3.1384 \(\int \frac{\sqrt{c+d x}}{(a+b x)^4} \, dx\)

Optimal. Leaf size=146 \[ -\frac{d^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{8 b^{3/2} (b c-a d)^{5/2}}+\frac{d^2 \sqrt{c+d x}}{8 b (a+b x) (b c-a d)^2}-\frac{d \sqrt{c+d x}}{12 b (a+b x)^2 (b c-a d)}-\frac{\sqrt{c+d x}}{3 b (a+b x)^3} \]

[Out]

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

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Rubi [A]  time = 0.256523, antiderivative size = 146, 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 \[ -\frac{d^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{8 b^{3/2} (b c-a d)^{5/2}}+\frac{d^2 \sqrt{c+d x}}{8 b (a+b x) (b c-a d)^2}-\frac{d \sqrt{c+d x}}{12 b (a+b x)^2 (b c-a d)}-\frac{\sqrt{c+d x}}{3 b (a+b x)^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 31.2702, size = 119, normalized size = 0.82 \[ \frac{d^{2} \sqrt{c + d x}}{8 b \left (a + b x\right ) \left (a d - b c\right )^{2}} + \frac{d \sqrt{c + d x}}{12 b \left (a + b x\right )^{2} \left (a d - b c\right )} - \frac{\sqrt{c + d x}}{3 b \left (a + b x\right )^{3}} + \frac{d^{3} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{8 b^{\frac{3}{2}} \left (a d - b c\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(1/2)/(b*x+a)**4,x)

[Out]

d**2*sqrt(c + d*x)/(8*b*(a + b*x)*(a*d - b*c)**2) + d*sqrt(c + d*x)/(12*b*(a + b
*x)**2*(a*d - b*c)) - sqrt(c + d*x)/(3*b*(a + b*x)**3) + d**3*atan(sqrt(b)*sqrt(
c + d*x)/sqrt(a*d - b*c))/(8*b**(3/2)*(a*d - b*c)**(5/2))

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Mathematica [A]  time = 0.177177, size = 130, normalized size = 0.89 \[ \sqrt{c+d x} \left (\frac{d^2}{8 b (a+b x) (b c-a d)^2}-\frac{d}{12 b (a+b x)^2 (b c-a d)}-\frac{1}{3 b (a+b x)^3}\right )-\frac{d^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{8 b^{3/2} (b c-a d)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[c + d*x]*(-1/(3*b*(a + b*x)^3) - d/(12*b*(b*c - a*d)*(a + b*x)^2) + d^2/(8*
b*(b*c - a*d)^2*(a + b*x))) - (d^3*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*
d]])/(8*b^(3/2)*(b*c - a*d)^(5/2))

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Maple [A]  time = 0.018, size = 170, normalized size = 1.2 \[{\frac{{d}^{3}b}{8\, \left ( bdx+ad \right ) ^{3} \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ) } \left ( dx+c \right ) ^{{\frac{5}{2}}}}+{\frac{{d}^{3}}{3\, \left ( bdx+ad \right ) ^{3} \left ( ad-bc \right ) } \left ( dx+c \right ) ^{{\frac{3}{2}}}}-{\frac{{d}^{3}}{8\, \left ( bdx+ad \right ) ^{3}b}\sqrt{dx+c}}+{\frac{{d}^{3}}{8\,b \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ) }\arctan \left ({b\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(1/2)/(b*x+a)^4,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.228869, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (3 \, b^{2} d^{2} x^{2} - 8 \, b^{2} c^{2} + 14 \, a b c d - 3 \, a^{2} d^{2} - 2 \,{\left (b^{2} c d - 4 \, a b d^{2}\right )} x\right )} \sqrt{b^{2} c - a b d} \sqrt{d x + c} + 3 \,{\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )} \log \left (\frac{\sqrt{b^{2} c - a b d}{\left (b d x + 2 \, b c - a d\right )} - 2 \,{\left (b^{2} c - a b d\right )} \sqrt{d x + c}}{b x + a}\right )}{48 \,{\left (a^{3} b^{3} c^{2} - 2 \, a^{4} b^{2} c d + a^{5} b d^{2} +{\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} x^{3} + 3 \,{\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} x^{2} + 3 \,{\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2}\right )} x\right )} \sqrt{b^{2} c - a b d}}, \frac{{\left (3 \, b^{2} d^{2} x^{2} - 8 \, b^{2} c^{2} + 14 \, a b c d - 3 \, a^{2} d^{2} - 2 \,{\left (b^{2} c d - 4 \, a b d^{2}\right )} x\right )} \sqrt{-b^{2} c + a b d} \sqrt{d x + c} - 3 \,{\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )} \arctan \left (-\frac{b c - a d}{\sqrt{-b^{2} c + a b d} \sqrt{d x + c}}\right )}{24 \,{\left (a^{3} b^{3} c^{2} - 2 \, a^{4} b^{2} c d + a^{5} b d^{2} +{\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} x^{3} + 3 \,{\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} x^{2} + 3 \,{\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2}\right )} x\right )} \sqrt{-b^{2} c + a b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/48*(2*(3*b^2*d^2*x^2 - 8*b^2*c^2 + 14*a*b*c*d - 3*a^2*d^2 - 2*(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^2 - 2*a^4*b^2*c*d + a^5*b*d^
2 + (b^6*c^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*x^3 + 3*(a*b^5*c^2 - 2*a^2*b^4*c*d + a
^3*b^3*d^2)*x^2 + 3*(a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2)*x)*sqrt(b^2*c -
a*b*d)), 1/24*((3*b^2*d^2*x^2 - 8*b^2*c^2 + 14*a*b*c*d - 3*a^2*d^2 - 2*(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^2 - 2*a^4*b^2*c*d + a^5*b*d^2 + (b^6*c^2 - 2*a*b^5*c*d +
a^2*b^4*d^2)*x^3 + 3*(a*b^5*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*x^2 + 3*(a^2*b^4
*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2)*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)**(1/2)/(b*x+a)**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.224561, size = 279, normalized size = 1.91 \[ \frac{d^{3} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{8 \,{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} \sqrt{-b^{2} c + a b d}} + \frac{3 \,{\left (d x + c\right )}^{\frac{5}{2}} b^{2} d^{3} - 8 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{2} c d^{3} - 3 \, \sqrt{d x + c} b^{2} c^{2} d^{3} + 8 \,{\left (d x + c\right )}^{\frac{3}{2}} a b d^{4} + 6 \, \sqrt{d x + c} a b c d^{4} - 3 \, \sqrt{d x + c} a^{2} d^{5}}{24 \,{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}{\left ({\left (d x + c\right )} b - b c + a d\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x + c)/(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^2 - 2*a*b^2*c*d + a
^2*b*d^2)*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*sq
rt(d*x + c)*a*b*c*d^4 - 3*sqrt(d*x + c)*a^2*d^5)/((b^3*c^2 - 2*a*b^2*c*d + a^2*b
*d^2)*((d*x + c)*b - b*c + a*d)^3)