3.1711 \(\int \frac{(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=294 \[ -\frac{(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{315 e^4 (a+b x) \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{315 e^4 (a+b x) \sqrt{d+e x}}{64 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{105 e^3 (d+e x)^{3/2}}{64 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.420883, antiderivative size = 294, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{315 e^4 (a+b x) \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{315 e^4 (a+b x) \sqrt{d+e x}}{64 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{105 e^3 (d+e x)^{3/2}}{64 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(9/2)/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

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

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.58492, size = 179, normalized size = 0.61 \[ \frac{(a+b x)^5 \left (-\frac{315 e^4 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{11/2}}-\frac{\sqrt{d+e x} \left (325 e^3 (a+b x)^3 (b d-a e)+210 e^2 (a+b x)^2 (b d-a e)^2+88 e (a+b x) (b d-a e)^3+16 (b d-a e)^4-128 e^4 (a+b x)^4\right )}{b^5 (a+b x)^4}\right )}{64 \left ((a+b x)^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(9/2)/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

((a + b*x)^5*(-((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))/(b^5*(a + b*x)^4)) - (315*e^4*Sqrt[b*d - a*e]*ArcTanh[(Sqrt[b]*Sqr
t[d + e*x])/Sqrt[b*d - a*e]])/b^(11/2)))/(64*((a + b*x)^2)^(5/2))

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Maple [B]  time = 0.032, size = 892, normalized size = 3. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(9/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/64*(1260*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^3*a*b^4*d*e^4+512*(b*(a
*e-b*d))^(1/2)*(e*x+d)^(1/2)*x^3*a*b^3*e^4-1530*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2
)*a*b^3*d*e-1260*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x*a^4*b*e^5+643*(b*
(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a^3*b*e^3+315*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d)
)^(1/2))*a^4*b*d*e^4-1890*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^2*a^3*b^
2*e^5+765*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*a^2*b^2*e^2+1890*arctan((e*x+d)^(1/2
)*b/(b*(a*e-b*d))^(1/2))*x^2*a^2*b^3*d*e^4+768*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)
*x^2*a^2*b^2*e^4+1260*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x*a^3*b^2*d*e^
4-1929*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a^2*b^2*d*e^2+1929*(b*(a*e-b*d))^(1/2)*
(e*x+d)^(3/2)*a*b^3*d^2*e-315*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^4*a*
b^4*e^5+315*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^4*b^5*d*e^4+128*(b*(a*
e-b*d))^(1/2)*(e*x+d)^(1/2)*x^4*b^4*e^4-1260*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d)
)^(1/2))*x^3*a^2*b^3*e^5+325*(b*(a*e-b*d))^(1/2)*(e*x+d)^(7/2)*a*b^3*e+315*(b*(a
*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^4*e^4+187*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*b^4*d
^4-643*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*b^4*d^3-315*arctan((e*x+d)^(1/2)*b/(b*(
a*e-b*d))^(1/2))*a^5*e^5-325*(b*(a*e-b*d))^(1/2)*(e*x+d)^(7/2)*b^4*d+765*(b*(a*e
-b*d))^(1/2)*(e*x+d)^(5/2)*b^4*d^2+512*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x*a^3*b
*e^4-748*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^3*b*d*e^3+1122*(b*(a*e-b*d))^(1/2)*
(e*x+d)^(1/2)*a^2*b^2*d^2*e^2-748*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a*b^3*d^3*e)
*(b*x+a)/(b*(a*e-b*d))^(1/2)/b^5/((b*x+a)^2)^(5/2)

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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((e*x + d)^(9/2)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224558, size = 1, normalized size = 0. \[ \left [\frac{315 \,{\left (b^{4} e^{4} x^{4} + 4 \, a b^{3} e^{4} x^{3} + 6 \, a^{2} b^{2} e^{4} x^{2} + 4 \, a^{3} b e^{4} x + a^{4} e^{4}\right )} \sqrt{\frac{b d - a e}{b}} \log \left (\frac{b e x + 2 \, b d - a e - 2 \, \sqrt{e x + d} b \sqrt{\frac{b d - a e}{b}}}{b x + a}\right ) + 2 \,{\left (128 \, b^{4} e^{4} x^{4} - 16 \, b^{4} d^{4} - 24 \, a b^{3} d^{3} e - 42 \, a^{2} b^{2} d^{2} e^{2} - 105 \, a^{3} b d e^{3} + 315 \, a^{4} e^{4} -{\left (325 \, b^{4} d e^{3} - 837 \, a b^{3} e^{4}\right )} x^{3} - 3 \,{\left (70 \, b^{4} d^{2} e^{2} + 185 \, a b^{3} d e^{3} - 511 \, a^{2} b^{2} e^{4}\right )} x^{2} -{\left (88 \, b^{4} d^{3} e + 156 \, a b^{3} d^{2} e^{2} + 399 \, a^{2} b^{2} d e^{3} - 1155 \, a^{3} b e^{4}\right )} x\right )} \sqrt{e x + d}}{128 \,{\left (b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}\right )}}, -\frac{315 \,{\left (b^{4} e^{4} x^{4} + 4 \, a b^{3} e^{4} x^{3} + 6 \, a^{2} b^{2} e^{4} x^{2} + 4 \, a^{3} b e^{4} x + a^{4} e^{4}\right )} \sqrt{-\frac{b d - a e}{b}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{b d - a e}{b}}}\right ) -{\left (128 \, b^{4} e^{4} x^{4} - 16 \, b^{4} d^{4} - 24 \, a b^{3} d^{3} e - 42 \, a^{2} b^{2} d^{2} e^{2} - 105 \, a^{3} b d e^{3} + 315 \, a^{4} e^{4} -{\left (325 \, b^{4} d e^{3} - 837 \, a b^{3} e^{4}\right )} x^{3} - 3 \,{\left (70 \, b^{4} d^{2} e^{2} + 185 \, a b^{3} d e^{3} - 511 \, a^{2} b^{2} e^{4}\right )} x^{2} -{\left (88 \, b^{4} d^{3} e + 156 \, a b^{3} d^{2} e^{2} + 399 \, a^{2} b^{2} d e^{3} - 1155 \, a^{3} b e^{4}\right )} x\right )} \sqrt{e x + d}}{64 \,{\left (b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(9/2)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),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((e*x+d)**(9/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.252999, size = 564, normalized size = 1.92 \[ -\frac{315 \,{\left (b d e^{4} - a e^{5}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{64 \, \sqrt{-b^{2} d + a b e} b^{5}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} - \frac{2 \, \sqrt{x e + d} e^{4}}{b^{5}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} + \frac{325 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{4} d e^{4} - 765 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{4} d^{2} e^{4} + 643 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{4} d^{3} e^{4} - 187 \, \sqrt{x e + d} b^{4} d^{4} e^{4} - 325 \,{\left (x e + d\right )}^{\frac{7}{2}} a b^{3} e^{5} + 1530 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{3} d e^{5} - 1929 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{3} d^{2} e^{5} + 748 \, \sqrt{x e + d} a b^{3} d^{3} e^{5} - 765 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{2} b^{2} e^{6} + 1929 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{2} d e^{6} - 1122 \, \sqrt{x e + d} a^{2} b^{2} d^{2} e^{6} - 643 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} b e^{7} + 748 \, \sqrt{x e + d} a^{3} b d e^{7} - 187 \, \sqrt{x e + d} a^{4} e^{8}}{64 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{4} b^{5}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(9/2)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),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*sign(-(x*e + d)*b*e + b*d*e - a*e^2)) - 2*sqrt(x*e + d)*e^4/(b^
5*sign(-(x*e + d)*b*e + b*d*e - a*e^2)) + 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 - 76
5*(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*sqrt(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*sign(-(x*e
+ d)*b*e + b*d*e - a*e^2))