3.2121 \(\int \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{17/2}} \, dx\)

Optimal. Leaf size=376 \[ -\frac{30 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{11 e^7 (a+b x) (d+e x)^{11/2}}+\frac{12 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{13 e^7 (a+b x) (d+e x)^{13/2}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{15 e^7 (a+b x) (d+e x)^{15/2}}-\frac{2 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^{3/2}}+\frac{12 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^7 (a+b x) (d+e x)^{5/2}}-\frac{30 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{7 e^7 (a+b x) (d+e x)^{7/2}}+\frac{40 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{9 e^7 (a+b x) (d+e x)^{9/2}} \]

[Out]

(-2*(b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(15*e^7*(a + b*x)*(d + e*x)^(15
/2)) + (12*b*(b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^7*(a + b*x)*(d +
e*x)^(13/2)) - (30*b^2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a
+ b*x)*(d + e*x)^(11/2)) + (40*b^3*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/
(9*e^7*(a + b*x)*(d + e*x)^(9/2)) - (30*b^4*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b
^2*x^2])/(7*e^7*(a + b*x)*(d + e*x)^(7/2)) + (12*b^5*(b*d - a*e)*Sqrt[a^2 + 2*a*
b*x + b^2*x^2])/(5*e^7*(a + b*x)*(d + e*x)^(5/2)) - (2*b^6*Sqrt[a^2 + 2*a*b*x +
b^2*x^2])/(3*e^7*(a + b*x)*(d + e*x)^(3/2))

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Rubi [A]  time = 0.391254, antiderivative size = 376, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086 \[ -\frac{30 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{11 e^7 (a+b x) (d+e x)^{11/2}}+\frac{12 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{13 e^7 (a+b x) (d+e x)^{13/2}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{15 e^7 (a+b x) (d+e x)^{15/2}}-\frac{2 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^{3/2}}+\frac{12 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^7 (a+b x) (d+e x)^{5/2}}-\frac{30 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{7 e^7 (a+b x) (d+e x)^{7/2}}+\frac{40 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{9 e^7 (a+b x) (d+e x)^{9/2}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^(17/2),x]

[Out]

(-2*(b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(15*e^7*(a + b*x)*(d + e*x)^(15
/2)) + (12*b*(b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^7*(a + b*x)*(d +
e*x)^(13/2)) - (30*b^2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a
+ b*x)*(d + e*x)^(11/2)) + (40*b^3*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/
(9*e^7*(a + b*x)*(d + e*x)^(9/2)) - (30*b^4*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b
^2*x^2])/(7*e^7*(a + b*x)*(d + e*x)^(7/2)) + (12*b^5*(b*d - a*e)*Sqrt[a^2 + 2*a*
b*x + b^2*x^2])/(5*e^7*(a + b*x)*(d + e*x)^(5/2)) - (2*b^6*Sqrt[a^2 + 2*a*b*x +
b^2*x^2])/(3*e^7*(a + b*x)*(d + e*x)^(3/2))

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Rubi in Sympy [A]  time = 50.5932, size = 299, normalized size = 0.8 \[ - \frac{1024 b^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{9009 e^{6} \left (d + e x\right )^{\frac{5}{2}}} + \frac{2048 b^{5} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{45045 e^{7} \left (a + b x\right ) \left (d + e x\right )^{\frac{5}{2}}} - \frac{256 b^{4} \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{9009 e^{5} \left (d + e x\right )^{\frac{7}{2}}} - \frac{128 b^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{1287 e^{4} \left (d + e x\right )^{\frac{9}{2}}} - \frac{16 b^{2} \left (5 a + 5 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{715 e^{3} \left (d + e x\right )^{\frac{11}{2}}} - \frac{8 b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{65 e^{2} \left (d + e x\right )^{\frac{13}{2}}} - \frac{2 \left (a + b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{15 e \left (d + e x\right )^{\frac{15}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**(17/2),x)

[Out]

-1024*b**5*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(9009*e**6*(d + e*x)**(5/2)) + 2048*
b**5*(a*e - b*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(45045*e**7*(a + b*x)*(d + e*x
)**(5/2)) - 256*b**4*(3*a + 3*b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(9009*e**5*(
d + e*x)**(7/2)) - 128*b**3*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(1287*e**4*(d +
e*x)**(9/2)) - 16*b**2*(5*a + 5*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(715*e*
*3*(d + e*x)**(11/2)) - 8*b*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(65*e**2*(d + e*
x)**(13/2)) - 2*(a + b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(15*e*(d + e*x)**(
15/2))

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Mathematica [A]  time = 0.322125, size = 163, normalized size = 0.43 \[ \frac{2 \sqrt{(a+b x)^2} \left (54054 b^5 (d+e x)^5 (b d-a e)-96525 b^4 (d+e x)^4 (b d-a e)^2+100100 b^3 (d+e x)^3 (b d-a e)^3-61425 b^2 (d+e x)^2 (b d-a e)^4+20790 b (d+e x) (b d-a e)^5-3003 (b d-a e)^6-15015 b^6 (d+e x)^6\right )}{45045 e^7 (a+b x) (d+e x)^{15/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^(17/2),x]

[Out]

(2*Sqrt[(a + b*x)^2]*(-3003*(b*d - a*e)^6 + 20790*b*(b*d - a*e)^5*(d + e*x) - 61
425*b^2*(b*d - a*e)^4*(d + e*x)^2 + 100100*b^3*(b*d - a*e)^3*(d + e*x)^3 - 96525
*b^4*(b*d - a*e)^2*(d + e*x)^4 + 54054*b^5*(b*d - a*e)*(d + e*x)^5 - 15015*b^6*(
d + e*x)^6))/(45045*e^7*(a + b*x)*(d + e*x)^(15/2))

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Maple [A]  time = 0.013, size = 393, normalized size = 1.1 \[ -{\frac{30030\,{x}^{6}{b}^{6}{e}^{6}+108108\,{x}^{5}a{b}^{5}{e}^{6}+72072\,{x}^{5}{b}^{6}d{e}^{5}+193050\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}+154440\,{x}^{4}a{b}^{5}d{e}^{5}+102960\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}+200200\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}+171600\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+137280\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}+91520\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+122850\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}+109200\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+93600\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}+74880\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}+49920\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+41580\,x{a}^{5}b{e}^{6}+37800\,x{a}^{4}{b}^{2}d{e}^{5}+33600\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}+28800\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+23040\,xa{b}^{5}{d}^{4}{e}^{2}+15360\,x{b}^{6}{d}^{5}e+6006\,{a}^{6}{e}^{6}+5544\,{a}^{5}bd{e}^{5}+5040\,{b}^{2}{a}^{4}{d}^{2}{e}^{4}+4480\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+3840\,{d}^{4}{e}^{2}{a}^{2}{b}^{4}+3072\,{d}^{5}a{b}^{5}e+2048\,{b}^{6}{d}^{6}}{45045\, \left ( bx+a \right ) ^{5}{e}^{7}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{15}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(17/2),x)

[Out]

-2/45045/(e*x+d)^(15/2)*(15015*b^6*e^6*x^6+54054*a*b^5*e^6*x^5+36036*b^6*d*e^5*x
^5+96525*a^2*b^4*e^6*x^4+77220*a*b^5*d*e^5*x^4+51480*b^6*d^2*e^4*x^4+100100*a^3*
b^3*e^6*x^3+85800*a^2*b^4*d*e^5*x^3+68640*a*b^5*d^2*e^4*x^3+45760*b^6*d^3*e^3*x^
3+61425*a^4*b^2*e^6*x^2+54600*a^3*b^3*d*e^5*x^2+46800*a^2*b^4*d^2*e^4*x^2+37440*
a*b^5*d^3*e^3*x^2+24960*b^6*d^4*e^2*x^2+20790*a^5*b*e^6*x+18900*a^4*b^2*d*e^5*x+
16800*a^3*b^3*d^2*e^4*x+14400*a^2*b^4*d^3*e^3*x+11520*a*b^5*d^4*e^2*x+7680*b^6*d
^5*e*x+3003*a^6*e^6+2772*a^5*b*d*e^5+2520*a^4*b^2*d^2*e^4+2240*a^3*b^3*d^3*e^3+1
920*a^2*b^4*d^4*e^2+1536*a*b^5*d^5*e+1024*b^6*d^6)*((b*x+a)^2)^(5/2)/e^7/(b*x+a)
^5

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Maxima [A]  time = 0.771889, size = 1022, normalized size = 2.72 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(b*x + a)/(e*x + d)^(17/2),x, algorithm="maxima")

[Out]

-2/45045*(9009*b^5*e^5*x^5 + 256*b^5*d^5 + 640*a*b^4*d^4*e + 1120*a^2*b^3*d^3*e^
2 + 1680*a^3*b^2*d^2*e^3 + 2310*a^4*b*d*e^4 + 3003*a^5*e^5 + 6435*(2*b^5*d*e^4 +
5*a*b^4*e^5)*x^4 + 1430*(8*b^5*d^2*e^3 + 20*a*b^4*d*e^4 + 35*a^2*b^3*e^5)*x^3 +
390*(16*b^5*d^3*e^2 + 40*a*b^4*d^2*e^3 + 70*a^2*b^3*d*e^4 + 105*a^3*b^2*e^5)*x^
2 + 15*(128*b^5*d^4*e + 320*a*b^4*d^3*e^2 + 560*a^2*b^3*d^2*e^3 + 840*a^3*b^2*d*
e^4 + 1155*a^4*b*e^5)*x)*a/((e^13*x^7 + 7*d*e^12*x^6 + 21*d^2*e^11*x^5 + 35*d^3*
e^10*x^4 + 35*d^4*e^9*x^3 + 21*d^5*e^8*x^2 + 7*d^6*e^7*x + d^7*e^6)*sqrt(e*x + d
)) - 2/45045*(15015*b^5*e^6*x^6 + 1024*b^5*d^6 + 1280*a*b^4*d^5*e + 1280*a^2*b^3
*d^4*e^2 + 1120*a^3*b^2*d^3*e^3 + 840*a^4*b*d^2*e^4 + 462*a^5*d*e^5 + 9009*(4*b^
5*d*e^5 + 5*a*b^4*e^6)*x^5 + 12870*(4*b^5*d^2*e^4 + 5*a*b^4*d*e^5 + 5*a^2*b^3*e^
6)*x^4 + 1430*(32*b^5*d^3*e^3 + 40*a*b^4*d^2*e^4 + 40*a^2*b^3*d*e^5 + 35*a^3*b^2
*e^6)*x^3 + 195*(128*b^5*d^4*e^2 + 160*a*b^4*d^3*e^3 + 160*a^2*b^3*d^2*e^4 + 140
*a^3*b^2*d*e^5 + 105*a^4*b*e^6)*x^2 + 15*(512*b^5*d^5*e + 640*a*b^4*d^4*e^2 + 64
0*a^2*b^3*d^3*e^3 + 560*a^3*b^2*d^2*e^4 + 420*a^4*b*d*e^5 + 231*a^5*e^6)*x)*b/((
e^14*x^7 + 7*d*e^13*x^6 + 21*d^2*e^12*x^5 + 35*d^3*e^11*x^4 + 35*d^4*e^10*x^3 +
21*d^5*e^9*x^2 + 7*d^6*e^8*x + d^7*e^7)*sqrt(e*x + d))

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Fricas [A]  time = 0.286697, size = 583, normalized size = 1.55 \[ -\frac{2 \,{\left (15015 \, b^{6} e^{6} x^{6} + 1024 \, b^{6} d^{6} + 1536 \, a b^{5} d^{5} e + 1920 \, a^{2} b^{4} d^{4} e^{2} + 2240 \, a^{3} b^{3} d^{3} e^{3} + 2520 \, a^{4} b^{2} d^{2} e^{4} + 2772 \, a^{5} b d e^{5} + 3003 \, a^{6} e^{6} + 18018 \,{\left (2 \, b^{6} d e^{5} + 3 \, a b^{5} e^{6}\right )} x^{5} + 6435 \,{\left (8 \, b^{6} d^{2} e^{4} + 12 \, a b^{5} d e^{5} + 15 \, a^{2} b^{4} e^{6}\right )} x^{4} + 2860 \,{\left (16 \, b^{6} d^{3} e^{3} + 24 \, a b^{5} d^{2} e^{4} + 30 \, a^{2} b^{4} d e^{5} + 35 \, a^{3} b^{3} e^{6}\right )} x^{3} + 195 \,{\left (128 \, b^{6} d^{4} e^{2} + 192 \, a b^{5} d^{3} e^{3} + 240 \, a^{2} b^{4} d^{2} e^{4} + 280 \, a^{3} b^{3} d e^{5} + 315 \, a^{4} b^{2} e^{6}\right )} x^{2} + 30 \,{\left (256 \, b^{6} d^{5} e + 384 \, a b^{5} d^{4} e^{2} + 480 \, a^{2} b^{4} d^{3} e^{3} + 560 \, a^{3} b^{3} d^{2} e^{4} + 630 \, a^{4} b^{2} d e^{5} + 693 \, a^{5} b e^{6}\right )} x\right )}}{45045 \,{\left (e^{14} x^{7} + 7 \, d e^{13} x^{6} + 21 \, d^{2} e^{12} x^{5} + 35 \, d^{3} e^{11} x^{4} + 35 \, d^{4} e^{10} x^{3} + 21 \, d^{5} e^{9} x^{2} + 7 \, d^{6} e^{8} x + d^{7} e^{7}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(b*x + a)/(e*x + d)^(17/2),x, algorithm="fricas")

[Out]

-2/45045*(15015*b^6*e^6*x^6 + 1024*b^6*d^6 + 1536*a*b^5*d^5*e + 1920*a^2*b^4*d^4
*e^2 + 2240*a^3*b^3*d^3*e^3 + 2520*a^4*b^2*d^2*e^4 + 2772*a^5*b*d*e^5 + 3003*a^6
*e^6 + 18018*(2*b^6*d*e^5 + 3*a*b^5*e^6)*x^5 + 6435*(8*b^6*d^2*e^4 + 12*a*b^5*d*
e^5 + 15*a^2*b^4*e^6)*x^4 + 2860*(16*b^6*d^3*e^3 + 24*a*b^5*d^2*e^4 + 30*a^2*b^4
*d*e^5 + 35*a^3*b^3*e^6)*x^3 + 195*(128*b^6*d^4*e^2 + 192*a*b^5*d^3*e^3 + 240*a^
2*b^4*d^2*e^4 + 280*a^3*b^3*d*e^5 + 315*a^4*b^2*e^6)*x^2 + 30*(256*b^6*d^5*e + 3
84*a*b^5*d^4*e^2 + 480*a^2*b^4*d^3*e^3 + 560*a^3*b^3*d^2*e^4 + 630*a^4*b^2*d*e^5
+ 693*a^5*b*e^6)*x)/((e^14*x^7 + 7*d*e^13*x^6 + 21*d^2*e^12*x^5 + 35*d^3*e^11*x
^4 + 35*d^4*e^10*x^3 + 21*d^5*e^9*x^2 + 7*d^6*e^8*x + d^7*e^7)*sqrt(e*x + d))

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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)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**(17/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.318406, size = 829, normalized size = 2.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(b*x + a)/(e*x + d)^(17/2),x, algorithm="giac")

[Out]

-2/45045*(15015*(x*e + d)^6*b^6*sign(b*x + a) - 54054*(x*e + d)^5*b^6*d*sign(b*x
+ a) + 96525*(x*e + d)^4*b^6*d^2*sign(b*x + a) - 100100*(x*e + d)^3*b^6*d^3*sig
n(b*x + a) + 61425*(x*e + d)^2*b^6*d^4*sign(b*x + a) - 20790*(x*e + d)*b^6*d^5*s
ign(b*x + a) + 3003*b^6*d^6*sign(b*x + a) + 54054*(x*e + d)^5*a*b^5*e*sign(b*x +
a) - 193050*(x*e + d)^4*a*b^5*d*e*sign(b*x + a) + 300300*(x*e + d)^3*a*b^5*d^2*
e*sign(b*x + a) - 245700*(x*e + d)^2*a*b^5*d^3*e*sign(b*x + a) + 103950*(x*e + d
)*a*b^5*d^4*e*sign(b*x + a) - 18018*a*b^5*d^5*e*sign(b*x + a) + 96525*(x*e + d)^
4*a^2*b^4*e^2*sign(b*x + a) - 300300*(x*e + d)^3*a^2*b^4*d*e^2*sign(b*x + a) + 3
68550*(x*e + d)^2*a^2*b^4*d^2*e^2*sign(b*x + a) - 207900*(x*e + d)*a^2*b^4*d^3*e
^2*sign(b*x + a) + 45045*a^2*b^4*d^4*e^2*sign(b*x + a) + 100100*(x*e + d)^3*a^3*
b^3*e^3*sign(b*x + a) - 245700*(x*e + d)^2*a^3*b^3*d*e^3*sign(b*x + a) + 207900*
(x*e + d)*a^3*b^3*d^2*e^3*sign(b*x + a) - 60060*a^3*b^3*d^3*e^3*sign(b*x + a) +
61425*(x*e + d)^2*a^4*b^2*e^4*sign(b*x + a) - 103950*(x*e + d)*a^4*b^2*d*e^4*sig
n(b*x + a) + 45045*a^4*b^2*d^2*e^4*sign(b*x + a) + 20790*(x*e + d)*a^5*b*e^5*sig
n(b*x + a) - 18018*a^5*b*d*e^5*sign(b*x + a) + 3003*a^6*e^6*sign(b*x + a))*e^(-7
)/(x*e + d)^(15/2)