3.2084 \(\int \frac{(a+b x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=171 \[ -\frac{315 e^4 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{11/2}}-\frac{105 e^3 (d+e x)^{3/2}}{64 b^4 (a+b x)}-\frac{21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x)^2}-\frac{3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^3}-\frac{(d+e x)^{9/2}}{4 b (a+b x)^4}+\frac{315 e^4 \sqrt{d+e x}}{64 b^5} \]

[Out]

(315*e^4*Sqrt[d + e*x])/(64*b^5) - (105*e^3*(d + e*x)^(3/2))/(64*b^4*(a + b*x))
- (21*e^2*(d + e*x)^(5/2))/(32*b^3*(a + b*x)^2) - (3*e*(d + e*x)^(7/2))/(8*b^2*(
a + b*x)^3) - (d + e*x)^(9/2)/(4*b*(a + b*x)^4) - (315*e^4*Sqrt[b*d - a*e]*ArcTa
nh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*b^(11/2))

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Rubi [A]  time = 0.222603, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152 \[ -\frac{315 e^4 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{11/2}}-\frac{105 e^3 (d+e x)^{3/2}}{64 b^4 (a+b x)}-\frac{21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x)^2}-\frac{3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^3}-\frac{(d+e x)^{9/2}}{4 b (a+b x)^4}+\frac{315 e^4 \sqrt{d+e x}}{64 b^5} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(d + e*x)^(9/2))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(315*e^4*Sqrt[d + e*x])/(64*b^5) - (105*e^3*(d + e*x)^(3/2))/(64*b^4*(a + b*x))
- (21*e^2*(d + e*x)^(5/2))/(32*b^3*(a + b*x)^2) - (3*e*(d + e*x)^(7/2))/(8*b^2*(
a + b*x)^3) - (d + e*x)^(9/2)/(4*b*(a + b*x)^4) - (315*e^4*Sqrt[b*d - a*e]*ArcTa
nh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*b^(11/2))

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Rubi in Sympy [A]  time = 66.0466, size = 156, normalized size = 0.91 \[ - \frac{\left (d + e x\right )^{\frac{9}{2}}}{4 b \left (a + b x\right )^{4}} - \frac{3 e \left (d + e x\right )^{\frac{7}{2}}}{8 b^{2} \left (a + b x\right )^{3}} - \frac{21 e^{2} \left (d + e x\right )^{\frac{5}{2}}}{32 b^{3} \left (a + b x\right )^{2}} - \frac{105 e^{3} \left (d + e x\right )^{\frac{3}{2}}}{64 b^{4} \left (a + b x\right )} + \frac{315 e^{4} \sqrt{d + e x}}{64 b^{5}} - \frac{315 e^{4} \sqrt{a e - b d} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{64 b^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(e*x+d)**(9/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

-(d + e*x)**(9/2)/(4*b*(a + b*x)**4) - 3*e*(d + e*x)**(7/2)/(8*b**2*(a + b*x)**3
) - 21*e**2*(d + e*x)**(5/2)/(32*b**3*(a + b*x)**2) - 105*e**3*(d + e*x)**(3/2)/
(64*b**4*(a + b*x)) + 315*e**4*sqrt(d + e*x)/(64*b**5) - 315*e**4*sqrt(a*e - b*d
)*atan(sqrt(b)*sqrt(d + e*x)/sqrt(a*e - b*d))/(64*b**(11/2))

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Mathematica [A]  time = 0.3666, size = 161, normalized size = 0.94 \[ -\frac{315 e^4 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{11/2}}-\frac{\sqrt{d+e x} \left (325 e^3 (a+b x)^3 (b d-a e)+210 e^2 (a+b x)^2 (b d-a e)^2+88 e (a+b x) (b d-a e)^3+16 (b d-a e)^4-128 e^4 (a+b x)^4\right )}{64 b^5 (a+b x)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(d + e*x)^(9/2))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

-(Sqrt[d + e*x]*(16*(b*d - a*e)^4 + 88*e*(b*d - a*e)^3*(a + b*x) + 210*e^2*(b*d
- a*e)^2*(a + b*x)^2 + 325*e^3*(b*d - a*e)*(a + b*x)^3 - 128*e^4*(a + b*x)^4))/(
64*b^5*(a + b*x)^4) - (315*e^4*Sqrt[b*d - a*e]*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/S
qrt[b*d - a*e]])/(64*b^(11/2))

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Maple [B]  time = 0.031, size = 497, normalized size = 2.9 \[ 2\,{\frac{{e}^{4}\sqrt{ex+d}}{{b}^{5}}}+{\frac{325\,a{e}^{5}}{64\,{b}^{2} \left ( bex+ae \right ) ^{4}} \left ( ex+d \right ) ^{{\frac{7}{2}}}}-{\frac{325\,{e}^{4}d}{64\,b \left ( bex+ae \right ) ^{4}} \left ( ex+d \right ) ^{{\frac{7}{2}}}}+{\frac{765\,{a}^{2}{e}^{6}}{64\,{b}^{3} \left ( bex+ae \right ) ^{4}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{765\,a{e}^{5}d}{32\,{b}^{2} \left ( bex+ae \right ) ^{4}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}+{\frac{765\,{e}^{4}{d}^{2}}{64\,b \left ( bex+ae \right ) ^{4}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}+{\frac{643\,{a}^{3}{e}^{7}}{64\,{b}^{4} \left ( bex+ae \right ) ^{4}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{1929\,{a}^{2}{e}^{6}d}{64\,{b}^{3} \left ( bex+ae \right ) ^{4}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{1929\,a{e}^{5}{d}^{2}}{64\,{b}^{2} \left ( bex+ae \right ) ^{4}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{643\,{e}^{4}{d}^{3}}{64\,b \left ( bex+ae \right ) ^{4}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{187\,{a}^{4}{e}^{8}}{64\,{b}^{5} \left ( bex+ae \right ) ^{4}}\sqrt{ex+d}}-{\frac{187\,{a}^{3}{e}^{7}d}{16\,{b}^{4} \left ( bex+ae \right ) ^{4}}\sqrt{ex+d}}+{\frac{561\,{a}^{2}{e}^{6}{d}^{2}}{32\,{b}^{3} \left ( bex+ae \right ) ^{4}}\sqrt{ex+d}}-{\frac{187\,a{e}^{5}{d}^{3}}{16\,{b}^{2} \left ( bex+ae \right ) ^{4}}\sqrt{ex+d}}+{\frac{187\,{e}^{4}{d}^{4}}{64\,b \left ( bex+ae \right ) ^{4}}\sqrt{ex+d}}-{\frac{315\,a{e}^{5}}{64\,{b}^{5}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}}+{\frac{315\,{e}^{4}d}{64\,{b}^{4}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(e*x+d)^(9/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

2*e^4*(e*x+d)^(1/2)/b^5+325/64*e^5/b^2/(b*e*x+a*e)^4*(e*x+d)^(7/2)*a-325/64*e^4/
b/(b*e*x+a*e)^4*(e*x+d)^(7/2)*d+765/64*e^6/b^3/(b*e*x+a*e)^4*(e*x+d)^(5/2)*a^2-7
65/32*e^5/b^2/(b*e*x+a*e)^4*(e*x+d)^(5/2)*a*d+765/64*e^4/b/(b*e*x+a*e)^4*(e*x+d)
^(5/2)*d^2+643/64*e^7/b^4/(b*e*x+a*e)^4*(e*x+d)^(3/2)*a^3-1929/64*e^6/b^3/(b*e*x
+a*e)^4*(e*x+d)^(3/2)*a^2*d+1929/64*e^5/b^2/(b*e*x+a*e)^4*(e*x+d)^(3/2)*a*d^2-64
3/64*e^4/b/(b*e*x+a*e)^4*(e*x+d)^(3/2)*d^3+187/64*e^8/b^5/(b*e*x+a*e)^4*(e*x+d)^
(1/2)*a^4-187/16*e^7/b^4/(b*e*x+a*e)^4*(e*x+d)^(1/2)*a^3*d+561/32*e^6/b^3/(b*e*x
+a*e)^4*(e*x+d)^(1/2)*a^2*d^2-187/16*e^5/b^2/(b*e*x+a*e)^4*(e*x+d)^(1/2)*a*d^3+1
87/64*e^4/b/(b*e*x+a*e)^4*(e*x+d)^(1/2)*d^4-315/64*e^5/b^5/(b*(a*e-b*d))^(1/2)*a
rctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a+315/64*e^4/b^4/(b*(a*e-b*d))^(1/2)*
arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*d

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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)^(9/2)/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.314319, size = 1, normalized size = 0.01 \[ \left [\frac{315 \,{\left (b^{4} e^{4} x^{4} + 4 \, a b^{3} e^{4} x^{3} + 6 \, a^{2} b^{2} e^{4} x^{2} + 4 \, a^{3} b e^{4} x + a^{4} e^{4}\right )} \sqrt{\frac{b d - a e}{b}} \log \left (\frac{b e x + 2 \, b d - a e - 2 \, \sqrt{e x + d} b \sqrt{\frac{b d - a e}{b}}}{b x + a}\right ) + 2 \,{\left (128 \, b^{4} e^{4} x^{4} - 16 \, b^{4} d^{4} - 24 \, a b^{3} d^{3} e - 42 \, a^{2} b^{2} d^{2} e^{2} - 105 \, a^{3} b d e^{3} + 315 \, a^{4} e^{4} -{\left (325 \, b^{4} d e^{3} - 837 \, a b^{3} e^{4}\right )} x^{3} - 3 \,{\left (70 \, b^{4} d^{2} e^{2} + 185 \, a b^{3} d e^{3} - 511 \, a^{2} b^{2} e^{4}\right )} x^{2} -{\left (88 \, b^{4} d^{3} e + 156 \, a b^{3} d^{2} e^{2} + 399 \, a^{2} b^{2} d e^{3} - 1155 \, a^{3} b e^{4}\right )} x\right )} \sqrt{e x + d}}{128 \,{\left (b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}\right )}}, -\frac{315 \,{\left (b^{4} e^{4} x^{4} + 4 \, a b^{3} e^{4} x^{3} + 6 \, a^{2} b^{2} e^{4} x^{2} + 4 \, a^{3} b e^{4} x + a^{4} e^{4}\right )} \sqrt{-\frac{b d - a e}{b}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{b d - a e}{b}}}\right ) -{\left (128 \, b^{4} e^{4} x^{4} - 16 \, b^{4} d^{4} - 24 \, a b^{3} d^{3} e - 42 \, a^{2} b^{2} d^{2} e^{2} - 105 \, a^{3} b d e^{3} + 315 \, a^{4} e^{4} -{\left (325 \, b^{4} d e^{3} - 837 \, a b^{3} e^{4}\right )} x^{3} - 3 \,{\left (70 \, b^{4} d^{2} e^{2} + 185 \, a b^{3} d e^{3} - 511 \, a^{2} b^{2} e^{4}\right )} x^{2} -{\left (88 \, b^{4} d^{3} e + 156 \, a b^{3} d^{2} e^{2} + 399 \, a^{2} b^{2} d e^{3} - 1155 \, a^{3} b e^{4}\right )} x\right )} \sqrt{e x + d}}{64 \,{\left (b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(e*x + d)^(9/2)/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="fricas")

[Out]

[1/128*(315*(b^4*e^4*x^4 + 4*a*b^3*e^4*x^3 + 6*a^2*b^2*e^4*x^2 + 4*a^3*b*e^4*x +
a^4*e^4)*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*(128*b^4*e^4*x^4 - 16*b^4*d^4 - 24*a*b^3*d^3*e -
42*a^2*b^2*d^2*e^2 - 105*a^3*b*d*e^3 + 315*a^4*e^4 - (325*b^4*d*e^3 - 837*a*b^3*
e^4)*x^3 - 3*(70*b^4*d^2*e^2 + 185*a*b^3*d*e^3 - 511*a^2*b^2*e^4)*x^2 - (88*b^4*
d^3*e + 156*a*b^3*d^2*e^2 + 399*a^2*b^2*d*e^3 - 1155*a^3*b*e^4)*x)*sqrt(e*x + d)
)/(b^9*x^4 + 4*a*b^8*x^3 + 6*a^2*b^7*x^2 + 4*a^3*b^6*x + a^4*b^5), -1/64*(315*(b
^4*e^4*x^4 + 4*a*b^3*e^4*x^3 + 6*a^2*b^2*e^4*x^2 + 4*a^3*b*e^4*x + a^4*e^4)*sqrt
(-(b*d - a*e)/b)*arctan(sqrt(e*x + d)/sqrt(-(b*d - a*e)/b)) - (128*b^4*e^4*x^4 -
16*b^4*d^4 - 24*a*b^3*d^3*e - 42*a^2*b^2*d^2*e^2 - 105*a^3*b*d*e^3 + 315*a^4*e^
4 - (325*b^4*d*e^3 - 837*a*b^3*e^4)*x^3 - 3*(70*b^4*d^2*e^2 + 185*a*b^3*d*e^3 -
511*a^2*b^2*e^4)*x^2 - (88*b^4*d^3*e + 156*a*b^3*d^2*e^2 + 399*a^2*b^2*d*e^3 - 1
155*a^3*b*e^4)*x)*sqrt(e*x + d))/(b^9*x^4 + 4*a*b^8*x^3 + 6*a^2*b^7*x^2 + 4*a^3*
b^6*x + a^4*b^5)]

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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)**(9/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.302851, size = 463, normalized size = 2.71 \[ \frac{315 \,{\left (b d e^{4} - a e^{5}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{64 \, \sqrt{-b^{2} d + a b e} b^{5}} + \frac{2 \, \sqrt{x e + d} e^{4}}{b^{5}} - \frac{325 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{4} d e^{4} - 765 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{4} d^{2} e^{4} + 643 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{4} d^{3} e^{4} - 187 \, \sqrt{x e + d} b^{4} d^{4} e^{4} - 325 \,{\left (x e + d\right )}^{\frac{7}{2}} a b^{3} e^{5} + 1530 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{3} d e^{5} - 1929 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{3} d^{2} e^{5} + 748 \, \sqrt{x e + d} a b^{3} d^{3} e^{5} - 765 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{2} b^{2} e^{6} + 1929 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{2} d e^{6} - 1122 \, \sqrt{x e + d} a^{2} b^{2} d^{2} e^{6} - 643 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} b e^{7} + 748 \, \sqrt{x e + d} a^{3} b d e^{7} - 187 \, \sqrt{x e + d} a^{4} e^{8}}{64 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{4} b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(e*x + d)^(9/2)/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="giac")

[Out]

315/64*(b*d*e^4 - a*e^5)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2
*d + a*b*e)*b^5) + 2*sqrt(x*e + d)*e^4/b^5 - 1/64*(325*(x*e + d)^(7/2)*b^4*d*e^4
- 765*(x*e + d)^(5/2)*b^4*d^2*e^4 + 643*(x*e + d)^(3/2)*b^4*d^3*e^4 - 187*sqrt(
x*e + d)*b^4*d^4*e^4 - 325*(x*e + d)^(7/2)*a*b^3*e^5 + 1530*(x*e + d)^(5/2)*a*b^
3*d*e^5 - 1929*(x*e + d)^(3/2)*a*b^3*d^2*e^5 + 748*sqrt(x*e + d)*a*b^3*d^3*e^5 -
765*(x*e + d)^(5/2)*a^2*b^2*e^6 + 1929*(x*e + d)^(3/2)*a^2*b^2*d*e^6 - 1122*sqr
t(x*e + d)*a^2*b^2*d^2*e^6 - 643*(x*e + d)^(3/2)*a^3*b*e^7 + 748*sqrt(x*e + d)*a
^3*b*d*e^7 - 187*sqrt(x*e + d)*a^4*e^8)/(((x*e + d)*b - b*d + a*e)^4*b^5)