3.1422 \(\int \frac{1}{(a+b x)^4 \sqrt{c+d x}} \, dx\)

Optimal. Leaf size=147 \[ \frac{5 d^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{8 \sqrt{b} (b c-a d)^{7/2}}-\frac{5 d^2 \sqrt{c+d x}}{8 (a+b x) (b c-a d)^3}+\frac{5 d \sqrt{c+d x}}{12 (a+b x)^2 (b c-a d)^2}-\frac{\sqrt{c+d x}}{3 (a+b x)^3 (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.158621, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{5 d^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{8 \sqrt{b} (b c-a d)^{7/2}}-\frac{5 d^2 \sqrt{c+d x}}{8 (a+b x) (b c-a d)^3}+\frac{5 d \sqrt{c+d x}}{12 (a+b x)^2 (b c-a d)^2}-\frac{\sqrt{c+d x}}{3 (a+b x)^3 (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^4*Sqrt[c + d*x]),x]

[Out]

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

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Rubi in Sympy [A]  time = 31.4412, size = 128, normalized size = 0.87 \[ \frac{5 d^{2} \sqrt{c + d x}}{8 \left (a + b x\right ) \left (a d - b c\right )^{3}} + \frac{5 d \sqrt{c + d x}}{12 \left (a + b x\right )^{2} \left (a d - b c\right )^{2}} + \frac{\sqrt{c + d x}}{3 \left (a + b x\right )^{3} \left (a d - b c\right )} + \frac{5 d^{3} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{8 \sqrt{b} \left (a d - b c\right )^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*d**2*sqrt(c + d*x)/(8*(a + b*x)*(a*d - b*c)**3) + 5*d*sqrt(c + d*x)/(12*(a + b
*x)**2*(a*d - b*c)**2) + sqrt(c + d*x)/(3*(a + b*x)**3*(a*d - b*c)) + 5*d**3*ata
n(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c))/(8*sqrt(b)*(a*d - b*c)**(7/2))

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Mathematica [A]  time = 0.199741, size = 128, normalized size = 0.87 \[ \frac{5 d^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{8 \sqrt{b} (b c-a d)^{7/2}}-\frac{\sqrt{c+d x} \left (33 a^2 d^2+2 a b d (20 d x-13 c)+b^2 \left (8 c^2-10 c d x+15 d^2 x^2\right )\right )}{24 (a+b x)^3 (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^4*Sqrt[c + d*x]),x]

[Out]

-(Sqrt[c + d*x]*(33*a^2*d^2 + 2*a*b*d*(-13*c + 20*d*x) + b^2*(8*c^2 - 10*c*d*x +
15*d^2*x^2)))/(24*(b*c - a*d)^3*(a + b*x)^3) + (5*d^3*ArcTanh[(Sqrt[b]*Sqrt[c +
d*x])/Sqrt[b*c - a*d]])/(8*Sqrt[b]*(b*c - a*d)^(7/2))

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Maple [A]  time = 0.011, size = 147, normalized size = 1. \[{\frac{{d}^{3}}{ \left ( 3\,ad-3\,bc \right ) \left ( bdx+ad \right ) ^{3}}\sqrt{dx+c}}+{\frac{5\,{d}^{3}}{12\, \left ( ad-bc \right ) ^{2} \left ( bdx+ad \right ) ^{2}}\sqrt{dx+c}}+{\frac{5\,{d}^{3}}{8\, \left ( ad-bc \right ) ^{3} \left ( bdx+ad \right ) }\sqrt{dx+c}}+{\frac{5\,{d}^{3}}{8\, \left ( ad-bc \right ) ^{3}}\arctan \left ({b\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*d^3*(d*x+c)^(1/2)/(a*d-b*c)/(b*d*x+a*d)^3+5/12*d^3/(a*d-b*c)^2*(d*x+c)^(1/2)
/(b*d*x+a*d)^2+5/8*d^3/(a*d-b*c)^3*(d*x+c)^(1/2)/(b*d*x+a*d)+5/8*d^3/(a*d-b*c)^3
/((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(1/((b*x + a)^4*sqrt(d*x + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.222177, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (15 \, b^{2} d^{2} x^{2} + 8 \, b^{2} c^{2} - 26 \, a b c d + 33 \, a^{2} d^{2} - 10 \,{\left (b^{2} c d - 4 \, a b d^{2}\right )} x\right )} \sqrt{b^{2} c - a b d} \sqrt{d x + c} + 15 \,{\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )} \log \left (\frac{\sqrt{b^{2} c - a b d}{\left (b d x + 2 \, b c - a d\right )} - 2 \,{\left (b^{2} c - a b d\right )} \sqrt{d x + c}}{b x + a}\right )}{48 \,{\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3} +{\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} x^{3} + 3 \,{\left (a b^{5} c^{3} - 3 \, a^{2} b^{4} c^{2} d + 3 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} x^{2} + 3 \,{\left (a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d + 3 \, a^{4} b^{2} c d^{2} - a^{5} b d^{3}\right )} x\right )} \sqrt{b^{2} c - a b d}}, -\frac{{\left (15 \, b^{2} d^{2} x^{2} + 8 \, b^{2} c^{2} - 26 \, a b c d + 33 \, a^{2} d^{2} - 10 \,{\left (b^{2} c d - 4 \, a b d^{2}\right )} x\right )} \sqrt{-b^{2} c + a b d} \sqrt{d x + c} - 15 \,{\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )} \arctan \left (-\frac{b c - a d}{\sqrt{-b^{2} c + a b d} \sqrt{d x + c}}\right )}{24 \,{\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3} +{\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} x^{3} + 3 \,{\left (a b^{5} c^{3} - 3 \, a^{2} b^{4} c^{2} d + 3 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} x^{2} + 3 \,{\left (a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d + 3 \, a^{4} b^{2} c d^{2} - a^{5} b d^{3}\right )} x\right )} \sqrt{-b^{2} c + a b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^4*sqrt(d*x + c)),x, algorithm="fricas")

[Out]

[-1/48*(2*(15*b^2*d^2*x^2 + 8*b^2*c^2 - 26*a*b*c*d + 33*a^2*d^2 - 10*(b^2*c*d -
4*a*b*d^2)*x)*sqrt(b^2*c - a*b*d)*sqrt(d*x + c) + 15*(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^3 - 3*a^4*b^2*c^2*d + 3
*a^5*b*c*d^2 - a^6*d^3 + (b^6*c^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^
3)*x^3 + 3*(a*b^5*c^3 - 3*a^2*b^4*c^2*d + 3*a^3*b^3*c*d^2 - a^4*b^2*d^3)*x^2 + 3
*(a^2*b^4*c^3 - 3*a^3*b^3*c^2*d + 3*a^4*b^2*c*d^2 - a^5*b*d^3)*x)*sqrt(b^2*c - a
*b*d)), -1/24*((15*b^2*d^2*x^2 + 8*b^2*c^2 - 26*a*b*c*d + 33*a^2*d^2 - 10*(b^2*c
*d - 4*a*b*d^2)*x)*sqrt(-b^2*c + a*b*d)*sqrt(d*x + c) - 15*(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)*s
qrt(d*x + c))))/((a^3*b^3*c^3 - 3*a^4*b^2*c^2*d + 3*a^5*b*c*d^2 - a^6*d^3 + (b^6
*c^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)*x^3 + 3*(a*b^5*c^3 - 3*a^2
*b^4*c^2*d + 3*a^3*b^3*c*d^2 - a^4*b^2*d^3)*x^2 + 3*(a^2*b^4*c^3 - 3*a^3*b^3*c^2
*d + 3*a^4*b^2*c*d^2 - a^5*b*d^3)*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(1/(b*x+a)**4/(d*x+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.222052, size = 312, normalized size = 2.12 \[ -\frac{5 \, d^{3} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{8 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt{-b^{2} c + a b d}} - \frac{15 \,{\left (d x + c\right )}^{\frac{5}{2}} b^{2} d^{3} - 40 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{2} c d^{3} + 33 \, \sqrt{d x + c} b^{2} c^{2} d^{3} + 40 \,{\left (d x + c\right )}^{\frac{3}{2}} a b d^{4} - 66 \, \sqrt{d x + c} a b c d^{4} + 33 \, \sqrt{d x + c} a^{2} d^{5}}{24 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}{\left ({\left (d x + c\right )} b - b c + a d\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^4*sqrt(d*x + c)),x, algorithm="giac")

[Out]

-5/8*d^3*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/((b^3*c^3 - 3*a*b^2*c^2*d
+ 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(-b^2*c + a*b*d)) - 1/24*(15*(d*x + c)^(5/2)*b^2*
d^3 - 40*(d*x + c)^(3/2)*b^2*c*d^3 + 33*sqrt(d*x + c)*b^2*c^2*d^3 + 40*(d*x + c)
^(3/2)*a*b*d^4 - 66*sqrt(d*x + c)*a*b*c*d^4 + 33*sqrt(d*x + c)*a^2*d^5)/((b^3*c^
3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*((d*x + c)*b - b*c + a*d)^3)