3.1653 \(\int \frac{(d+e x)^{15/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\)
Optimal. Leaf size=253 \[ -\frac{9009 e^5 (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{17/2}}+\frac{9009 e^5 \sqrt{d+e x} (b d-a e)^2}{128 b^8}+\frac{3003 e^5 (d+e x)^{3/2} (b d-a e)}{128 b^7}-\frac{1287 e^4 (d+e x)^{7/2}}{128 b^5 (a+b x)}-\frac{143 e^3 (d+e x)^{9/2}}{64 b^4 (a+b x)^2}-\frac{13 e^2 (d+e x)^{11/2}}{16 b^3 (a+b x)^3}-\frac{3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac{(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac{9009 e^5 (d+e x)^{5/2}}{640 b^6} \]
[Out]
(9009*e^5*(b*d - a*e)^2*Sqrt[d + e*x])/(128*b^8) + (3003*e^5*(b*d - a*e)*(d + e*
x)^(3/2))/(128*b^7) + (9009*e^5*(d + e*x)^(5/2))/(640*b^6) - (1287*e^4*(d + e*x)
^(7/2))/(128*b^5*(a + b*x)) - (143*e^3*(d + e*x)^(9/2))/(64*b^4*(a + b*x)^2) - (
13*e^2*(d + e*x)^(11/2))/(16*b^3*(a + b*x)^3) - (3*e*(d + e*x)^(13/2))/(8*b^2*(a
+ b*x)^4) - (d + e*x)^(15/2)/(5*b*(a + b*x)^5) - (9009*e^5*(b*d - a*e)^(5/2)*Ar
cTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*b^(17/2))
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Rubi [A] time = 0.5059, antiderivative size = 253, normalized size of antiderivative = 1.,
number of steps used = 11, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179
\[ -\frac{9009 e^5 (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{17/2}}+\frac{9009 e^5 \sqrt{d+e x} (b d-a e)^2}{128 b^8}+\frac{3003 e^5 (d+e x)^{3/2} (b d-a e)}{128 b^7}-\frac{1287 e^4 (d+e x)^{7/2}}{128 b^5 (a+b x)}-\frac{143 e^3 (d+e x)^{9/2}}{64 b^4 (a+b x)^2}-\frac{13 e^2 (d+e x)^{11/2}}{16 b^3 (a+b x)^3}-\frac{3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac{(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac{9009 e^5 (d+e x)^{5/2}}{640 b^6} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^(15/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
(9009*e^5*(b*d - a*e)^2*Sqrt[d + e*x])/(128*b^8) + (3003*e^5*(b*d - a*e)*(d + e*
x)^(3/2))/(128*b^7) + (9009*e^5*(d + e*x)^(5/2))/(640*b^6) - (1287*e^4*(d + e*x)
^(7/2))/(128*b^5*(a + b*x)) - (143*e^3*(d + e*x)^(9/2))/(64*b^4*(a + b*x)^2) - (
13*e^2*(d + e*x)^(11/2))/(16*b^3*(a + b*x)^3) - (3*e*(d + e*x)^(13/2))/(8*b^2*(a
+ b*x)^4) - (d + e*x)^(15/2)/(5*b*(a + b*x)^5) - (9009*e^5*(b*d - a*e)^(5/2)*Ar
cTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*b^(17/2))
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Rubi in Sympy [A] time = 112.261, size = 235, normalized size = 0.93 \[ - \frac{\left (d + e x\right )^{\frac{15}{2}}}{5 b \left (a + b x\right )^{5}} - \frac{3 e \left (d + e x\right )^{\frac{13}{2}}}{8 b^{2} \left (a + b x\right )^{4}} - \frac{13 e^{2} \left (d + e x\right )^{\frac{11}{2}}}{16 b^{3} \left (a + b x\right )^{3}} - \frac{143 e^{3} \left (d + e x\right )^{\frac{9}{2}}}{64 b^{4} \left (a + b x\right )^{2}} - \frac{1287 e^{4} \left (d + e x\right )^{\frac{7}{2}}}{128 b^{5} \left (a + b x\right )} + \frac{9009 e^{5} \left (d + e x\right )^{\frac{5}{2}}}{640 b^{6}} - \frac{3003 e^{5} \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )}{128 b^{7}} + \frac{9009 e^{5} \sqrt{d + e x} \left (a e - b d\right )^{2}}{128 b^{8}} - \frac{9009 e^{5} \left (a e - b d\right )^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{128 b^{\frac{17}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(15/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
-(d + e*x)**(15/2)/(5*b*(a + b*x)**5) - 3*e*(d + e*x)**(13/2)/(8*b**2*(a + b*x)*
*4) - 13*e**2*(d + e*x)**(11/2)/(16*b**3*(a + b*x)**3) - 143*e**3*(d + e*x)**(9/
2)/(64*b**4*(a + b*x)**2) - 1287*e**4*(d + e*x)**(7/2)/(128*b**5*(a + b*x)) + 90
09*e**5*(d + e*x)**(5/2)/(640*b**6) - 3003*e**5*(d + e*x)**(3/2)*(a*e - b*d)/(12
8*b**7) + 9009*e**5*sqrt(d + e*x)*(a*e - b*d)**2/(128*b**8) - 9009*e**5*(a*e - b
*d)**(5/2)*atan(sqrt(b)*sqrt(d + e*x)/sqrt(a*e - b*d))/(128*b**(17/2))
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Mathematica [A] time = 0.878064, size = 249, normalized size = 0.98 \[ -\frac{\sqrt{d+e x} \left (-256 e^5 (a+b x)^5 \left (105 a^2 e^2-220 a b d e+116 b^2 d^2\right )-256 b^2 e^7 x^2 (a+b x)^5-512 b e^6 x (a+b x)^5 (6 b d-5 a e)+26635 e^4 (a+b x)^4 (b d-a e)^3+12110 e^3 (a+b x)^3 (b d-a e)^4+4648 e^2 (a+b x)^2 (b d-a e)^5+1136 e (a+b x) (b d-a e)^6+128 (b d-a e)^7\right )}{640 b^8 (a+b x)^5}-\frac{9009 e^5 (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{17/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^(15/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
-(Sqrt[d + e*x]*(128*(b*d - a*e)^7 + 1136*e*(b*d - a*e)^6*(a + b*x) + 4648*e^2*(
b*d - a*e)^5*(a + b*x)^2 + 12110*e^3*(b*d - a*e)^4*(a + b*x)^3 + 26635*e^4*(b*d
- a*e)^3*(a + b*x)^4 - 256*e^5*(116*b^2*d^2 - 220*a*b*d*e + 105*a^2*e^2)*(a + b*
x)^5 - 512*b*e^6*(6*b*d - 5*a*e)*x*(a + b*x)^5 - 256*b^2*e^7*x^2*(a + b*x)^5))/(
640*b^8*(a + b*x)^5) - (9009*e^5*(b*d - a*e)^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x
])/Sqrt[b*d - a*e]])/(128*b^(17/2))
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Maple [B] time = 0.054, size = 1164, normalized size = 4.6 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(15/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)
[Out]
4*e^5/b^6*d*(e*x+d)^(3/2)+42*e^5/b^6*d^2*(e*x+d)^(1/2)-4*e^6/b^7*a*(e*x+d)^(3/2)
+42*e^7/b^8*a^2*(e*x+d)^(1/2)-127155/128*e^9/b^5/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a^4
*d^3-9443/16*e^8/b^4/(b*e*x+a*e)^5*(e*x+d)^(7/2)*a^3*d+28329/32*e^7/b^3/(b*e*x+a
*e)^5*(e*x+d)^(7/2)*a^2*d^2-2002*e^7/b^3/(b*e*x+a*e)^5*(e*x+d)^(5/2)*a^2*d^3+100
1*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(5/2)*a*d^4-23511/32*e^10/b^6/(b*e*x+a*e)^5*(e*x
+d)^(3/2)*a^5*d+117555/64*e^9/b^5/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a^4*d^2-39185/16*e
^8/b^4/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a^3*d^3+117555/64*e^7/b^3/(b*e*x+a*e)^5*(e*x+
d)^(3/2)*a^2*d^4-23511/32*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a*d^5-25431/128*e^
11/b^7/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a^6*d+76293/128*e^10/b^6/(b*e*x+a*e)^5*(e*x+d
)^(1/2)*a^5*d^2+127155/128*e^8/b^4/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a^3*d^4+27027/128
*e^7/b^7/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^2*d-2
7027/128*e^6/b^6/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))
*a*d^2+2002*e^8/b^4/(b*e*x+a*e)^5*(e*x+d)^(5/2)*a^3*d^2-9443/16*e^6/b^2/(b*e*x+a
*e)^5*(e*x+d)^(7/2)*a*d^3-76293/128*e^7/b^3/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a^2*d^5+
25431/128*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a*d^6+15981/128*e^6/b^2/(b*e*x+a*e
)^5*(e*x+d)^(9/2)*a*d^2-1001*e^9/b^5/(b*e*x+a*e)^5*(e*x+d)^(5/2)*a^4*d-15981/128
*e^7/b^3/(b*e*x+a*e)^5*(e*x+d)^(9/2)*a^2*d+2/5*e^5*(e*x+d)^(5/2)/b^6+9009/128*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))*d^3-5327/1
28*e^5/b/(b*e*x+a*e)^5*(e*x+d)^(9/2)*d^3-1001/5*e^5/b/(b*e*x+a*e)^5*(e*x+d)^(5/2
)*d^5+7837/64*e^5/b/(b*e*x+a*e)^5*(e*x+d)^(3/2)*d^6-3633/128*e^5/b/(b*e*x+a*e)^5
*(e*x+d)^(1/2)*d^7+9443/64*e^5/b/(b*e*x+a*e)^5*(e*x+d)^(7/2)*d^4-9009/128*e^8/b^
8/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^3+3633/128*e
^12/b^8/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a^7+5327/128*e^8/b^4/(b*e*x+a*e)^5*(e*x+d)^(
9/2)*a^3+1001/5*e^10/b^6/(b*e*x+a*e)^5*(e*x+d)^(5/2)*a^5+7837/64*e^11/b^7/(b*e*x
+a*e)^5*(e*x+d)^(3/2)*a^6+9443/64*e^9/b^5/(b*e*x+a*e)^5*(e*x+d)^(7/2)*a^4-84*e^6
/b^7*d*a*(e*x+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)^(15/2)/(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 = 0.233592, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(15/2)/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="fricas")
[Out]
[1/1280*(45045*(a^5*b^2*d^2*e^5 - 2*a^6*b*d*e^6 + a^7*e^7 + (b^7*d^2*e^5 - 2*a*b
^6*d*e^6 + a^2*b^5*e^7)*x^5 + 5*(a*b^6*d^2*e^5 - 2*a^2*b^5*d*e^6 + a^3*b^4*e^7)*
x^4 + 10*(a^2*b^5*d^2*e^5 - 2*a^3*b^4*d*e^6 + a^4*b^3*e^7)*x^3 + 10*(a^3*b^4*d^2
*e^5 - 2*a^4*b^3*d*e^6 + a^5*b^2*e^7)*x^2 + 5*(a^4*b^3*d^2*e^5 - 2*a^5*b^2*d*e^6
+ a^6*b*e^7)*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*(256*b^7*e^7*x^7 - 128*b^7*d^7 - 240*a*b^6
*d^6*e - 520*a^2*b^5*d^5*e^2 - 1430*a^3*b^4*d^4*e^3 - 6435*a^4*b^3*d^3*e^4 + 690
69*a^5*b^2*d^2*e^5 - 105105*a^6*b*d*e^6 + 45045*a^7*e^7 + 256*(12*b^7*d*e^6 - 5*
a*b^6*e^7)*x^6 + 256*(116*b^7*d^2*e^5 - 160*a*b^6*d*e^6 + 65*a^2*b^5*e^7)*x^5 -
5*(5327*b^7*d^3*e^4 - 45677*a*b^6*d^2*e^5 + 66157*a^2*b^5*d*e^6 - 27599*a^3*b^4*
e^7)*x^4 - 10*(1211*b^7*d^4*e^3 + 5810*a*b^6*d^3*e^4 - 54392*a^2*b^5*d^2*e^5 + 8
0366*a^3*b^4*d*e^6 - 33891*a^4*b^3*e^7)*x^3 - 2*(2324*b^7*d^5*e^2 + 6545*a*b^6*d
^4*e^3 + 30485*a^2*b^5*d^3*e^4 - 302445*a^3*b^4*d^2*e^5 + 452595*a^4*b^3*d*e^6 -
192192*a^5*b^2*e^7)*x^2 - 2*(568*b^7*d^6*e + 1240*a*b^6*d^5*e^2 + 3445*a^2*b^5*
d^4*e^3 + 15730*a^3*b^4*d^3*e^4 - 163020*a^4*b^3*d^2*e^5 + 246246*a^5*b^2*d*e^6
- 105105*a^6*b*e^7)*x)*sqrt(e*x + d))/(b^13*x^5 + 5*a*b^12*x^4 + 10*a^2*b^11*x^3
+ 10*a^3*b^10*x^2 + 5*a^4*b^9*x + a^5*b^8), -1/640*(45045*(a^5*b^2*d^2*e^5 - 2*
a^6*b*d*e^6 + a^7*e^7 + (b^7*d^2*e^5 - 2*a*b^6*d*e^6 + a^2*b^5*e^7)*x^5 + 5*(a*b
^6*d^2*e^5 - 2*a^2*b^5*d*e^6 + a^3*b^4*e^7)*x^4 + 10*(a^2*b^5*d^2*e^5 - 2*a^3*b^
4*d*e^6 + a^4*b^3*e^7)*x^3 + 10*(a^3*b^4*d^2*e^5 - 2*a^4*b^3*d*e^6 + a^5*b^2*e^7
)*x^2 + 5*(a^4*b^3*d^2*e^5 - 2*a^5*b^2*d*e^6 + a^6*b*e^7)*x)*sqrt(-(b*d - a*e)/b
)*arctan(sqrt(e*x + d)/sqrt(-(b*d - a*e)/b)) - (256*b^7*e^7*x^7 - 128*b^7*d^7 -
240*a*b^6*d^6*e - 520*a^2*b^5*d^5*e^2 - 1430*a^3*b^4*d^4*e^3 - 6435*a^4*b^3*d^3*
e^4 + 69069*a^5*b^2*d^2*e^5 - 105105*a^6*b*d*e^6 + 45045*a^7*e^7 + 256*(12*b^7*d
*e^6 - 5*a*b^6*e^7)*x^6 + 256*(116*b^7*d^2*e^5 - 160*a*b^6*d*e^6 + 65*a^2*b^5*e^
7)*x^5 - 5*(5327*b^7*d^3*e^4 - 45677*a*b^6*d^2*e^5 + 66157*a^2*b^5*d*e^6 - 27599
*a^3*b^4*e^7)*x^4 - 10*(1211*b^7*d^4*e^3 + 5810*a*b^6*d^3*e^4 - 54392*a^2*b^5*d^
2*e^5 + 80366*a^3*b^4*d*e^6 - 33891*a^4*b^3*e^7)*x^3 - 2*(2324*b^7*d^5*e^2 + 654
5*a*b^6*d^4*e^3 + 30485*a^2*b^5*d^3*e^4 - 302445*a^3*b^4*d^2*e^5 + 452595*a^4*b^
3*d*e^6 - 192192*a^5*b^2*e^7)*x^2 - 2*(568*b^7*d^6*e + 1240*a*b^6*d^5*e^2 + 3445
*a^2*b^5*d^4*e^3 + 15730*a^3*b^4*d^3*e^4 - 163020*a^4*b^3*d^2*e^5 + 246246*a^5*b
^2*d*e^6 - 105105*a^6*b*e^7)*x)*sqrt(e*x + d))/(b^13*x^5 + 5*a*b^12*x^4 + 10*a^2
*b^11*x^3 + 10*a^3*b^10*x^2 + 5*a^4*b^9*x + a^5*b^8)]
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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)**(15/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
Timed out
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GIAC/XCAS [A] time = 0.24968, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(15/2)/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="giac")
[Out]
Done