3.2138 \(\int \frac{(a+b x) (d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=241 \[ -\frac{(d+e x)^{7/2}}{3 b \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}-\frac{7 e (d+e x)^{5/2}}{12 b^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 e^3 (a+b x) \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{35 e^3 (a+b x) \sqrt{d+e x}}{8 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 e^2 (d+e x)^{3/2}}{24 b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

-(d + e*x)^(7/2)/(3*b*(a^2 + 2*a*b*x + b^2*x^2)^(3/2)) + (35*e^3*(a + b*x)*Sqrt[
d + e*x])/(8*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*e^2*(d + e*x)^(3/2))/(24*b
^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (7*e*(d + e*x)^(5/2))/(12*b^2*(a + b*x)*Sqrt
[a^2 + 2*a*b*x + b^2*x^2]) - (35*e^3*Sqrt[b*d - a*e]*(a + b*x)*ArcTanh[(Sqrt[b]*
Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*b^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 0.402907, antiderivative size = 241, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171 \[ -\frac{(d+e x)^{7/2}}{3 b \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}-\frac{7 e (d+e x)^{5/2}}{12 b^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 e^3 (a+b x) \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{35 e^3 (a+b x) \sqrt{d+e x}}{8 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 e^2 (d+e x)^{3/2}}{24 b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(d + e*x)^(7/2))/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

-(d + e*x)^(7/2)/(3*b*(a^2 + 2*a*b*x + b^2*x^2)^(3/2)) + (35*e^3*(a + b*x)*Sqrt[
d + e*x])/(8*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*e^2*(d + e*x)^(3/2))/(24*b
^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (7*e*(d + e*x)^(5/2))/(12*b^2*(a + b*x)*Sqrt
[a^2 + 2*a*b*x + b^2*x^2]) - (35*e^3*Sqrt[b*d - a*e]*(a + b*x)*ArcTanh[(Sqrt[b]*
Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*b^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

_______________________________________________________________________________________

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)**(7/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Exception raised: RecursionError

_______________________________________________________________________________________

Mathematica [A]  time = 0.370023, size = 157, normalized size = 0.65 \[ \frac{(a+b x) \left (-\frac{35 e^3 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{9/2}}-\frac{\sqrt{d+e x} \left (87 e^2 (a+b x)^2 (b d-a e)+38 e (a+b x) (b d-a e)^2+8 (b d-a e)^3-48 e^3 (a+b x)^3\right )}{3 b^4 (a+b x)^3}\right )}{8 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(d + e*x)^(7/2))/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

((a + b*x)*(-(Sqrt[d + e*x]*(8*(b*d - a*e)^3 + 38*e*(b*d - a*e)^2*(a + b*x) + 87
*e^2*(b*d - a*e)*(a + b*x)^2 - 48*e^3*(a + b*x)^3))/(3*b^4*(a + b*x)^3) - (35*e^
3*Sqrt[b*d - a*e]*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/b^(9/2)))/(8
*Sqrt[(a + b*x)^2])

_______________________________________________________________________________________

Maple [B]  time = 0.029, size = 638, normalized size = 2.7 \[{\frac{ \left ( bx+a \right ) ^{2}}{24\,{b}^{4}} \left ( -105\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){x}^{3}a{b}^{3}{e}^{4}+105\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){x}^{3}{b}^{4}d{e}^{3}+48\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{x}^{3}{b}^{3}{e}^{3}-315\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){x}^{2}{a}^{2}{b}^{2}{e}^{4}+315\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){x}^{2}a{b}^{3}d{e}^{3}+87\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{5/2}a{b}^{2}e-87\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{5/2}{b}^{3}d+144\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{x}^{2}a{b}^{2}{e}^{3}-315\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) x{a}^{3}b{e}^{4}+315\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) x{a}^{2}{b}^{2}d{e}^{3}+136\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{3/2}{a}^{2}b{e}^{2}-272\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{3/2}a{b}^{2}de+136\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{3/2}{b}^{3}{d}^{2}+144\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}x{a}^{2}b{e}^{3}-105\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){a}^{4}{e}^{4}+105\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){a}^{3}bd{e}^{3}+105\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{a}^{3}{e}^{3}-171\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{a}^{2}bd{e}^{2}+171\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}a{b}^{2}{d}^{2}e-57\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{b}^{3}{d}^{3} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/24*(-105*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^3*a*b^3*e^4+105*arctan(
(e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^3*b^4*d*e^3+48*(b*(a*e-b*d))^(1/2)*(e*x+d
)^(1/2)*x^3*b^3*e^3-315*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^2*a^2*b^2*
e^4+315*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^2*a*b^3*d*e^3+87*(b*(a*e-b
*d))^(1/2)*(e*x+d)^(5/2)*a*b^2*e-87*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*b^3*d+144*
(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x^2*a*b^2*e^3-315*arctan((e*x+d)^(1/2)*b/(b*(a
*e-b*d))^(1/2))*x*a^3*b*e^4+315*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x*a^
2*b^2*d*e^3+136*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a^2*b*e^2-272*(b*(a*e-b*d))^(1
/2)*(e*x+d)^(3/2)*a*b^2*d*e+136*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*b^3*d^2+144*(b
*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x*a^2*b*e^3-105*arctan((e*x+d)^(1/2)*b/(b*(a*e-b
*d))^(1/2))*a^4*e^4+105*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^3*b*d*e^3+
105*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^3*e^3-171*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1
/2)*a^2*b*d*e^2+171*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a*b^2*d^2*e-57*(b*(a*e-b*d
))^(1/2)*(e*x+d)^(1/2)*b^3*d^3)*(b*x+a)^2/(b*(a*e-b*d))^(1/2)/b^4/((b*x+a)^2)^(5
/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)^(7/2)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.303905, size = 1, normalized size = 0. \[ \left [\frac{105 \,{\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \sqrt{\frac{b d - a e}{b}} \log \left (\frac{b e x + 2 \, b d - a e - 2 \, \sqrt{e x + d} b \sqrt{\frac{b d - a e}{b}}}{b x + a}\right ) + 2 \,{\left (48 \, b^{3} e^{3} x^{3} - 8 \, b^{3} d^{3} - 14 \, a b^{2} d^{2} e - 35 \, a^{2} b d e^{2} + 105 \, a^{3} e^{3} - 3 \,{\left (29 \, b^{3} d e^{2} - 77 \, a b^{2} e^{3}\right )} x^{2} - 2 \,{\left (19 \, b^{3} d^{2} e + 49 \, a b^{2} d e^{2} - 140 \, a^{2} b e^{3}\right )} x\right )} \sqrt{e x + d}}{48 \,{\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}}, -\frac{105 \,{\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \sqrt{-\frac{b d - a e}{b}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{b d - a e}{b}}}\right ) -{\left (48 \, b^{3} e^{3} x^{3} - 8 \, b^{3} d^{3} - 14 \, a b^{2} d^{2} e - 35 \, a^{2} b d e^{2} + 105 \, a^{3} e^{3} - 3 \,{\left (29 \, b^{3} d e^{2} - 77 \, a b^{2} e^{3}\right )} x^{2} - 2 \,{\left (19 \, b^{3} d^{2} e + 49 \, a b^{2} d e^{2} - 140 \, a^{2} b e^{3}\right )} x\right )} \sqrt{e x + d}}{24 \,{\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(e*x + d)^(7/2)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="fricas")

[Out]

[1/48*(105*(b^3*e^3*x^3 + 3*a*b^2*e^3*x^2 + 3*a^2*b*e^3*x + a^3*e^3)*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*(48*b^3*e^3*x^3 - 8*b^3*d^3 - 14*a*b^2*d^2*e - 35*a^2*b*d*e^2 + 105*a^
3*e^3 - 3*(29*b^3*d*e^2 - 77*a*b^2*e^3)*x^2 - 2*(19*b^3*d^2*e + 49*a*b^2*d*e^2 -
140*a^2*b*e^3)*x)*sqrt(e*x + d))/(b^7*x^3 + 3*a*b^6*x^2 + 3*a^2*b^5*x + a^3*b^4
), -1/24*(105*(b^3*e^3*x^3 + 3*a*b^2*e^3*x^2 + 3*a^2*b*e^3*x + a^3*e^3)*sqrt(-(b
*d - a*e)/b)*arctan(sqrt(e*x + d)/sqrt(-(b*d - a*e)/b)) - (48*b^3*e^3*x^3 - 8*b^
3*d^3 - 14*a*b^2*d^2*e - 35*a^2*b*d*e^2 + 105*a^3*e^3 - 3*(29*b^3*d*e^2 - 77*a*b
^2*e^3)*x^2 - 2*(19*b^3*d^2*e + 49*a*b^2*d*e^2 - 140*a^2*b*e^3)*x)*sqrt(e*x + d)
)/(b^7*x^3 + 3*a*b^6*x^2 + 3*a^2*b^5*x + a^3*b^4)]

_______________________________________________________________________________________

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)**(7/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.316464, size = 436, normalized size = 1.81 \[ -\frac{35 \,{\left (b d e^{3} - a e^{4}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{8 \, \sqrt{-b^{2} d + a b e} b^{4}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} - \frac{2 \, \sqrt{x e + d} e^{3}}{b^{4}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} + \frac{87 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{3} d e^{3} - 136 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} d^{2} e^{3} + 57 \, \sqrt{x e + d} b^{3} d^{3} e^{3} - 87 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{2} e^{4} + 272 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{2} d e^{4} - 171 \, \sqrt{x e + d} a b^{2} d^{2} e^{4} - 136 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b e^{5} + 171 \, \sqrt{x e + d} a^{2} b d e^{5} - 57 \, \sqrt{x e + d} a^{3} e^{6}}{24 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{3} b^{4}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(e*x + d)^(7/2)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="giac")

[Out]

-35/8*(b*d*e^3 - a*e^4)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*
d + a*b*e)*b^4*sign(-(x*e + d)*b*e + b*d*e - a*e^2)) - 2*sqrt(x*e + d)*e^3/(b^4*
sign(-(x*e + d)*b*e + b*d*e - a*e^2)) + 1/24*(87*(x*e + d)^(5/2)*b^3*d*e^3 - 136
*(x*e + d)^(3/2)*b^3*d^2*e^3 + 57*sqrt(x*e + d)*b^3*d^3*e^3 - 87*(x*e + d)^(5/2)
*a*b^2*e^4 + 272*(x*e + d)^(3/2)*a*b^2*d*e^4 - 171*sqrt(x*e + d)*a*b^2*d^2*e^4 -
136*(x*e + d)^(3/2)*a^2*b*e^5 + 171*sqrt(x*e + d)*a^2*b*d*e^5 - 57*sqrt(x*e + d
)*a^3*e^6)/(((x*e + d)*b - b*d + a*e)^3*b^4*sign(-(x*e + d)*b*e + b*d*e - a*e^2)
)