3.1656 \(\int \frac{(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=178 \[ -\frac{63 e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{11/2} \sqrt{b d-a e}}-\frac{63 e^4 \sqrt{d+e x}}{128 b^5 (a+b x)}-\frac{21 e^3 (d+e x)^{3/2}}{64 b^4 (a+b x)^2}-\frac{21 e^2 (d+e x)^{5/2}}{80 b^3 (a+b x)^3}-\frac{9 e (d+e x)^{7/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{9/2}}{5 b (a+b x)^5} \]

[Out]

(-63*e^4*Sqrt[d + e*x])/(128*b^5*(a + b*x)) - (21*e^3*(d + e*x)^(3/2))/(64*b^4*(
a + b*x)^2) - (21*e^2*(d + e*x)^(5/2))/(80*b^3*(a + b*x)^3) - (9*e*(d + e*x)^(7/
2))/(40*b^2*(a + b*x)^4) - (d + e*x)^(9/2)/(5*b*(a + b*x)^5) - (63*e^5*ArcTanh[(
Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*b^(11/2)*Sqrt[b*d - a*e])

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Rubi [A]  time = 0.271623, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{63 e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{11/2} \sqrt{b d-a e}}-\frac{63 e^4 \sqrt{d+e x}}{128 b^5 (a+b x)}-\frac{21 e^3 (d+e x)^{3/2}}{64 b^4 (a+b x)^2}-\frac{21 e^2 (d+e x)^{5/2}}{80 b^3 (a+b x)^3}-\frac{9 e (d+e x)^{7/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{9/2}}{5 b (a+b x)^5} \]

Antiderivative was successfully verified.

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

[Out]

(-63*e^4*Sqrt[d + e*x])/(128*b^5*(a + b*x)) - (21*e^3*(d + e*x)^(3/2))/(64*b^4*(
a + b*x)^2) - (21*e^2*(d + e*x)^(5/2))/(80*b^3*(a + b*x)^3) - (9*e*(d + e*x)^(7/
2))/(40*b^2*(a + b*x)^4) - (d + e*x)^(9/2)/(5*b*(a + b*x)^5) - (63*e^5*ArcTanh[(
Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*b^(11/2)*Sqrt[b*d - a*e])

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Rubi in Sympy [A]  time = 67.6612, size = 163, normalized size = 0.92 \[ - \frac{\left (d + e x\right )^{\frac{9}{2}}}{5 b \left (a + b x\right )^{5}} - \frac{9 e \left (d + e x\right )^{\frac{7}{2}}}{40 b^{2} \left (a + b x\right )^{4}} - \frac{21 e^{2} \left (d + e x\right )^{\frac{5}{2}}}{80 b^{3} \left (a + b x\right )^{3}} - \frac{21 e^{3} \left (d + e x\right )^{\frac{3}{2}}}{64 b^{4} \left (a + b x\right )^{2}} - \frac{63 e^{4} \sqrt{d + e x}}{128 b^{5} \left (a + b x\right )} + \frac{63 e^{5} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{128 b^{\frac{11}{2}} \sqrt{a e - b d}} \]

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)**3,x)

[Out]

-(d + e*x)**(9/2)/(5*b*(a + b*x)**5) - 9*e*(d + e*x)**(7/2)/(40*b**2*(a + b*x)**
4) - 21*e**2*(d + e*x)**(5/2)/(80*b**3*(a + b*x)**3) - 21*e**3*(d + e*x)**(3/2)/
(64*b**4*(a + b*x)**2) - 63*e**4*sqrt(d + e*x)/(128*b**5*(a + b*x)) + 63*e**5*at
an(sqrt(b)*sqrt(d + e*x)/sqrt(a*e - b*d))/(128*b**(11/2)*sqrt(a*e - b*d))

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Mathematica [A]  time = 0.399184, size = 161, normalized size = 0.9 \[ -\frac{63 e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{11/2} \sqrt{b d-a e}}-\frac{\sqrt{d+e x} \left (1490 e^3 (a+b x)^3 (b d-a e)+1368 e^2 (a+b x)^2 (b d-a e)^2+656 e (a+b x) (b d-a e)^3+128 (b d-a e)^4+965 e^4 (a+b x)^4\right )}{640 b^5 (a+b x)^5} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[d + e*x]*(128*(b*d - a*e)^4 + 656*e*(b*d - a*e)^3*(a + b*x) + 1368*e^2*(b
*d - a*e)^2*(a + b*x)^2 + 1490*e^3*(b*d - a*e)*(a + b*x)^3 + 965*e^4*(a + b*x)^4
))/(640*b^5*(a + b*x)^5) - (63*e^5*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*
e]])/(128*b^(11/2)*Sqrt[b*d - a*e])

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Maple [B]  time = 0.029, size = 463, normalized size = 2.6 \[ -{\frac{193\,{e}^{5}}{128\, \left ( bex+ae \right ) ^{5}b} \left ( ex+d \right ) ^{{\frac{9}{2}}}}-{\frac{237\,{e}^{6}a}{64\, \left ( bex+ae \right ) ^{5}{b}^{2}} \left ( ex+d \right ) ^{{\frac{7}{2}}}}+{\frac{237\,{e}^{5}d}{64\, \left ( bex+ae \right ) ^{5}b} \left ( ex+d \right ) ^{{\frac{7}{2}}}}-{\frac{21\,{a}^{2}{e}^{7}}{5\, \left ( bex+ae \right ) ^{5}{b}^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}+{\frac{42\,{e}^{6}ad}{5\, \left ( bex+ae \right ) ^{5}{b}^{2}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{21\,{e}^{5}{d}^{2}}{5\, \left ( bex+ae \right ) ^{5}b} \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{147\,{e}^{8}{a}^{3}}{64\, \left ( bex+ae \right ) ^{5}{b}^{4}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{441\,{a}^{2}{e}^{7}d}{64\, \left ( bex+ae \right ) ^{5}{b}^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{441\,{e}^{6}a{d}^{2}}{64\, \left ( bex+ae \right ) ^{5}{b}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{147\,{e}^{5}{d}^{3}}{64\, \left ( bex+ae \right ) ^{5}b} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{63\,{e}^{9}{a}^{4}}{128\, \left ( bex+ae \right ) ^{5}{b}^{5}}\sqrt{ex+d}}+{\frac{63\,{e}^{8}{a}^{3}d}{32\, \left ( bex+ae \right ) ^{5}{b}^{4}}\sqrt{ex+d}}-{\frac{189\,{a}^{2}{e}^{7}{d}^{2}}{64\, \left ( bex+ae \right ) ^{5}{b}^{3}}\sqrt{ex+d}}+{\frac{63\,{e}^{6}a{d}^{3}}{32\, \left ( bex+ae \right ) ^{5}{b}^{2}}\sqrt{ex+d}}-{\frac{63\,{e}^{5}{d}^{4}}{128\, \left ( bex+ae \right ) ^{5}b}\sqrt{ex+d}}+{\frac{63\,{e}^{5}}{128\,{b}^{5}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \]

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)^3,x)

[Out]

-193/128*e^5/(b*e*x+a*e)^5/b*(e*x+d)^(9/2)-237/64*e^6/(b*e*x+a*e)^5/b^2*(e*x+d)^
(7/2)*a+237/64*e^5/(b*e*x+a*e)^5/b*(e*x+d)^(7/2)*d-21/5*e^7/(b*e*x+a*e)^5/b^3*(e
*x+d)^(5/2)*a^2+42/5*e^6/(b*e*x+a*e)^5/b^2*(e*x+d)^(5/2)*a*d-21/5*e^5/(b*e*x+a*e
)^5/b*(e*x+d)^(5/2)*d^2-147/64*e^8/(b*e*x+a*e)^5/b^4*(e*x+d)^(3/2)*a^3+441/64*e^
7/(b*e*x+a*e)^5/b^3*(e*x+d)^(3/2)*a^2*d-441/64*e^6/(b*e*x+a*e)^5/b^2*(e*x+d)^(3/
2)*a*d^2+147/64*e^5/(b*e*x+a*e)^5/b*(e*x+d)^(3/2)*d^3-63/128*e^9/(b*e*x+a*e)^5/b
^5*(e*x+d)^(1/2)*a^4+63/32*e^8/(b*e*x+a*e)^5/b^4*(e*x+d)^(1/2)*a^3*d-189/64*e^7/
(b*e*x+a*e)^5/b^3*(e*x+d)^(1/2)*a^2*d^2+63/32*e^6/(b*e*x+a*e)^5/b^2*(e*x+d)^(1/2
)*a*d^3-63/128*e^5/(b*e*x+a*e)^5/b*(e*x+d)^(1/2)*d^4+63/128*e^5/b^5/(b*(a*e-b*d)
)^(1/2)*arctan((e*x+d)^(1/2)*b/(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((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 = 0.226903, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (965 \, b^{4} e^{4} x^{4} + 128 \, b^{4} d^{4} + 144 \, a b^{3} d^{3} e + 168 \, a^{2} b^{2} d^{2} e^{2} + 210 \, a^{3} b d e^{3} + 315 \, a^{4} e^{4} + 10 \,{\left (149 \, b^{4} d e^{3} + 237 \, a b^{3} e^{4}\right )} x^{3} + 6 \,{\left (228 \, b^{4} d^{2} e^{2} + 289 \, a b^{3} d e^{3} + 448 \, a^{2} b^{2} e^{4}\right )} x^{2} + 2 \,{\left (328 \, b^{4} d^{3} e + 384 \, a b^{3} d^{2} e^{2} + 483 \, a^{2} b^{2} d e^{3} + 735 \, a^{3} b e^{4}\right )} x\right )} \sqrt{b^{2} d - a b e} \sqrt{e x + d} - 315 \,{\left (b^{5} e^{5} x^{5} + 5 \, a b^{4} e^{5} x^{4} + 10 \, a^{2} b^{3} e^{5} x^{3} + 10 \, a^{3} b^{2} e^{5} x^{2} + 5 \, a^{4} b e^{5} x + a^{5} e^{5}\right )} \log \left (\frac{\sqrt{b^{2} d - a b e}{\left (b e x + 2 \, b d - a e\right )} - 2 \,{\left (b^{2} d - a b e\right )} \sqrt{e x + d}}{b x + a}\right )}{1280 \,{\left (b^{10} x^{5} + 5 \, a b^{9} x^{4} + 10 \, a^{2} b^{8} x^{3} + 10 \, a^{3} b^{7} x^{2} + 5 \, a^{4} b^{6} x + a^{5} b^{5}\right )} \sqrt{b^{2} d - a b e}}, -\frac{{\left (965 \, b^{4} e^{4} x^{4} + 128 \, b^{4} d^{4} + 144 \, a b^{3} d^{3} e + 168 \, a^{2} b^{2} d^{2} e^{2} + 210 \, a^{3} b d e^{3} + 315 \, a^{4} e^{4} + 10 \,{\left (149 \, b^{4} d e^{3} + 237 \, a b^{3} e^{4}\right )} x^{3} + 6 \,{\left (228 \, b^{4} d^{2} e^{2} + 289 \, a b^{3} d e^{3} + 448 \, a^{2} b^{2} e^{4}\right )} x^{2} + 2 \,{\left (328 \, b^{4} d^{3} e + 384 \, a b^{3} d^{2} e^{2} + 483 \, a^{2} b^{2} d e^{3} + 735 \, a^{3} b e^{4}\right )} x\right )} \sqrt{-b^{2} d + a b e} \sqrt{e x + d} + 315 \,{\left (b^{5} e^{5} x^{5} + 5 \, a b^{4} e^{5} x^{4} + 10 \, a^{2} b^{3} e^{5} x^{3} + 10 \, a^{3} b^{2} e^{5} x^{2} + 5 \, a^{4} b e^{5} x + a^{5} e^{5}\right )} \arctan \left (-\frac{b d - a e}{\sqrt{-b^{2} d + a b e} \sqrt{e x + d}}\right )}{640 \,{\left (b^{10} x^{5} + 5 \, a b^{9} x^{4} + 10 \, a^{2} b^{8} x^{3} + 10 \, a^{3} b^{7} x^{2} + 5 \, a^{4} b^{6} x + a^{5} b^{5}\right )} \sqrt{-b^{2} d + a b e}}\right ] \]

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)^3,x, algorithm="fricas")

[Out]

[-1/1280*(2*(965*b^4*e^4*x^4 + 128*b^4*d^4 + 144*a*b^3*d^3*e + 168*a^2*b^2*d^2*e
^2 + 210*a^3*b*d*e^3 + 315*a^4*e^4 + 10*(149*b^4*d*e^3 + 237*a*b^3*e^4)*x^3 + 6*
(228*b^4*d^2*e^2 + 289*a*b^3*d*e^3 + 448*a^2*b^2*e^4)*x^2 + 2*(328*b^4*d^3*e + 3
84*a*b^3*d^2*e^2 + 483*a^2*b^2*d*e^3 + 735*a^3*b*e^4)*x)*sqrt(b^2*d - a*b*e)*sqr
t(e*x + d) - 315*(b^5*e^5*x^5 + 5*a*b^4*e^5*x^4 + 10*a^2*b^3*e^5*x^3 + 10*a^3*b^
2*e^5*x^2 + 5*a^4*b*e^5*x + a^5*e^5)*log((sqrt(b^2*d - a*b*e)*(b*e*x + 2*b*d - a
*e) - 2*(b^2*d - a*b*e)*sqrt(e*x + d))/(b*x + a)))/((b^10*x^5 + 5*a*b^9*x^4 + 10
*a^2*b^8*x^3 + 10*a^3*b^7*x^2 + 5*a^4*b^6*x + a^5*b^5)*sqrt(b^2*d - a*b*e)), -1/
640*((965*b^4*e^4*x^4 + 128*b^4*d^4 + 144*a*b^3*d^3*e + 168*a^2*b^2*d^2*e^2 + 21
0*a^3*b*d*e^3 + 315*a^4*e^4 + 10*(149*b^4*d*e^3 + 237*a*b^3*e^4)*x^3 + 6*(228*b^
4*d^2*e^2 + 289*a*b^3*d*e^3 + 448*a^2*b^2*e^4)*x^2 + 2*(328*b^4*d^3*e + 384*a*b^
3*d^2*e^2 + 483*a^2*b^2*d*e^3 + 735*a^3*b*e^4)*x)*sqrt(-b^2*d + a*b*e)*sqrt(e*x
+ d) + 315*(b^5*e^5*x^5 + 5*a*b^4*e^5*x^4 + 10*a^2*b^3*e^5*x^3 + 10*a^3*b^2*e^5*
x^2 + 5*a^4*b*e^5*x + a^5*e^5)*arctan(-(b*d - a*e)/(sqrt(-b^2*d + a*b*e)*sqrt(e*
x + d))))/((b^10*x^5 + 5*a*b^9*x^4 + 10*a^2*b^8*x^3 + 10*a^3*b^7*x^2 + 5*a^4*b^6
*x + a^5*b^5)*sqrt(-b^2*d + a*b*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((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 = 0.236398, size = 451, normalized size = 2.53 \[ \frac{63 \, \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{5}}{128 \, \sqrt{-b^{2} d + a b e} b^{5}} - \frac{965 \,{\left (x e + d\right )}^{\frac{9}{2}} b^{4} e^{5} - 2370 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{4} d e^{5} + 2688 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{4} d^{2} e^{5} - 1470 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{4} d^{3} e^{5} + 315 \, \sqrt{x e + d} b^{4} d^{4} e^{5} + 2370 \,{\left (x e + d\right )}^{\frac{7}{2}} a b^{3} e^{6} - 5376 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{3} d e^{6} + 4410 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{3} d^{2} e^{6} - 1260 \, \sqrt{x e + d} a b^{3} d^{3} e^{6} + 2688 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{2} b^{2} e^{7} - 4410 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{2} d e^{7} + 1890 \, \sqrt{x e + d} a^{2} b^{2} d^{2} e^{7} + 1470 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} b e^{8} - 1260 \, \sqrt{x e + d} a^{3} b d e^{8} + 315 \, \sqrt{x e + d} a^{4} e^{9}}{640 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{5} b^{5}} \]

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)^3,x, algorithm="giac")

[Out]

63/128*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^5/(sqrt(-b^2*d + a*b*e)*b^
5) - 1/640*(965*(x*e + d)^(9/2)*b^4*e^5 - 2370*(x*e + d)^(7/2)*b^4*d*e^5 + 2688*
(x*e + d)^(5/2)*b^4*d^2*e^5 - 1470*(x*e + d)^(3/2)*b^4*d^3*e^5 + 315*sqrt(x*e +
d)*b^4*d^4*e^5 + 2370*(x*e + d)^(7/2)*a*b^3*e^6 - 5376*(x*e + d)^(5/2)*a*b^3*d*e
^6 + 4410*(x*e + d)^(3/2)*a*b^3*d^2*e^6 - 1260*sqrt(x*e + d)*a*b^3*d^3*e^6 + 268
8*(x*e + d)^(5/2)*a^2*b^2*e^7 - 4410*(x*e + d)^(3/2)*a^2*b^2*d*e^7 + 1890*sqrt(x
*e + d)*a^2*b^2*d^2*e^7 + 1470*(x*e + d)^(3/2)*a^3*b*e^8 - 1260*sqrt(x*e + d)*a^
3*b*d*e^8 + 315*sqrt(x*e + d)*a^4*e^9)/(((x*e + d)*b - b*d + a*e)^5*b^5)