3.1693 \(\int \frac{(d+e x)^{5/2}}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=212 \[ \frac{2 (a+b x) (d+e x)^{3/2} (b d-a e)}{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (a+b x) (d+e x)^{5/2}}{5 b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 (a+b x) (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (a+b x) \sqrt{d+e x} (b d-a e)^2}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.321017, antiderivative size = 212, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{2 (a+b x) (d+e x)^{3/2} (b d-a e)}{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (a+b x) (d+e x)^{5/2}}{5 b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 (a+b x) (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (a+b x) \sqrt{d+e x} (b d-a e)^2}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.230632, size = 127, normalized size = 0.6 \[ \frac{2 (a+b x) \left (\sqrt{b} \sqrt{d+e x} \left (15 a^2 e^2-5 a b e (7 d+e x)+b^2 \left (23 d^2+11 d e x+3 e^2 x^2\right )\right )-15 (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )\right )}{15 b^{7/2} \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.012, size = 309, normalized size = 1.5 \[{\frac{2\,bx+2\,a}{15\,{b}^{3}} \left ( 3\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{5/2}{b}^{2}-5\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{3/2}abe+5\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{3/2}{b}^{2}d-15\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){a}^{3}{e}^{3}+45\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){a}^{2}bd{e}^{2}-45\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) a{b}^{2}{d}^{2}e+15\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){b}^{3}{d}^{3}+15\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{a}^{2}{e}^{2}-30\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}abde+15\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{b}^{2}{d}^{2} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(5/2)/((b*x+a)^2)^(1/2),x)

[Out]

2/15*(b*x+a)*(3*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*b^2-5*(b*(a*e-b*d))^(1/2)*(e*x
+d)^(3/2)*a*b*e+5*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*b^2*d-15*arctan((e*x+d)^(1/2
)*b/(b*(a*e-b*d))^(1/2))*a^3*e^3+45*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*
a^2*b*d*e^2-45*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a*b^2*d^2*e+15*arctan
((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*b^3*d^3+15*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/
2)*a^2*e^2-30*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a*b*d*e+15*(b*(a*e-b*d))^(1/2)*(
e*x+d)^(1/2)*b^2*d^2)/((b*x+a)^2)^(1/2)/b^3/(b*(a*e-b*d))^(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((e*x + d)^(5/2)/sqrt((b*x + a)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.219308, size = 1, normalized size = 0. \[ \left [\frac{15 \,{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \sqrt{\frac{b d - a e}{b}} \log \left (\frac{b e x + 2 \, b d - a e - 2 \, \sqrt{e x + d} b \sqrt{\frac{b d - a e}{b}}}{b x + a}\right ) + 2 \,{\left (3 \, b^{2} e^{2} x^{2} + 23 \, b^{2} d^{2} - 35 \, a b d e + 15 \, a^{2} e^{2} +{\left (11 \, b^{2} d e - 5 \, a b e^{2}\right )} x\right )} \sqrt{e x + d}}{15 \, b^{3}}, -\frac{2 \,{\left (15 \,{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \sqrt{-\frac{b d - a e}{b}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{b d - a e}{b}}}\right ) -{\left (3 \, b^{2} e^{2} x^{2} + 23 \, b^{2} d^{2} - 35 \, a b d e + 15 \, a^{2} e^{2} +{\left (11 \, b^{2} d e - 5 \, a b e^{2}\right )} x\right )} \sqrt{e x + d}\right )}}{15 \, b^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(5/2)/sqrt((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 [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.219605, size = 324, normalized size = 1.53 \[ \frac{2 \,{\left (b^{3} d^{3}{\rm sign}\left (b x + a\right ) - 3 \, a b^{2} d^{2} e{\rm sign}\left (b x + a\right ) + 3 \, a^{2} b d e^{2}{\rm sign}\left (b x + a\right ) - a^{3} e^{3}{\rm sign}\left (b x + a\right )\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e} b^{3}} + \frac{2 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{4}{\rm sign}\left (b x + a\right ) + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{4} d{\rm sign}\left (b x + a\right ) + 15 \, \sqrt{x e + d} b^{4} d^{2}{\rm sign}\left (b x + a\right ) - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{3} e{\rm sign}\left (b x + a\right ) - 30 \, \sqrt{x e + d} a b^{3} d e{\rm sign}\left (b x + a\right ) + 15 \, \sqrt{x e + d} a^{2} b^{2} e^{2}{\rm sign}\left (b x + a\right )\right )}}{15 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(b^3*d^3*sign(b*x + a) - 3*a*b^2*d^2*e*sign(b*x + a) + 3*a^2*b*d*e^2*sign(b*x
+ a) - a^3*e^3*sign(b*x + a))*arctan(sqrt(x*e + d)*b/sqrt(-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*sign(b*x + a) + 5*(x*e + d)^
(3/2)*b^4*d*sign(b*x + a) + 15*sqrt(x*e + d)*b^4*d^2*sign(b*x + a) - 5*(x*e + d)
^(3/2)*a*b^3*e*sign(b*x + a) - 30*sqrt(x*e + d)*a*b^3*d*e*sign(b*x + a) + 15*sqr
t(x*e + d)*a^2*b^2*e^2*sign(b*x + a))/b^5