3.2130 \(\int \frac{(a+b x) (d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=250 \[ -\frac{(d+e x)^{7/2}}{b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 e (a+b x) (d+e x)^{5/2}}{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{7 e (a+b x) (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 e (a+b x) \sqrt{d+e x} (b d-a e)^2}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 e (a+b x) (d+e x)^{3/2} (b d-a e)}{3 b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

(7*e*(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])
+ (7*e*(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]) + (7*e*(a + b*x)*(d + e*x)^(5/2))/(5*b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (d
+ e*x)^(7/2)/(b*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (7*e*(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 = 0.470227, antiderivative size = 250, 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 \[ -\frac{(d+e x)^{7/2}}{b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 e (a+b x) (d+e x)^{5/2}}{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{7 e (a+b x) (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 e (a+b x) \sqrt{d+e x} (b d-a e)^2}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 e (a+b x) (d+e x)^{3/2} (b d-a e)}{3 b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(d + e*x)^(7/2))/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

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

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.528274, size = 155, normalized size = 0.62 \[ \frac{(a+b x) \left (\frac{\sqrt{d+e x} \left (2 e \left (45 a^2 e^2-100 a b d e+58 b^2 d^2\right )+4 b e^2 x (8 b d-5 a e)-\frac{15 (b d-a e)^3}{a+b x}+6 b^2 e^3 x^2\right )}{15 b^4}-\frac{7 e (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{9/2}}\right )}{\sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(d + e*x)^(7/2))/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

((a + b*x)*((Sqrt[d + e*x]*(2*e*(58*b^2*d^2 - 100*a*b*d*e + 45*a^2*e^2) + 4*b*e^
2*(8*b*d - 5*a*e)*x + 6*b^2*e^3*x^2 - (15*(b*d - a*e)^3)/(a + b*x)))/(15*b^4) -
(7*e*(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 = 0.025, size = 662, normalized size = 2.7 \[{\frac{ \left ( bx+a \right ) ^{2}}{15\,{b}^{4}} \left ( 6\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{5/2}x{b}^{3}e+6\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{5/2}a{b}^{2}e-20\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{3/2}xa{b}^{2}{e}^{2}+20\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{3/2}x{b}^{3}de-105\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) x{a}^{3}b{e}^{4}+315\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) x{a}^{2}{b}^{2}d{e}^{3}-315\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) xa{b}^{3}{d}^{2}{e}^{2}+105\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) x{b}^{4}{d}^{3}e-20\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{3/2}{a}^{2}b{e}^{2}+20\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{3/2}a{b}^{2}de+90\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}x{a}^{2}b{e}^{3}-180\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}xa{b}^{2}d{e}^{2}+90\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}x{b}^{3}{d}^{2}e-105\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){a}^{4}{e}^{4}+315\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){a}^{3}bd{e}^{3}-315\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){a}^{2}{b}^{2}{d}^{2}{e}^{2}+105\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) a{b}^{3}{d}^{3}e+105\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{a}^{3}{e}^{3}-225\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{a}^{2}bd{e}^{2}+135\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}a{b}^{2}{d}^{2}e-15\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{b}^{3}{d}^{3} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

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)^(3/2),x)

[Out]

1/15*(6*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*x*b^3*e+6*(b*(a*e-b*d))^(1/2)*(e*x+d)^
(5/2)*a*b^2*e-20*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*x*a*b^2*e^2+20*(b*(a*e-b*d))^
(1/2)*(e*x+d)^(3/2)*x*b^3*d*e-105*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^3-315*ar
ctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x*a*b^3*d^2*e^2+105*arctan((e*x+d)^(1/
2)*b/(b*(a*e-b*d))^(1/2))*x*b^4*d^3*e-20*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a^2*b
*e^2+20*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a*b^2*d*e+90*(b*(a*e-b*d))^(1/2)*(e*x+
d)^(1/2)*x*a^2*b*e^3-180*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x*a*b^2*d*e^2+90*(b*(
a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x*b^3*d^2*e-105*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d
))^(1/2))*a^4*e^4+315*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^3*b*d*e^3-31
5*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^2*b^2*d^2*e^2+105*arctan((e*x+d)
^(1/2)*b/(b*(a*e-b*d))^(1/2))*a*b^3*d^3*e+105*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*
a^3*e^3-225*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^2*b*d*e^2+135*(b*(a*e-b*d))^(1/2
)*(e*x+d)^(1/2)*a*b^2*d^2*e-15*(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)^(3/2)

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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)^(7/2)/(b^2*x^2 + 2*a*b*x + a^2)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.323394, size = 1, normalized size = 0. \[ \left [\frac{105 \,{\left (a b^{2} d^{2} e - 2 \, a^{2} b d e^{2} + a^{3} e^{3} +{\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x\right )} \sqrt{\frac{b d - a e}{b}} \log \left (\frac{b e x + 2 \, b d - a e - 2 \, \sqrt{e x + d} b \sqrt{\frac{b d - a e}{b}}}{b x + a}\right ) + 2 \,{\left (6 \, b^{3} e^{3} x^{3} - 15 \, b^{3} d^{3} + 161 \, a b^{2} d^{2} e - 245 \, a^{2} b d e^{2} + 105 \, a^{3} e^{3} + 2 \,{\left (16 \, b^{3} d e^{2} - 7 \, a b^{2} e^{3}\right )} x^{2} + 2 \,{\left (58 \, b^{3} d^{2} e - 84 \, a b^{2} d e^{2} + 35 \, a^{2} b e^{3}\right )} x\right )} \sqrt{e x + d}}{30 \,{\left (b^{5} x + a b^{4}\right )}}, -\frac{105 \,{\left (a b^{2} d^{2} e - 2 \, a^{2} b d e^{2} + a^{3} e^{3} +{\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x\right )} \sqrt{-\frac{b d - a e}{b}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{b d - a e}{b}}}\right ) -{\left (6 \, b^{3} e^{3} x^{3} - 15 \, b^{3} d^{3} + 161 \, a b^{2} d^{2} e - 245 \, a^{2} b d e^{2} + 105 \, a^{3} e^{3} + 2 \,{\left (16 \, b^{3} d e^{2} - 7 \, a b^{2} e^{3}\right )} x^{2} + 2 \,{\left (58 \, b^{3} d^{2} e - 84 \, a b^{2} d e^{2} + 35 \, a^{2} b e^{3}\right )} x\right )} \sqrt{e x + d}}{15 \,{\left (b^{5} x + a b^{4}\right )}}\right ] \]

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)^(3/2),x, algorithm="fricas")

[Out]

[1/30*(105*(a*b^2*d^2*e - 2*a^2*b*d*e^2 + a^3*e^3 + (b^3*d^2*e - 2*a*b^2*d*e^2 +
a^2*b*e^3)*x)*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*(6*b^3*e^3*x^3 - 15*b^3*d^3 + 161*a*b^2*d^2*
e - 245*a^2*b*d*e^2 + 105*a^3*e^3 + 2*(16*b^3*d*e^2 - 7*a*b^2*e^3)*x^2 + 2*(58*b
^3*d^2*e - 84*a*b^2*d*e^2 + 35*a^2*b*e^3)*x)*sqrt(e*x + d))/(b^5*x + a*b^4), -1/
15*(105*(a*b^2*d^2*e - 2*a^2*b*d*e^2 + a^3*e^3 + (b^3*d^2*e - 2*a*b^2*d*e^2 + a^
2*b*e^3)*x)*sqrt(-(b*d - a*e)/b)*arctan(sqrt(e*x + d)/sqrt(-(b*d - a*e)/b)) - (6
*b^3*e^3*x^3 - 15*b^3*d^3 + 161*a*b^2*d^2*e - 245*a^2*b*d*e^2 + 105*a^3*e^3 + 2*
(16*b^3*d*e^2 - 7*a*b^2*e^3)*x^2 + 2*(58*b^3*d^2*e - 84*a*b^2*d*e^2 + 35*a^2*b*e
^3)*x)*sqrt(e*x + d))/(b^5*x + a*b^4)]

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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)**(7/2)/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.33149, size = 487, normalized size = 1.95 \[ -\frac{7 \,{\left (b^{3} d^{3} e^{2} - 3 \, a b^{2} d^{2} e^{3} + 3 \, a^{2} b d e^{4} - a^{3} e^{5}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{\left (-1\right )}}{\sqrt{-b^{2} d + a b e} b^{4}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} + \frac{{\left (\sqrt{x e + d} b^{3} d^{3} e^{2} - 3 \, \sqrt{x e + d} a b^{2} d^{2} e^{3} + 3 \, \sqrt{x e + d} a^{2} b d e^{4} - \sqrt{x e + d} a^{3} e^{5}\right )} e^{\left (-1\right )}}{{\left ({\left (x e + d\right )} b - b d + a e\right )} b^{4}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} - \frac{2 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{8} e^{6} + 10 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{8} d e^{6} + 45 \, \sqrt{x e + d} b^{8} d^{2} e^{6} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{7} e^{7} - 90 \, \sqrt{x e + d} a b^{7} d e^{7} + 45 \, \sqrt{x e + d} a^{2} b^{6} e^{8}\right )} e^{\left (-5\right )}}{15 \, b^{10}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} \]

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)^(3/2),x, algorithm="giac")

[Out]

-7*(b^3*d^3*e^2 - 3*a*b^2*d^2*e^3 + 3*a^2*b*d*e^4 - a^3*e^5)*arctan(sqrt(x*e + d
)*b/sqrt(-b^2*d + a*b*e))*e^(-1)/(sqrt(-b^2*d + a*b*e)*b^4*sign(-(x*e + d)*b*e +
b*d*e - a*e^2)) + (sqrt(x*e + d)*b^3*d^3*e^2 - 3*sqrt(x*e + d)*a*b^2*d^2*e^3 +
3*sqrt(x*e + d)*a^2*b*d*e^4 - sqrt(x*e + d)*a^3*e^5)*e^(-1)/(((x*e + d)*b - b*d
+ a*e)*b^4*sign(-(x*e + d)*b*e + b*d*e - a*e^2)) - 2/15*(3*(x*e + d)^(5/2)*b^8*e
^6 + 10*(x*e + d)^(3/2)*b^8*d*e^6 + 45*sqrt(x*e + d)*b^8*d^2*e^6 - 10*(x*e + d)^
(3/2)*a*b^7*e^7 - 90*sqrt(x*e + d)*a*b^7*d*e^7 + 45*sqrt(x*e + d)*a^2*b^6*e^8)*e
^(-5)/(b^10*sign(-(x*e + d)*b*e + b*d*e - a*e^2))