3.1859 \(\int \frac{(A+B x) (d+e x)^{5/2}}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=289 \[ \frac{2 (a+b x) (d+e x)^{5/2} (A b-a B)}{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 (a+b x) (A b-a B) (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (a+b x) \sqrt{d+e x} (A b-a B) (b d-a e)^2}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (a+b x) (d+e x)^{3/2} (A b-a B) (b d-a e)}{3 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B (a+b x) (d+e x)^{7/2}}{7 b e \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

(2*(A*b - a*B)*(b*d - a*e)^2*(a + b*x)*Sqrt[d + e*x])/(b^4*Sqrt[a^2 + 2*a*b*x +
b^2*x^2]) + (2*(A*b - a*B)*(b*d - a*e)*(a + b*x)*(d + e*x)^(3/2))/(3*b^3*Sqrt[a^
2 + 2*a*b*x + b^2*x^2]) + (2*(A*b - a*B)*(a + b*x)*(d + e*x)^(5/2))/(5*b^2*Sqrt[
a^2 + 2*a*b*x + b^2*x^2]) + (2*B*(a + b*x)*(d + e*x)^(7/2))/(7*b*e*Sqrt[a^2 + 2*
a*b*x + b^2*x^2]) - (2*(A*b - a*B)*(b*d - a*e)^(5/2)*(a + b*x)*ArcTanh[(Sqrt[b]*
Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.603408, antiderivative size = 289, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{2 (a+b x) (d+e x)^{5/2} (A b-a B)}{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 (a+b x) (A b-a B) (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (a+b x) \sqrt{d+e x} (A b-a B) (b d-a e)^2}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (a+b x) (d+e x)^{3/2} (A b-a B) (b d-a e)}{3 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B (a+b x) (d+e x)^{7/2}}{7 b e \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(A*b - a*B)*(b*d - a*e)^2*(a + b*x)*Sqrt[d + e*x])/(b^4*Sqrt[a^2 + 2*a*b*x +
b^2*x^2]) + (2*(A*b - a*B)*(b*d - a*e)*(a + b*x)*(d + e*x)^(3/2))/(3*b^3*Sqrt[a^
2 + 2*a*b*x + b^2*x^2]) + (2*(A*b - a*B)*(a + b*x)*(d + e*x)^(5/2))/(5*b^2*Sqrt[
a^2 + 2*a*b*x + b^2*x^2]) + (2*B*(a + b*x)*(d + e*x)^(7/2))/(7*b*e*Sqrt[a^2 + 2*
a*b*x + b^2*x^2]) - (2*(A*b - a*B)*(b*d - a*e)^(5/2)*(a + b*x)*ArcTanh[(Sqrt[b]*
Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b^(9/2)*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)**(5/2)/((b*x+a)**2)**(1/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.513545, size = 202, normalized size = 0.7 \[ \frac{(a+b x) \left (\frac{2 \sqrt{d+e x} \left (-105 a^3 B e^3+35 a^2 b e^2 (3 A e+7 B d+B e x)-7 a b^2 e \left (5 A e (7 d+e x)+B \left (23 d^2+11 d e x+3 e^2 x^2\right )\right )+b^3 \left (7 A e \left (23 d^2+11 d e x+3 e^2 x^2\right )+15 B (d+e x)^3\right )\right )}{105 b^4 e}-\frac{2 (A b-a B) (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{9/2}}\right )}{\sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)*((2*Sqrt[d + e*x]*(-105*a^3*B*e^3 + 35*a^2*b*e^2*(7*B*d + 3*A*e + B*e
*x) - 7*a*b^2*e*(5*A*e*(7*d + e*x) + B*(23*d^2 + 11*d*e*x + 3*e^2*x^2)) + b^3*(1
5*B*(d + e*x)^3 + 7*A*e*(23*d^2 + 11*d*e*x + 3*e^2*x^2))))/(105*b^4*e) - (2*(A*b
- a*B)*(b*d - a*e)^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/b^(9
/2)))/Sqrt[(a + b*x)^2]

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Maple [B]  time = 0.018, size = 671, normalized size = 2.3 \[{\frac{2\,bx+2\,a}{105\,e{b}^{4}} \left ( 15\,B\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{7/2}{b}^{3}+21\,A\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{5/2}{b}^{3}e-21\,B\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{5/2}a{b}^{2}e-35\,A\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{3/2}a{b}^{2}{e}^{2}+35\,A\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{3/2}{b}^{3}de-105\,A\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){a}^{3}b{e}^{4}+315\,A\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){a}^{2}{b}^{2}d{e}^{3}-315\,A\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) a{b}^{3}{d}^{2}{e}^{2}+105\,A\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){b}^{4}{d}^{3}e+35\,B\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{3/2}{a}^{2}b{e}^{2}-35\,B\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{3/2}a{b}^{2}de+105\,B\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){a}^{4}{e}^{4}-315\,B\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){a}^{3}bd{e}^{3}+315\,B\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){a}^{2}{b}^{2}{d}^{2}{e}^{2}-105\,B\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) a{b}^{3}{d}^{3}e+105\,A\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{a}^{2}b{e}^{3}-210\,A\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}a{b}^{2}d{e}^{2}+105\,A\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{b}^{3}{d}^{2}e-105\,B\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{a}^{3}{e}^{3}+210\,B\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{a}^{2}bd{e}^{2}-105\,B\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}a{b}^{2}{d}^{2}e \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/105*(b*x+a)*(15*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(7/2)*b^3+21*A*(b*(a*e-b*d))^(1/
2)*(e*x+d)^(5/2)*b^3*e-21*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*a*b^2*e-35*A*(b*(a
*e-b*d))^(1/2)*(e*x+d)^(3/2)*a*b^2*e^2+35*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*b^
3*d*e-105*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^3*b*e^4+315*A*arctan((
e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^2*b^2*d*e^3-315*A*arctan((e*x+d)^(1/2)*b/(
b*(a*e-b*d))^(1/2))*a*b^3*d^2*e^2+105*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/
2))*b^4*d^3*e+35*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a^2*b*e^2-35*B*(b*(a*e-b*d)
)^(1/2)*(e*x+d)^(3/2)*a*b^2*d*e+105*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2)
)*a^4*e^4-315*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^3*b*d*e^3+315*B*ar
ctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^2*b^2*d^2*e^2-105*B*arctan((e*x+d)^(
1/2)*b/(b*(a*e-b*d))^(1/2))*a*b^3*d^3*e+105*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*
a^2*b*e^3-210*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a*b^2*d*e^2+105*A*(b*(a*e-b*d)
)^(1/2)*(e*x+d)^(1/2)*b^3*d^2*e-105*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^3*e^3+
210*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^2*b*d*e^2-105*B*(b*(a*e-b*d))^(1/2)*(e
*x+d)^(1/2)*a*b^2*d^2*e)/((b*x+a)^2)^(1/2)/e/b^4/(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)^(5/2)/sqrt((b*x + a)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.304618, size = 1, normalized size = 0. \[ \left [\frac{105 \,{\left ({\left (B a b^{2} - A b^{3}\right )} d^{2} e - 2 \,{\left (B a^{2} b - A a b^{2}\right )} d e^{2} +{\left (B a^{3} - A a^{2} b\right )} e^{3}\right )} \sqrt{\frac{b d - a e}{b}} \log \left (\frac{b e x + 2 \, b d - a e + 2 \, \sqrt{e x + d} b \sqrt{\frac{b d - a e}{b}}}{b x + a}\right ) + 2 \,{\left (15 \, B b^{3} e^{3} x^{3} + 15 \, B b^{3} d^{3} - 161 \,{\left (B a b^{2} - A b^{3}\right )} d^{2} e + 245 \,{\left (B a^{2} b - A a b^{2}\right )} d e^{2} - 105 \,{\left (B a^{3} - A a^{2} b\right )} e^{3} + 3 \,{\left (15 \, B b^{3} d e^{2} - 7 \,{\left (B a b^{2} - A b^{3}\right )} e^{3}\right )} x^{2} +{\left (45 \, B b^{3} d^{2} e - 77 \,{\left (B a b^{2} - A b^{3}\right )} d e^{2} + 35 \,{\left (B a^{2} b - A a b^{2}\right )} e^{3}\right )} x\right )} \sqrt{e x + d}}{105 \, b^{4} e}, \frac{2 \,{\left (105 \,{\left ({\left (B a b^{2} - A b^{3}\right )} d^{2} e - 2 \,{\left (B a^{2} b - A a b^{2}\right )} d e^{2} +{\left (B a^{3} - A a^{2} b\right )} e^{3}\right )} \sqrt{-\frac{b d - a e}{b}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{b d - a e}{b}}}\right ) +{\left (15 \, B b^{3} e^{3} x^{3} + 15 \, B b^{3} d^{3} - 161 \,{\left (B a b^{2} - A b^{3}\right )} d^{2} e + 245 \,{\left (B a^{2} b - A a b^{2}\right )} d e^{2} - 105 \,{\left (B a^{3} - A a^{2} b\right )} e^{3} + 3 \,{\left (15 \, B b^{3} d e^{2} - 7 \,{\left (B a b^{2} - A b^{3}\right )} e^{3}\right )} x^{2} +{\left (45 \, B b^{3} d^{2} e - 77 \,{\left (B a b^{2} - A b^{3}\right )} d e^{2} + 35 \,{\left (B a^{2} b - A a b^{2}\right )} e^{3}\right )} x\right )} \sqrt{e x + d}\right )}}{105 \, b^{4} e}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/105*(105*((B*a*b^2 - A*b^3)*d^2*e - 2*(B*a^2*b - A*a*b^2)*d*e^2 + (B*a^3 - A*
a^2*b)*e^3)*sqrt((b*d - a*e)/b)*log((b*e*x + 2*b*d - a*e + 2*sqrt(e*x + d)*b*sqr
t((b*d - a*e)/b))/(b*x + a)) + 2*(15*B*b^3*e^3*x^3 + 15*B*b^3*d^3 - 161*(B*a*b^2
- A*b^3)*d^2*e + 245*(B*a^2*b - A*a*b^2)*d*e^2 - 105*(B*a^3 - A*a^2*b)*e^3 + 3*
(15*B*b^3*d*e^2 - 7*(B*a*b^2 - A*b^3)*e^3)*x^2 + (45*B*b^3*d^2*e - 77*(B*a*b^2 -
A*b^3)*d*e^2 + 35*(B*a^2*b - A*a*b^2)*e^3)*x)*sqrt(e*x + d))/(b^4*e), 2/105*(10
5*((B*a*b^2 - A*b^3)*d^2*e - 2*(B*a^2*b - A*a*b^2)*d*e^2 + (B*a^3 - A*a^2*b)*e^3
)*sqrt(-(b*d - a*e)/b)*arctan(sqrt(e*x + d)/sqrt(-(b*d - a*e)/b)) + (15*B*b^3*e^
3*x^3 + 15*B*b^3*d^3 - 161*(B*a*b^2 - A*b^3)*d^2*e + 245*(B*a^2*b - A*a*b^2)*d*e
^2 - 105*(B*a^3 - A*a^2*b)*e^3 + 3*(15*B*b^3*d*e^2 - 7*(B*a*b^2 - A*b^3)*e^3)*x^
2 + (45*B*b^3*d^2*e - 77*(B*a*b^2 - A*b^3)*d*e^2 + 35*(B*a^2*b - A*a*b^2)*e^3)*x
)*sqrt(e*x + d))/(b^4*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)**(5/2)/((b*x+a)**2)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.30412, size = 671, normalized size = 2.32 \[ -\frac{2 \,{\left (B a b^{3} d^{3}{\rm sign}\left (b x + a\right ) - A b^{4} d^{3}{\rm sign}\left (b x + a\right ) - 3 \, B a^{2} b^{2} d^{2} e{\rm sign}\left (b x + a\right ) + 3 \, A a b^{3} d^{2} e{\rm sign}\left (b x + a\right ) + 3 \, B a^{3} b d e^{2}{\rm sign}\left (b x + a\right ) - 3 \, A a^{2} b^{2} d e^{2}{\rm sign}\left (b x + a\right ) - B a^{4} e^{3}{\rm sign}\left (b x + a\right ) + A a^{3} b e^{3}{\rm sign}\left (b x + a\right )\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e} b^{4}} + \frac{2 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} B b^{6} e^{6}{\rm sign}\left (b x + a\right ) - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} B a b^{5} e^{7}{\rm sign}\left (b x + a\right ) + 21 \,{\left (x e + d\right )}^{\frac{5}{2}} A b^{6} e^{7}{\rm sign}\left (b x + a\right ) - 35 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b^{5} d e^{7}{\rm sign}\left (b x + a\right ) + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{6} d e^{7}{\rm sign}\left (b x + a\right ) - 105 \, \sqrt{x e + d} B a b^{5} d^{2} e^{7}{\rm sign}\left (b x + a\right ) + 105 \, \sqrt{x e + d} A b^{6} d^{2} e^{7}{\rm sign}\left (b x + a\right ) + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} B a^{2} b^{4} e^{8}{\rm sign}\left (b x + a\right ) - 35 \,{\left (x e + d\right )}^{\frac{3}{2}} A a b^{5} e^{8}{\rm sign}\left (b x + a\right ) + 210 \, \sqrt{x e + d} B a^{2} b^{4} d e^{8}{\rm sign}\left (b x + a\right ) - 210 \, \sqrt{x e + d} A a b^{5} d e^{8}{\rm sign}\left (b x + a\right ) - 105 \, \sqrt{x e + d} B a^{3} b^{3} e^{9}{\rm sign}\left (b x + a\right ) + 105 \, \sqrt{x e + d} A a^{2} b^{4} e^{9}{\rm sign}\left (b x + a\right )\right )} e^{\left (-7\right )}}{105 \, b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(B*a*b^3*d^3*sign(b*x + a) - A*b^4*d^3*sign(b*x + a) - 3*B*a^2*b^2*d^2*e*sign
(b*x + a) + 3*A*a*b^3*d^2*e*sign(b*x + a) + 3*B*a^3*b*d*e^2*sign(b*x + a) - 3*A*
a^2*b^2*d*e^2*sign(b*x + a) - B*a^4*e^3*sign(b*x + a) + A*a^3*b*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^4) + 2/1
05*(15*(x*e + d)^(7/2)*B*b^6*e^6*sign(b*x + a) - 21*(x*e + d)^(5/2)*B*a*b^5*e^7*
sign(b*x + a) + 21*(x*e + d)^(5/2)*A*b^6*e^7*sign(b*x + a) - 35*(x*e + d)^(3/2)*
B*a*b^5*d*e^7*sign(b*x + a) + 35*(x*e + d)^(3/2)*A*b^6*d*e^7*sign(b*x + a) - 105
*sqrt(x*e + d)*B*a*b^5*d^2*e^7*sign(b*x + a) + 105*sqrt(x*e + d)*A*b^6*d^2*e^7*s
ign(b*x + a) + 35*(x*e + d)^(3/2)*B*a^2*b^4*e^8*sign(b*x + a) - 35*(x*e + d)^(3/
2)*A*a*b^5*e^8*sign(b*x + a) + 210*sqrt(x*e + d)*B*a^2*b^4*d*e^8*sign(b*x + a) -
210*sqrt(x*e + d)*A*a*b^5*d*e^8*sign(b*x + a) - 105*sqrt(x*e + d)*B*a^3*b^3*e^9
*sign(b*x + a) + 105*sqrt(x*e + d)*A*a^2*b^4*e^9*sign(b*x + a))*e^(-7)/b^7