3.1385 \(\int \frac{\sqrt{c+d x}}{(a+b x)^5} \, dx\)

Optimal. Leaf size=182 \[ \frac{5 d^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{64 b^{3/2} (b c-a d)^{7/2}}-\frac{5 d^3 \sqrt{c+d x}}{64 b (a+b x) (b c-a d)^3}+\frac{5 d^2 \sqrt{c+d x}}{96 b (a+b x)^2 (b c-a d)^2}-\frac{d \sqrt{c+d x}}{24 b (a+b x)^3 (b c-a d)}-\frac{\sqrt{c+d x}}{4 b (a+b x)^4} \]

[Out]

-Sqrt[c + d*x]/(4*b*(a + b*x)^4) - (d*Sqrt[c + d*x])/(24*b*(b*c - a*d)*(a + b*x)
^3) + (5*d^2*Sqrt[c + d*x])/(96*b*(b*c - a*d)^2*(a + b*x)^2) - (5*d^3*Sqrt[c + d
*x])/(64*b*(b*c - a*d)^3*(a + b*x)) + (5*d^4*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqr
t[b*c - a*d]])/(64*b^(3/2)*(b*c - a*d)^(7/2))

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Rubi [A]  time = 0.317586, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{5 d^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{64 b^{3/2} (b c-a d)^{7/2}}-\frac{5 d^3 \sqrt{c+d x}}{64 b (a+b x) (b c-a d)^3}+\frac{5 d^2 \sqrt{c+d x}}{96 b (a+b x)^2 (b c-a d)^2}-\frac{d \sqrt{c+d x}}{24 b (a+b x)^3 (b c-a d)}-\frac{\sqrt{c+d x}}{4 b (a+b x)^4} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[c + d*x]/(a + b*x)^5,x]

[Out]

-Sqrt[c + d*x]/(4*b*(a + b*x)^4) - (d*Sqrt[c + d*x])/(24*b*(b*c - a*d)*(a + b*x)
^3) + (5*d^2*Sqrt[c + d*x])/(96*b*(b*c - a*d)^2*(a + b*x)^2) - (5*d^3*Sqrt[c + d
*x])/(64*b*(b*c - a*d)^3*(a + b*x)) + (5*d^4*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqr
t[b*c - a*d]])/(64*b^(3/2)*(b*c - a*d)^(7/2))

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Rubi in Sympy [A]  time = 43.2389, size = 155, normalized size = 0.85 \[ \frac{5 d^{3} \sqrt{c + d x}}{64 b \left (a + b x\right ) \left (a d - b c\right )^{3}} + \frac{5 d^{2} \sqrt{c + d x}}{96 b \left (a + b x\right )^{2} \left (a d - b c\right )^{2}} + \frac{d \sqrt{c + d x}}{24 b \left (a + b x\right )^{3} \left (a d - b c\right )} - \frac{\sqrt{c + d x}}{4 b \left (a + b x\right )^{4}} + \frac{5 d^{4} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{64 b^{\frac{3}{2}} \left (a d - b c\right )^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(1/2)/(b*x+a)**5,x)

[Out]

5*d**3*sqrt(c + d*x)/(64*b*(a + b*x)*(a*d - b*c)**3) + 5*d**2*sqrt(c + d*x)/(96*
b*(a + b*x)**2*(a*d - b*c)**2) + d*sqrt(c + d*x)/(24*b*(a + b*x)**3*(a*d - b*c))
- sqrt(c + d*x)/(4*b*(a + b*x)**4) + 5*d**4*atan(sqrt(b)*sqrt(c + d*x)/sqrt(a*d
- b*c))/(64*b**(3/2)*(a*d - b*c)**(7/2))

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Mathematica [A]  time = 0.309561, size = 149, normalized size = 0.82 \[ \frac{5 d^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{64 b^{3/2} (b c-a d)^{7/2}}-\frac{\sqrt{c+d x} \left (10 d^2 (a+b x)^2 (a d-b c)+8 d (a+b x) (b c-a d)^2+48 (b c-a d)^3+15 d^3 (a+b x)^3\right )}{192 b (a+b x)^4 (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[c + d*x]/(a + b*x)^5,x]

[Out]

-(Sqrt[c + d*x]*(48*(b*c - a*d)^3 + 8*d*(b*c - a*d)^2*(a + b*x) + 10*d^2*(-(b*c)
+ a*d)*(a + b*x)^2 + 15*d^3*(a + b*x)^3))/(192*b*(b*c - a*d)^3*(a + b*x)^4) + (
5*d^4*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(64*b^(3/2)*(b*c - a*d)^
(7/2))

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Maple [A]  time = 0.02, size = 248, normalized size = 1.4 \[{\frac{5\,{d}^{4}{b}^{2}}{64\, \left ( bdx+ad \right ) ^{4} \left ({a}^{3}{d}^{3}-3\,{a}^{2}bc{d}^{2}+3\,a{b}^{2}{c}^{2}d-{b}^{3}{c}^{3} \right ) } \left ( dx+c \right ) ^{{\frac{7}{2}}}}+{\frac{55\,{d}^{4}b}{192\, \left ( bdx+ad \right ) ^{4} \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ) } \left ( dx+c \right ) ^{{\frac{5}{2}}}}+{\frac{73\,{d}^{4}}{192\, \left ( bdx+ad \right ) ^{4} \left ( ad-bc \right ) } \left ( dx+c \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{d}^{4}}{64\, \left ( bdx+ad \right ) ^{4}b}\sqrt{dx+c}}+{\frac{5\,{d}^{4}}{64\,b \left ({a}^{3}{d}^{3}-3\,{a}^{2}bc{d}^{2}+3\,a{b}^{2}{c}^{2}d-{b}^{3}{c}^{3} \right ) }\arctan \left ({b\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(1/2)/(b*x+a)^5,x)

[Out]

5/64*d^4/(b*d*x+a*d)^4*b^2/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*(d*x+c)
^(7/2)+55/192*d^4/(b*d*x+a*d)^4*b/(a^2*d^2-2*a*b*c*d+b^2*c^2)*(d*x+c)^(5/2)+73/1
92*d^4/(b*d*x+a*d)^4/(a*d-b*c)*(d*x+c)^(3/2)-5/64*d^4/(b*d*x+a*d)^4/b*(d*x+c)^(1
/2)+5/64*d^4/b/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*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(sqrt(d*x + c)/(b*x + a)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.232987, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x + c)/(b*x + a)^5,x, algorithm="fricas")

[Out]

[-1/384*(2*(15*b^3*d^3*x^3 + 48*b^3*c^3 - 136*a*b^2*c^2*d + 118*a^2*b*c*d^2 - 15
*a^3*d^3 - 5*(2*b^3*c*d^2 - 11*a*b^2*d^3)*x^2 + (8*b^3*c^2*d - 36*a*b^2*c*d^2 +
73*a^2*b*d^3)*x)*sqrt(b^2*c - a*b*d)*sqrt(d*x + c) + 15*(b^4*d^4*x^4 + 4*a*b^3*d
^4*x^3 + 6*a^2*b^2*d^4*x^2 + 4*a^3*b*d^4*x + a^4*d^4)*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^4*b^4*c^
3 - 3*a^5*b^3*c^2*d + 3*a^6*b^2*c*d^2 - a^7*b*d^3 + (b^8*c^3 - 3*a*b^7*c^2*d + 3
*a^2*b^6*c*d^2 - a^3*b^5*d^3)*x^4 + 4*(a*b^7*c^3 - 3*a^2*b^6*c^2*d + 3*a^3*b^5*c
*d^2 - a^4*b^4*d^3)*x^3 + 6*(a^2*b^6*c^3 - 3*a^3*b^5*c^2*d + 3*a^4*b^4*c*d^2 - a
^5*b^3*d^3)*x^2 + 4*(a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^5*b^3*c*d^2 - a^6*b^2*d
^3)*x)*sqrt(b^2*c - a*b*d)), -1/192*((15*b^3*d^3*x^3 + 48*b^3*c^3 - 136*a*b^2*c^
2*d + 118*a^2*b*c*d^2 - 15*a^3*d^3 - 5*(2*b^3*c*d^2 - 11*a*b^2*d^3)*x^2 + (8*b^3
*c^2*d - 36*a*b^2*c*d^2 + 73*a^2*b*d^3)*x)*sqrt(-b^2*c + a*b*d)*sqrt(d*x + c) -
15*(b^4*d^4*x^4 + 4*a*b^3*d^4*x^3 + 6*a^2*b^2*d^4*x^2 + 4*a^3*b*d^4*x + a^4*d^4)
*arctan(-(b*c - a*d)/(sqrt(-b^2*c + a*b*d)*sqrt(d*x + c))))/((a^4*b^4*c^3 - 3*a^
5*b^3*c^2*d + 3*a^6*b^2*c*d^2 - a^7*b*d^3 + (b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6
*c*d^2 - a^3*b^5*d^3)*x^4 + 4*(a*b^7*c^3 - 3*a^2*b^6*c^2*d + 3*a^3*b^5*c*d^2 - a
^4*b^4*d^3)*x^3 + 6*(a^2*b^6*c^3 - 3*a^3*b^5*c^2*d + 3*a^4*b^4*c*d^2 - a^5*b^3*d
^3)*x^2 + 4*(a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^5*b^3*c*d^2 - a^6*b^2*d^3)*x)*s
qrt(-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)**5,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.231156, size = 420, normalized size = 2.31 \[ -\frac{5 \, d^{4} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{64 \,{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} \sqrt{-b^{2} c + a b d}} - \frac{15 \,{\left (d x + c\right )}^{\frac{7}{2}} b^{3} d^{4} - 55 \,{\left (d x + c\right )}^{\frac{5}{2}} b^{3} c d^{4} + 73 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{3} c^{2} d^{4} + 15 \, \sqrt{d x + c} b^{3} c^{3} d^{4} + 55 \,{\left (d x + c\right )}^{\frac{5}{2}} a b^{2} d^{5} - 146 \,{\left (d x + c\right )}^{\frac{3}{2}} a b^{2} c d^{5} - 45 \, \sqrt{d x + c} a b^{2} c^{2} d^{5} + 73 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{2} b d^{6} + 45 \, \sqrt{d x + c} a^{2} b c d^{6} - 15 \, \sqrt{d x + c} a^{3} d^{7}}{192 \,{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )}{\left ({\left (d x + c\right )} b - b c + a d\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x + c)/(b*x + a)^5,x, algorithm="giac")

[Out]

-5/64*d^4*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/((b^4*c^3 - 3*a*b^3*c^2*d
+ 3*a^2*b^2*c*d^2 - a^3*b*d^3)*sqrt(-b^2*c + a*b*d)) - 1/192*(15*(d*x + c)^(7/2
)*b^3*d^4 - 55*(d*x + c)^(5/2)*b^3*c*d^4 + 73*(d*x + c)^(3/2)*b^3*c^2*d^4 + 15*s
qrt(d*x + c)*b^3*c^3*d^4 + 55*(d*x + c)^(5/2)*a*b^2*d^5 - 146*(d*x + c)^(3/2)*a*
b^2*c*d^5 - 45*sqrt(d*x + c)*a*b^2*c^2*d^5 + 73*(d*x + c)^(3/2)*a^2*b*d^6 + 45*s
qrt(d*x + c)*a^2*b*c*d^6 - 15*sqrt(d*x + c)*a^3*d^7)/((b^4*c^3 - 3*a*b^3*c^2*d +
3*a^2*b^2*c*d^2 - a^3*b*d^3)*((d*x + c)*b - b*c + a*d)^4)