b^{2} d^{3} + 8 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{2} c d^{3} - 3 \, \sqrt{d x + c} b^{2} c^{2} d^{3} - 8 \,{\left (d x + c\right )}^{\frac{3}{2}} a b d^{4} + 6 \, \sqrt{d x + c} a b c d^{4} - 3 \, \sqrt{d x + c} a^{2} d^{5}}{24 \,{\left (b^{3} c - a b^{2} d\right )}{\left ({\left (d x + c\right )} b - b c + a d\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(3/2)/(b*x + a)^4,x, algorithm="giac")

[Out]

-1/8*d^3*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/((b^3*c - a*b^2*d)*sqrt(-b
^2*c + a*b*d)) - 1/24*(3*(d*x + c)^(5/2)*b^2*d^3 + 8*(d*x + c)^(3/2)*b^2*c*d^3 -
3*sqrt(d*x + c)*b^2*c^2*d^3 - 8*(d*x + c)^(3/2)*a*b*d^4 + 6*sqrt(d*x + c)*a*b*c
*d^4 - 3*sqrt(d*x + c)*a^2*d^5)/((b^3*c - a*b^2*d)*((d*x + c)*b - b*c + a*d)^3)