3.1409 \(\int \frac{(c+d x)^{5/2}}{(a+b x)^5} \, dx\)

Optimal. Leaf size=162 \[ \frac{5 d^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{64 b^{7/2} (b c-a d)^{3/2}}-\frac{5 d^3 \sqrt{c+d x}}{64 b^3 (a+b x) (b c-a d)}-\frac{5 d^2 \sqrt{c+d x}}{32 b^3 (a+b x)^2}-\frac{5 d (c+d x)^{3/2}}{24 b^2 (a+b x)^3}-\frac{(c+d x)^{5/2}}{4 b (a+b x)^4} \]

[Out]

(-5*d^2*Sqrt[c + d*x])/(32*b^3*(a + b*x)^2) - (5*d^3*Sqrt[c + d*x])/(64*b^3*(b*c
- a*d)*(a + b*x)) - (5*d*(c + d*x)^(3/2))/(24*b^2*(a + b*x)^3) - (c + d*x)^(5/2
)/(4*b*(a + b*x)^4) + (5*d^4*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(
64*b^(7/2)*(b*c - a*d)^(3/2))

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Rubi [A]  time = 0.208954, antiderivative size = 162, 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 \[ \frac{5 d^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{64 b^{7/2} (b c-a d)^{3/2}}-\frac{5 d^3 \sqrt{c+d x}}{64 b^3 (a+b x) (b c-a d)}-\frac{5 d^2 \sqrt{c+d x}}{32 b^3 (a+b x)^2}-\frac{5 d (c+d x)^{3/2}}{24 b^2 (a+b x)^3}-\frac{(c+d x)^{5/2}}{4 b (a+b x)^4} \]

Antiderivative was successfully verified.

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

[Out]

(-5*d^2*Sqrt[c + d*x])/(32*b^3*(a + b*x)^2) - (5*d^3*Sqrt[c + d*x])/(64*b^3*(b*c
- a*d)*(a + b*x)) - (5*d*(c + d*x)^(3/2))/(24*b^2*(a + b*x)^3) - (c + d*x)^(5/2
)/(4*b*(a + b*x)^4) + (5*d^4*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(
64*b^(7/2)*(b*c - a*d)^(3/2))

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Rubi in Sympy [A]  time = 39.2165, size = 144, normalized size = 0.89 \[ - \frac{\left (c + d x\right )^{\frac{5}{2}}}{4 b \left (a + b x\right )^{4}} - \frac{5 d \left (c + d x\right )^{\frac{3}{2}}}{24 b^{2} \left (a + b x\right )^{3}} + \frac{5 d^{3} \sqrt{c + d x}}{64 b^{3} \left (a + b x\right ) \left (a d - b c\right )} - \frac{5 d^{2} \sqrt{c + d x}}{32 b^{3} \left (a + b x\right )^{2}} + \frac{5 d^{4} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{64 b^{\frac{7}{2}} \left (a d - b c\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(c + d*x)**(5/2)/(4*b*(a + b*x)**4) - 5*d*(c + d*x)**(3/2)/(24*b**2*(a + b*x)**
3) + 5*d**3*sqrt(c + d*x)/(64*b**3*(a + b*x)*(a*d - b*c)) - 5*d**2*sqrt(c + d*x)
/(32*b**3*(a + b*x)**2) + 5*d**4*atan(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c))/(64
*b**(7/2)*(a*d - b*c)**(3/2))

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Mathematica [A]  time = 0.248643, size = 149, normalized size = 0.92 \[ \frac{5 d^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{64 b^{7/2} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x} \left (118 d^2 (a+b x)^2 (b c-a d)+136 d (a+b x) (b c-a d)^2+48 (b c-a d)^3+15 d^3 (a+b x)^3\right )}{192 b^3 (a+b x)^4 (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[c + d*x]*(48*(b*c - a*d)^3 + 136*d*(b*c - a*d)^2*(a + b*x) + 118*d^2*(b*c
- a*d)*(a + b*x)^2 + 15*d^3*(a + b*x)^3))/(192*b^3*(b*c - a*d)*(a + b*x)^4) + (
5*d^4*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(64*b^(7/2)*(b*c - a*d)^
(3/2))

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Maple [A]  time = 0.021, size = 246, normalized size = 1.5 \[{\frac{5\,{d}^{4}}{64\, \left ( bdx+ad \right ) ^{4} \left ( ad-bc \right ) } \left ( dx+c \right ) ^{{\frac{7}{2}}}}-{\frac{73\,{d}^{4}}{192\, \left ( bdx+ad \right ) ^{4}b} \left ( dx+c \right ) ^{{\frac{5}{2}}}}-{\frac{55\,{d}^{5}a}{192\, \left ( bdx+ad \right ) ^{4}{b}^{2}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}+{\frac{55\,{d}^{4}c}{192\, \left ( bdx+ad \right ) ^{4}b} \left ( dx+c \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{d}^{6}{a}^{2}}{64\, \left ( bdx+ad \right ) ^{4}{b}^{3}}\sqrt{dx+c}}+{\frac{5\,{d}^{5}ac}{32\, \left ( bdx+ad \right ) ^{4}{b}^{2}}\sqrt{dx+c}}-{\frac{5\,{d}^{4}{c}^{2}}{64\, \left ( bdx+ad \right ) ^{4}b}\sqrt{dx+c}}+{\frac{5\,{d}^{4}}{ \left ( 64\,ad-64\,bc \right ){b}^{3}}\arctan \left ({b\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(5/2)/(b*x+a)^5,x)

[Out]

5/64*d^4/(b*d*x+a*d)^4/(a*d-b*c)*(d*x+c)^(7/2)-73/192*d^4/(b*d*x+a*d)^4/b*(d*x+c
)^(5/2)-55/192*d^5/(b*d*x+a*d)^4/b^2*(d*x+c)^(3/2)*a+55/192*d^4/(b*d*x+a*d)^4/b*
(d*x+c)^(3/2)*c-5/64*d^6/(b*d*x+a*d)^4/b^3*(d*x+c)^(1/2)*a^2+5/32*d^5/(b*d*x+a*d
)^4/b^2*(d*x+c)^(1/2)*a*c-5/64*d^4/(b*d*x+a*d)^4/b*(d*x+c)^(1/2)*c^2+5/64*d^4/(a
*d-b*c)/b^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((d*x + c)^(5/2)/(b*x + a)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.235507, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (15 \, b^{3} d^{3} x^{3} + 48 \, b^{3} c^{3} - 8 \, a b^{2} c^{2} d - 10 \, a^{2} b c d^{2} - 15 \, a^{3} d^{3} +{\left (118 \, b^{3} c d^{2} - 73 \, a b^{2} d^{3}\right )} x^{2} +{\left (136 \, b^{3} c^{2} d - 36 \, a b^{2} c d^{2} - 55 \, a^{2} b d^{3}\right )} x\right )} \sqrt{b^{2} c - a b d} \sqrt{d x + c} + 15 \,{\left (b^{4} d^{4} x^{4} + 4 \, a b^{3} d^{4} x^{3} + 6 \, a^{2} b^{2} d^{4} x^{2} + 4 \, a^{3} b d^{4} x + a^{4} d^{4}\right )} \log \left (\frac{\sqrt{b^{2} c - a b d}{\left (b d x + 2 \, b c - a d\right )} - 2 \,{\left (b^{2} c - a b d\right )} \sqrt{d x + c}}{b x + a}\right )}{384 \,{\left (a^{4} b^{4} c - a^{5} b^{3} d +{\left (b^{8} c - a b^{7} d\right )} x^{4} + 4 \,{\left (a b^{7} c - a^{2} b^{6} d\right )} x^{3} + 6 \,{\left (a^{2} b^{6} c - a^{3} b^{5} d\right )} x^{2} + 4 \,{\left (a^{3} b^{5} c - a^{4} b^{4} d\right )} x\right )} \sqrt{b^{2} c - a b d}}, -\frac{{\left (15 \, b^{3} d^{3} x^{3} + 48 \, b^{3} c^{3} - 8 \, a b^{2} c^{2} d - 10 \, a^{2} b c d^{2} - 15 \, a^{3} d^{3} +{\left (118 \, b^{3} c d^{2} - 73 \, a b^{2} d^{3}\right )} x^{2} +{\left (136 \, b^{3} c^{2} d - 36 \, a b^{2} c d^{2} - 55 \, a^{2} b d^{3}\right )} x\right )} \sqrt{-b^{2} c + a b d} \sqrt{d x + c} - 15 \,{\left (b^{4} d^{4} x^{4} + 4 \, a b^{3} d^{4} x^{3} + 6 \, a^{2} b^{2} d^{4} x^{2} + 4 \, a^{3} b d^{4} x + a^{4} d^{4}\right )} \arctan \left (-\frac{b c - a d}{\sqrt{-b^{2} c + a b d} \sqrt{d x + c}}\right )}{192 \,{\left (a^{4} b^{4} c - a^{5} b^{3} d +{\left (b^{8} c - a b^{7} d\right )} x^{4} + 4 \,{\left (a b^{7} c - a^{2} b^{6} d\right )} x^{3} + 6 \,{\left (a^{2} b^{6} c - a^{3} b^{5} d\right )} x^{2} + 4 \,{\left (a^{3} b^{5} c - a^{4} b^{4} d\right )} x\right )} \sqrt{-b^{2} c + a b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/2)/(b*x + a)^5,x, algorithm="fricas")

[Out]

[-1/384*(2*(15*b^3*d^3*x^3 + 48*b^3*c^3 - 8*a*b^2*c^2*d - 10*a^2*b*c*d^2 - 15*a^
3*d^3 + (118*b^3*c*d^2 - 73*a*b^2*d^3)*x^2 + (136*b^3*c^2*d - 36*a*b^2*c*d^2 - 5
5*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 -
a^5*b^3*d + (b^8*c - a*b^7*d)*x^4 + 4*(a*b^7*c - a^2*b^6*d)*x^3 + 6*(a^2*b^6*c
- a^3*b^5*d)*x^2 + 4*(a^3*b^5*c - a^4*b^4*d)*x)*sqrt(b^2*c - a*b*d)), -1/192*((1
5*b^3*d^3*x^3 + 48*b^3*c^3 - 8*a*b^2*c^2*d - 10*a^2*b*c*d^2 - 15*a^3*d^3 + (118*
b^3*c*d^2 - 73*a*b^2*d^3)*x^2 + (136*b^3*c^2*d - 36*a*b^2*c*d^2 - 55*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 - a^5*b^3*d + (b^8*c - a*b^7*d)*x^4 + 4*(a*b^7*c
- a^2*b^6*d)*x^3 + 6*(a^2*b^6*c - a^3*b^5*d)*x^2 + 4*(a^3*b^5*c - a^4*b^4*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)**(5/2)/(b*x+a)**5,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.238295, size = 350, normalized size = 2.16 \[ -\frac{5 \, d^{4} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{64 \,{\left (b^{4} c - a b^{3} d\right )} \sqrt{-b^{2} c + a b d}} - \frac{15 \,{\left (d x + c\right )}^{\frac{7}{2}} b^{3} d^{4} + 73 \,{\left (d x + c\right )}^{\frac{5}{2}} b^{3} c d^{4} - 55 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{3} c^{2} d^{4} + 15 \, \sqrt{d x + c} b^{3} c^{3} d^{4} - 73 \,{\left (d x + c\right )}^{\frac{5}{2}} a b^{2} d^{5} + 110 \,{\left (d x + c\right )}^{\frac{3}{2}} a b^{2} c d^{5} - 45 \, \sqrt{d x + c} a b^{2} c^{2} d^{5} - 55 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{2} b d^{6} + 45 \, \sqrt{d x + c} a^{2} b c d^{6} - 15 \, \sqrt{d x + c} a^{3} d^{7}}{192 \,{\left (b^{4} c - a b^{3} d\right )}{\left ({\left (d x + c\right )} b - b c + a d\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/2)/(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 - a*b^3*d)*sqrt(-
b^2*c + a*b*d)) - 1/192*(15*(d*x + c)^(7/2)*b^3*d^4 + 73*(d*x + c)^(5/2)*b^3*c*d
^4 - 55*(d*x + c)^(3/2)*b^3*c^2*d^4 + 15*sqrt(d*x + c)*b^3*c^3*d^4 - 73*(d*x + c
)^(5/2)*a*b^2*d^5 + 110*(d*x + c)^(3/2)*a*b^2*c*d^5 - 45*sqrt(d*x + c)*a*b^2*c^2
*d^5 - 55*(d*x + c)^(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)/((b^4*c - a*b^3*d)*((d*x + c)*b - b*c + a*d)^4)