3.2067 \(\int \frac{(a+b x) (d+e x)^{5/2}}{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=112 \[ -\frac{2 (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{7/2}}+\frac{2 \sqrt{d+e x} (b d-a e)^2}{b^3}+\frac{2 (d+e x)^{3/2} (b d-a e)}{3 b^2}+\frac{2 (d+e x)^{5/2}}{5 b} \]

[Out]

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

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Rubi [A]  time = 0.152019, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121 \[ -\frac{2 (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{7/2}}+\frac{2 \sqrt{d+e x} (b d-a e)^2}{b^3}+\frac{2 (d+e x)^{3/2} (b d-a e)}{3 b^2}+\frac{2 (d+e x)^{5/2}}{5 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 45.6926, size = 99, normalized size = 0.88 \[ \frac{2 \left (d + e x\right )^{\frac{5}{2}}}{5 b} - \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )}{3 b^{2}} + \frac{2 \sqrt{d + e x} \left (a e - b d\right )^{2}}{b^{3}} - \frac{2 \left (a e - b d\right )^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{b^{\frac{7}{2}}} \]

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),x)

[Out]

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

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Mathematica [A]  time = 0.130369, size = 108, normalized size = 0.96 \[ \frac{2 \sqrt{d+e x} \left (15 a^2 e^2-5 a b e (7 d+e x)+b^2 \left (23 d^2+11 d e x+3 e^2 x^2\right )\right )}{15 b^3}-\frac{2 (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(15*a^2*e^2 - 5*a*b*e*(7*d + e*x) + b^2*(23*d^2 + 11*d*e*x + 3*
e^2*x^2)))/(15*b^3) - (2*(b*d - a*e)^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[
b*d - a*e]])/b^(7/2)

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Maple [B]  time = 0.009, size = 263, normalized size = 2.4 \[{\frac{2}{5\,b} \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{2\,ae}{3\,{b}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{2\,d}{3\,b} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+2\,{\frac{{a}^{2}{e}^{2}\sqrt{ex+d}}{{b}^{3}}}-4\,{\frac{aed\sqrt{ex+d}}{{b}^{2}}}+2\,{\frac{{d}^{2}\sqrt{ex+d}}{b}}-2\,{\frac{{a}^{3}{e}^{3}}{{b}^{3}\sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) }+6\,{\frac{{a}^{2}d{e}^{2}}{{b}^{2}\sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) }-6\,{\frac{a{d}^{2}e}{b\sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) }+2\,{\frac{{d}^{3}}{\sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) } \]

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),x)

[Out]

2/5*(e*x+d)^(5/2)/b-2/3/b^2*a*e*(e*x+d)^(3/2)+2/3/b*d*(e*x+d)^(3/2)+2/b^3*a^2*e^
2*(e*x+d)^(1/2)-4/b^2*d*e*a*(e*x+d)^(1/2)+2/b*d^2*(e*x+d)^(1/2)-2/b^3/(b*(a*e-b*
d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^3*e^3+6/b^2/(b*(a*e-b*d)
)^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^2*d*e^2-6/b/(b*(a*e-b*d))^
(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a*d^2*e+2/(b*(a*e-b*d))^(1/2)*
arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*d^3

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.3026, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \sqrt{\frac{b d - a e}{b}} \log \left (\frac{b e x + 2 \, b d - a e - 2 \, \sqrt{e x + d} b \sqrt{\frac{b d - a e}{b}}}{b x + a}\right ) + 2 \,{\left (3 \, b^{2} e^{2} x^{2} + 23 \, b^{2} d^{2} - 35 \, a b d e + 15 \, a^{2} e^{2} +{\left (11 \, b^{2} d e - 5 \, a b e^{2}\right )} x\right )} \sqrt{e x + d}}{15 \, b^{3}}, -\frac{2 \,{\left (15 \,{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \sqrt{-\frac{b d - a e}{b}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{b d - a e}{b}}}\right ) -{\left (3 \, b^{2} e^{2} x^{2} + 23 \, b^{2} d^{2} - 35 \, a b d e + 15 \, a^{2} e^{2} +{\left (11 \, b^{2} d e - 5 \, a b e^{2}\right )} x\right )} \sqrt{e x + d}\right )}}{15 \, b^{3}}\right ] \]

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),x, algorithm="fricas")

[Out]

[1/15*(15*(b^2*d^2 - 2*a*b*d*e + a^2*e^2)*sqrt((b*d - a*e)/b)*log((b*e*x + 2*b*d
- a*e - 2*sqrt(e*x + d)*b*sqrt((b*d - a*e)/b))/(b*x + a)) + 2*(3*b^2*e^2*x^2 +
23*b^2*d^2 - 35*a*b*d*e + 15*a^2*e^2 + (11*b^2*d*e - 5*a*b*e^2)*x)*sqrt(e*x + d)
)/b^3, -2/15*(15*(b^2*d^2 - 2*a*b*d*e + a^2*e^2)*sqrt(-(b*d - a*e)/b)*arctan(sqr
t(e*x + d)/sqrt(-(b*d - a*e)/b)) - (3*b^2*e^2*x^2 + 23*b^2*d^2 - 35*a*b*d*e + 15
*a^2*e^2 + (11*b^2*d*e - 5*a*b*e^2)*x)*sqrt(e*x + d))/b^3]

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Sympy [A]  time = 128.022, size = 240, normalized size = 2.14 \[ \frac{2 \left (d + e x\right )^{\frac{5}{2}}}{5 b} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (- 2 a e + 2 b d\right )}{3 b^{2}} + \frac{\sqrt{d + e x} \left (2 a^{2} e^{2} - 4 a b d e + 2 b^{2} d^{2}\right )}{b^{3}} - \frac{2 \left (a e - b d\right )^{3} \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{a e - b d}{b}}} \right )}}{b \sqrt{\frac{a e - b d}{b}}} & \text{for}\: \frac{a e - b d}{b} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{- a e + b d}{b}}} \right )}}{b \sqrt{\frac{- a e + b d}{b}}} & \text{for}\: d + e x > \frac{- a e + b d}{b} \wedge \frac{a e - b d}{b} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{- a e + b d}{b}}} \right )}}{b \sqrt{\frac{- a e + b d}{b}}} & \text{for}\: \frac{a e - b d}{b} < 0 \wedge d + e x < \frac{- a e + b d}{b} \end{cases}\right )}{b^{3}} \]

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),x)

[Out]

2*(d + e*x)**(5/2)/(5*b) + (d + e*x)**(3/2)*(-2*a*e + 2*b*d)/(3*b**2) + sqrt(d +
e*x)*(2*a**2*e**2 - 4*a*b*d*e + 2*b**2*d**2)/b**3 - 2*(a*e - b*d)**3*Piecewise(
(atan(sqrt(d + e*x)/sqrt((a*e - b*d)/b))/(b*sqrt((a*e - b*d)/b)), (a*e - b*d)/b
> 0), (-acoth(sqrt(d + e*x)/sqrt((-a*e + b*d)/b))/(b*sqrt((-a*e + b*d)/b)), ((a*
e - b*d)/b < 0) & (d + e*x > (-a*e + b*d)/b)), (-atanh(sqrt(d + e*x)/sqrt((-a*e
+ b*d)/b))/(b*sqrt((-a*e + b*d)/b)), ((a*e - b*d)/b < 0) & (d + e*x < (-a*e + b*
d)/b)))/b**3

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GIAC/XCAS [A]  time = 0.281252, size = 243, normalized size = 2.17 \[ \frac{2 \,{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e} b^{3}} + \frac{2 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{4} + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{4} d + 15 \, \sqrt{x e + d} b^{4} d^{2} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{3} e - 30 \, \sqrt{x e + d} a b^{3} d e + 15 \, \sqrt{x e + d} a^{2} b^{2} e^{2}\right )}}{15 \, b^{5}} \]

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),x, algorithm="giac")

[Out]

2*(b^3*d^3 - 3*a*b^2*d^2*e + 3*a^2*b*d*e^2 - a^3*e^3)*arctan(sqrt(x*e + d)*b/sqr
t(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^3) + 2/15*(3*(x*e + d)^(5/2)*b^4 + 5*
(x*e + d)^(3/2)*b^4*d + 15*sqrt(x*e + d)*b^4*d^2 - 5*(x*e + d)^(3/2)*a*b^3*e - 3
0*sqrt(x*e + d)*a*b^3*d*e + 15*sqrt(x*e + d)*a^2*b^2*e^2)/b^5