3.1860 \(\int \frac{(A+B x) (d+e x)^{3/2}}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=230 \[ \frac{2 (a+b x) (d+e x)^{3/2} (A b-a B)}{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 (a+b x) (A b-a B) (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (a+b x) \sqrt{d+e x} (A b-a B) (b d-a e)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B (a+b x) (d+e x)^{5/2}}{5 b e \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

(2*(A*b - a*B)*(b*d - a*e)*(a + b*x)*Sqrt[d + e*x])/(b^3*Sqrt[a^2 + 2*a*b*x + b^
2*x^2]) + (2*(A*b - a*B)*(a + b*x)*(d + e*x)^(3/2))/(3*b^2*Sqrt[a^2 + 2*a*b*x +
b^2*x^2]) + (2*B*(a + b*x)*(d + e*x)^(5/2))/(5*b*e*Sqrt[a^2 + 2*a*b*x + b^2*x^2]
) - (2*(A*b - a*B)*(b*d - a*e)^(3/2)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/S
qrt[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 = 0.429408, antiderivative size = 230, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{2 (a+b x) (d+e x)^{3/2} (A b-a B)}{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 (a+b x) (A b-a B) (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (a+b x) \sqrt{d+e x} (A b-a B) (b d-a e)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B (a+b x) (d+e x)^{5/2}}{5 b e \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(A*b - a*B)*(b*d - a*e)*(a + b*x)*Sqrt[d + e*x])/(b^3*Sqrt[a^2 + 2*a*b*x + b^
2*x^2]) + (2*(A*b - a*B)*(a + b*x)*(d + e*x)^(3/2))/(3*b^2*Sqrt[a^2 + 2*a*b*x +
b^2*x^2]) + (2*B*(a + b*x)*(d + e*x)^(5/2))/(5*b*e*Sqrt[a^2 + 2*a*b*x + b^2*x^2]
) - (2*(A*b - a*B)*(b*d - a*e)^(3/2)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/S
qrt[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((B*x+A)*(e*x+d)**(3/2)/((b*x+a)**2)**(1/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.315785, size = 146, normalized size = 0.63 \[ \frac{(a+b x) \left (\frac{2 \sqrt{d+e x} \left (15 a^2 B e^2-5 a b e (3 A e+4 B d+B e x)+b^2 \left (5 A e (4 d+e x)+3 B (d+e x)^2\right )\right )}{15 b^3 e}-\frac{2 (A b-a B) (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{7/2}}\right )}{\sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)*((2*Sqrt[d + e*x]*(15*a^2*B*e^2 - 5*a*b*e*(4*B*d + 3*A*e + B*e*x) + b
^2*(3*B*(d + e*x)^2 + 5*A*e*(4*d + e*x))))/(15*b^3*e) - (2*(A*b - a*B)*(b*d - a*
e)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/b^(7/2)))/Sqrt[(a + b
*x)^2]

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Maple [B]  time = 0.016, size = 414, normalized size = 1.8 \[{\frac{2\,bx+2\,a}{15\,{b}^{3}e} \left ( 3\,B\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{5/2}{b}^{2}+5\,A\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{3/2}{b}^{2}e+15\,A\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){a}^{2}b{e}^{3}-30\,A\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) a{b}^{2}d{e}^{2}+15\,A\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){b}^{3}{d}^{2}e-5\,B\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{3/2}abe-15\,B\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){a}^{3}{e}^{3}+30\,B\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){a}^{2}bd{e}^{2}-15\,B\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) a{b}^{2}{d}^{2}e-15\,A\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}ab{e}^{2}+15\,A\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{b}^{2}de+15\,B\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{a}^{2}{e}^{2}-15\,B\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}abde \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^(3/2)/((b*x+a)^2)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.287441, size = 1, normalized size = 0. \[ \left [-\frac{15 \,{\left ({\left (B a b - A b^{2}\right )} d e -{\left (B a^{2} - A a b\right )} e^{2}\right )} \sqrt{\frac{b d - a e}{b}} \log \left (\frac{b e x + 2 \, b d - a e - 2 \, \sqrt{e x + d} b \sqrt{\frac{b d - a e}{b}}}{b x + a}\right ) - 2 \,{\left (3 \, B b^{2} e^{2} x^{2} + 3 \, B b^{2} d^{2} - 20 \,{\left (B a b - A b^{2}\right )} d e + 15 \,{\left (B a^{2} - A a b\right )} e^{2} +{\left (6 \, B b^{2} d e - 5 \,{\left (B a b - A b^{2}\right )} e^{2}\right )} x\right )} \sqrt{e x + d}}{15 \, b^{3} e}, \frac{2 \,{\left (15 \,{\left ({\left (B a b - A b^{2}\right )} d e -{\left (B a^{2} - A a b\right )} e^{2}\right )} \sqrt{-\frac{b d - a e}{b}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{b d - a e}{b}}}\right ) +{\left (3 \, B b^{2} e^{2} x^{2} + 3 \, B b^{2} d^{2} - 20 \,{\left (B a b - A b^{2}\right )} d e + 15 \,{\left (B a^{2} - A a b\right )} e^{2} +{\left (6 \, B b^{2} d e - 5 \,{\left (B a b - A b^{2}\right )} e^{2}\right )} x\right )} \sqrt{e x + d}\right )}}{15 \, b^{3} e}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/15*(15*((B*a*b - A*b^2)*d*e - (B*a^2 - A*a*b)*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
*b^2*e^2*x^2 + 3*B*b^2*d^2 - 20*(B*a*b - A*b^2)*d*e + 15*(B*a^2 - A*a*b)*e^2 + (
6*B*b^2*d*e - 5*(B*a*b - A*b^2)*e^2)*x)*sqrt(e*x + d))/(b^3*e), 2/15*(15*((B*a*b
- A*b^2)*d*e - (B*a^2 - A*a*b)*e^2)*sqrt(-(b*d - a*e)/b)*arctan(sqrt(e*x + d)/s
qrt(-(b*d - a*e)/b)) + (3*B*b^2*e^2*x^2 + 3*B*b^2*d^2 - 20*(B*a*b - A*b^2)*d*e +
15*(B*a^2 - A*a*b)*e^2 + (6*B*b^2*d*e - 5*(B*a*b - A*b^2)*e^2)*x)*sqrt(e*x + d)
)/(b^3*e)]

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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((B*x+A)*(e*x+d)**(3/2)/((b*x+a)**2)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.287699, size = 413, normalized size = 1.8 \[ -\frac{2 \,{\left (B a b^{2} d^{2}{\rm sign}\left (b x + a\right ) - A b^{3} d^{2}{\rm sign}\left (b x + a\right ) - 2 \, B a^{2} b d e{\rm sign}\left (b x + a\right ) + 2 \, A a b^{2} d e{\rm sign}\left (b x + a\right ) + B a^{3} e^{2}{\rm sign}\left (b x + a\right ) - A a^{2} b e^{2}{\rm sign}\left (b x + a\right )\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e} b^{3}} + \frac{2 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} B b^{4} e^{4}{\rm sign}\left (b x + a\right ) - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b^{3} e^{5}{\rm sign}\left (b x + a\right ) + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{4} e^{5}{\rm sign}\left (b x + a\right ) - 15 \, \sqrt{x e + d} B a b^{3} d e^{5}{\rm sign}\left (b x + a\right ) + 15 \, \sqrt{x e + d} A b^{4} d e^{5}{\rm sign}\left (b x + a\right ) + 15 \, \sqrt{x e + d} B a^{2} b^{2} e^{6}{\rm sign}\left (b x + a\right ) - 15 \, \sqrt{x e + d} A a b^{3} e^{6}{\rm sign}\left (b x + a\right )\right )} e^{\left (-5\right )}}{15 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(B*a*b^2*d^2*sign(b*x + a) - A*b^3*d^2*sign(b*x + a) - 2*B*a^2*b*d*e*sign(b*x
+ a) + 2*A*a*b^2*d*e*sign(b*x + a) + B*a^3*e^2*sign(b*x + a) - A*a^2*b*e^2*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*b^4*e^4*sign(b*x + a) - 5*(x*e + d)^(3/2)*B*a*b^3
*e^5*sign(b*x + a) + 5*(x*e + d)^(3/2)*A*b^4*e^5*sign(b*x + a) - 15*sqrt(x*e + d
)*B*a*b^3*d*e^5*sign(b*x + a) + 15*sqrt(x*e + d)*A*b^4*d*e^5*sign(b*x + a) + 15*
sqrt(x*e + d)*B*a^2*b^2*e^6*sign(b*x + a) - 15*sqrt(x*e + d)*A*a*b^3*e^6*sign(b*
x + a))*e^(-5)/b^5