3.2086 \(\int \frac{(a+b x) (d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\)

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

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.230086, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152 \[ \frac{5 e^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{7/2} (b d-a e)^{3/2}}-\frac{5 e^3 \sqrt{d+e x}}{64 b^3 (a+b x) (b d-a e)}-\frac{5 e^2 \sqrt{d+e x}}{32 b^3 (a+b x)^2}-\frac{5 e (d+e x)^{3/2}}{24 b^2 (a+b x)^3}-\frac{(d+e x)^{5/2}}{4 b (a+b x)^4} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(d + e*x)^(5/2))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 65.5475, size = 144, normalized size = 0.89 \[ - \frac{\left (d + e x\right )^{\frac{5}{2}}}{4 b \left (a + b x\right )^{4}} - \frac{5 e \left (d + e x\right )^{\frac{3}{2}}}{24 b^{2} \left (a + b x\right )^{3}} + \frac{5 e^{3} \sqrt{d + e x}}{64 b^{3} \left (a + b x\right ) \left (a e - b d\right )} - \frac{5 e^{2} \sqrt{d + e x}}{32 b^{3} \left (a + b x\right )^{2}} + \frac{5 e^{4} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{64 b^{\frac{7}{2}} \left (a e - b d\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(e*x+d)**(5/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.269413, size = 149, normalized size = 0.92 \[ \frac{5 e^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{7/2} (b d-a e)^{3/2}}-\frac{\sqrt{d+e x} \left (118 e^2 (a+b x)^2 (b d-a e)+136 e (a+b x) (b d-a e)^2+48 (b d-a e)^3+15 e^3 (a+b x)^3\right )}{192 b^3 (a+b x)^4 (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(d + e*x)^(5/2))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.022, size = 246, normalized size = 1.5 \[{\frac{5\,{e}^{4}}{64\, \left ( bex+ae \right ) ^{4} \left ( ae-bd \right ) } \left ( ex+d \right ) ^{{\frac{7}{2}}}}-{\frac{73\,{e}^{4}}{192\, \left ( bex+ae \right ) ^{4}b} \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{55\,a{e}^{5}}{192\, \left ( bex+ae \right ) ^{4}{b}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{55\,{e}^{4}d}{192\, \left ( bex+ae \right ) ^{4}b} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{a}^{2}{e}^{6}}{64\, \left ( bex+ae \right ) ^{4}{b}^{3}}\sqrt{ex+d}}+{\frac{5\,a{e}^{5}d}{32\, \left ( bex+ae \right ) ^{4}{b}^{2}}\sqrt{ex+d}}-{\frac{5\,{e}^{4}{d}^{2}}{64\, \left ( bex+ae \right ) ^{4}b}\sqrt{ex+d}}+{\frac{5\,{e}^{4}}{64\,{b}^{3} \left ( ae-bd \right ) }\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(e*x+d)^(5/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

5/64*e^4/(b*e*x+a*e)^4/(a*e-b*d)*(e*x+d)^(7/2)-73/192*e^4/(b*e*x+a*e)^4/b*(e*x+d
)^(5/2)-55/192*e^5/(b*e*x+a*e)^4/b^2*(e*x+d)^(3/2)*a+55/192*e^4/(b*e*x+a*e)^4/b*
(e*x+d)^(3/2)*d-5/64*e^6/(b*e*x+a*e)^4/b^3*(e*x+d)^(1/2)*a^2+5/32*e^5/(b*e*x+a*e
)^4/b^2*(e*x+d)^(1/2)*a*d-5/64*e^4/(b*e*x+a*e)^4/b*(e*x+d)^(1/2)*d^2+5/64*e^4/b^
3/(a*e-b*d)/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))

_______________________________________________________________________________________

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((b*x + a)*(e*x + d)^(5/2)/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.306193, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (15 \, b^{3} e^{3} x^{3} + 48 \, b^{3} d^{3} - 8 \, a b^{2} d^{2} e - 10 \, a^{2} b d e^{2} - 15 \, a^{3} e^{3} +{\left (118 \, b^{3} d e^{2} - 73 \, a b^{2} e^{3}\right )} x^{2} +{\left (136 \, b^{3} d^{2} e - 36 \, a b^{2} d e^{2} - 55 \, a^{2} b e^{3}\right )} x\right )} \sqrt{b^{2} d - a b e} \sqrt{e x + d} + 15 \,{\left (b^{4} e^{4} x^{4} + 4 \, a b^{3} e^{4} x^{3} + 6 \, a^{2} b^{2} e^{4} x^{2} + 4 \, a^{3} b e^{4} x + a^{4} e^{4}\right )} \log \left (\frac{\sqrt{b^{2} d - a b e}{\left (b e x + 2 \, b d - a e\right )} - 2 \,{\left (b^{2} d - a b e\right )} \sqrt{e x + d}}{b x + a}\right )}{384 \,{\left (a^{4} b^{4} d - a^{5} b^{3} e +{\left (b^{8} d - a b^{7} e\right )} x^{4} + 4 \,{\left (a b^{7} d - a^{2} b^{6} e\right )} x^{3} + 6 \,{\left (a^{2} b^{6} d - a^{3} b^{5} e\right )} x^{2} + 4 \,{\left (a^{3} b^{5} d - a^{4} b^{4} e\right )} x\right )} \sqrt{b^{2} d - a b e}}, -\frac{{\left (15 \, b^{3} e^{3} x^{3} + 48 \, b^{3} d^{3} - 8 \, a b^{2} d^{2} e - 10 \, a^{2} b d e^{2} - 15 \, a^{3} e^{3} +{\left (118 \, b^{3} d e^{2} - 73 \, a b^{2} e^{3}\right )} x^{2} +{\left (136 \, b^{3} d^{2} e - 36 \, a b^{2} d e^{2} - 55 \, a^{2} b e^{3}\right )} x\right )} \sqrt{-b^{2} d + a b e} \sqrt{e x + d} - 15 \,{\left (b^{4} e^{4} x^{4} + 4 \, a b^{3} e^{4} x^{3} + 6 \, a^{2} b^{2} e^{4} x^{2} + 4 \, a^{3} b e^{4} x + a^{4} e^{4}\right )} \arctan \left (-\frac{b d - a e}{\sqrt{-b^{2} d + a b e} \sqrt{e x + d}}\right )}{192 \,{\left (a^{4} b^{4} d - a^{5} b^{3} e +{\left (b^{8} d - a b^{7} e\right )} x^{4} + 4 \,{\left (a b^{7} d - a^{2} b^{6} e\right )} x^{3} + 6 \,{\left (a^{2} b^{6} d - a^{3} b^{5} e\right )} x^{2} + 4 \,{\left (a^{3} b^{5} d - a^{4} b^{4} e\right )} x\right )} \sqrt{-b^{2} d + a b e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(e*x + d)^(5/2)/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

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((b*x+a)*(e*x+d)**(5/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.295659, size = 358, normalized size = 2.21 \[ -\frac{5 \, \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{4}}{64 \,{\left (b^{4} d - a b^{3} e\right )} \sqrt{-b^{2} d + a b e}} - \frac{15 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{3} e^{4} + 73 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{3} d e^{4} - 55 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} d^{2} e^{4} + 15 \, \sqrt{x e + d} b^{3} d^{3} e^{4} - 73 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{2} e^{5} + 110 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{2} d e^{5} - 45 \, \sqrt{x e + d} a b^{2} d^{2} e^{5} - 55 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b e^{6} + 45 \, \sqrt{x e + d} a^{2} b d e^{6} - 15 \, \sqrt{x e + d} a^{3} e^{7}}{192 \,{\left (b^{4} d - a b^{3} e\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(e*x + d)^(5/2)/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="giac")

[Out]

-5/64*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^4/((b^4*d - a*b^3*e)*sqrt(-
b^2*d + a*b*e)) - 1/192*(15*(x*e + d)^(7/2)*b^3*e^4 + 73*(x*e + d)^(5/2)*b^3*d*e
^4 - 55*(x*e + d)^(3/2)*b^3*d^2*e^4 + 15*sqrt(x*e + d)*b^3*d^3*e^4 - 73*(x*e + d
)^(5/2)*a*b^2*e^5 + 110*(x*e + d)^(3/2)*a*b^2*d*e^5 - 45*sqrt(x*e + d)*a*b^2*d^2
*e^5 - 55*(x*e + d)^(3/2)*a^2*b*e^6 + 45*sqrt(x*e + d)*a^2*b*d*e^6 - 15*sqrt(x*e
+ d)*a^3*e^7)/((b^4*d - a*b^3*e)*((x*e + d)*b - b*d + a*e)^4)