3.215 \(\int \sqrt{x} (A+B x) \left (b x+c x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=170 \[ \frac{32 b^3 \left (b x+c x^2\right )^{7/2} (8 b B-15 A c)}{45045 c^5 x^{7/2}}-\frac{16 b^2 \left (b x+c x^2\right )^{7/2} (8 b B-15 A c)}{6435 c^4 x^{5/2}}+\frac{4 b \left (b x+c x^2\right )^{7/2} (8 b B-15 A c)}{715 c^3 x^{3/2}}-\frac{2 \left (b x+c x^2\right )^{7/2} (8 b B-15 A c)}{195 c^2 \sqrt{x}}+\frac{2 B \sqrt{x} \left (b x+c x^2\right )^{7/2}}{15 c} \]

[Out]

(32*b^3*(8*b*B - 15*A*c)*(b*x + c*x^2)^(7/2))/(45045*c^5*x^(7/2)) - (16*b^2*(8*b
*B - 15*A*c)*(b*x + c*x^2)^(7/2))/(6435*c^4*x^(5/2)) + (4*b*(8*b*B - 15*A*c)*(b*
x + c*x^2)^(7/2))/(715*c^3*x^(3/2)) - (2*(8*b*B - 15*A*c)*(b*x + c*x^2)^(7/2))/(
195*c^2*Sqrt[x]) + (2*B*Sqrt[x]*(b*x + c*x^2)^(7/2))/(15*c)

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Rubi [A]  time = 0.333893, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{32 b^3 \left (b x+c x^2\right )^{7/2} (8 b B-15 A c)}{45045 c^5 x^{7/2}}-\frac{16 b^2 \left (b x+c x^2\right )^{7/2} (8 b B-15 A c)}{6435 c^4 x^{5/2}}+\frac{4 b \left (b x+c x^2\right )^{7/2} (8 b B-15 A c)}{715 c^3 x^{3/2}}-\frac{2 \left (b x+c x^2\right )^{7/2} (8 b B-15 A c)}{195 c^2 \sqrt{x}}+\frac{2 B \sqrt{x} \left (b x+c x^2\right )^{7/2}}{15 c} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]*(A + B*x)*(b*x + c*x^2)^(5/2),x]

[Out]

(32*b^3*(8*b*B - 15*A*c)*(b*x + c*x^2)^(7/2))/(45045*c^5*x^(7/2)) - (16*b^2*(8*b
*B - 15*A*c)*(b*x + c*x^2)^(7/2))/(6435*c^4*x^(5/2)) + (4*b*(8*b*B - 15*A*c)*(b*
x + c*x^2)^(7/2))/(715*c^3*x^(3/2)) - (2*(8*b*B - 15*A*c)*(b*x + c*x^2)^(7/2))/(
195*c^2*Sqrt[x]) + (2*B*Sqrt[x]*(b*x + c*x^2)^(7/2))/(15*c)

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Rubi in Sympy [A]  time = 21.2014, size = 167, normalized size = 0.98 \[ \frac{2 B \sqrt{x} \left (b x + c x^{2}\right )^{\frac{7}{2}}}{15 c} - \frac{32 b^{3} \left (15 A c - 8 B b\right ) \left (b x + c x^{2}\right )^{\frac{7}{2}}}{45045 c^{5} x^{\frac{7}{2}}} + \frac{16 b^{2} \left (15 A c - 8 B b\right ) \left (b x + c x^{2}\right )^{\frac{7}{2}}}{6435 c^{4} x^{\frac{5}{2}}} - \frac{4 b \left (15 A c - 8 B b\right ) \left (b x + c x^{2}\right )^{\frac{7}{2}}}{715 c^{3} x^{\frac{3}{2}}} + \frac{2 \left (15 A c - 8 B b\right ) \left (b x + c x^{2}\right )^{\frac{7}{2}}}{195 c^{2} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**(5/2)*x**(1/2),x)

[Out]

2*B*sqrt(x)*(b*x + c*x**2)**(7/2)/(15*c) - 32*b**3*(15*A*c - 8*B*b)*(b*x + c*x**
2)**(7/2)/(45045*c**5*x**(7/2)) + 16*b**2*(15*A*c - 8*B*b)*(b*x + c*x**2)**(7/2)
/(6435*c**4*x**(5/2)) - 4*b*(15*A*c - 8*B*b)*(b*x + c*x**2)**(7/2)/(715*c**3*x**
(3/2)) + 2*(15*A*c - 8*B*b)*(b*x + c*x**2)**(7/2)/(195*c**2*sqrt(x))

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Mathematica [A]  time = 0.108218, size = 101, normalized size = 0.59 \[ \frac{2 (b+c x)^3 \sqrt{x (b+c x)} \left (-16 b^3 c (15 A+28 B x)+168 b^2 c^2 x (5 A+6 B x)-42 b c^3 x^2 (45 A+44 B x)+231 c^4 x^3 (15 A+13 B x)+128 b^4 B\right )}{45045 c^5 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x]*(A + B*x)*(b*x + c*x^2)^(5/2),x]

[Out]

(2*(b + c*x)^3*Sqrt[x*(b + c*x)]*(128*b^4*B + 168*b^2*c^2*x*(5*A + 6*B*x) + 231*
c^4*x^3*(15*A + 13*B*x) - 16*b^3*c*(15*A + 28*B*x) - 42*b*c^3*x^2*(45*A + 44*B*x
)))/(45045*c^5*Sqrt[x])

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Maple [A]  time = 0.01, size = 107, normalized size = 0.6 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -3003\,B{x}^{4}{c}^{4}-3465\,A{c}^{4}{x}^{3}+1848\,Bb{c}^{3}{x}^{3}+1890\,Ab{c}^{3}{x}^{2}-1008\,B{b}^{2}{c}^{2}{x}^{2}-840\,A{b}^{2}{c}^{2}x+448\,B{b}^{3}cx+240\,A{b}^{3}c-128\,{b}^{4}B \right ) }{45045\,{c}^{5}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}{x}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^(5/2)*x^(1/2),x)

[Out]

-2/45045*(c*x+b)*(-3003*B*c^4*x^4-3465*A*c^4*x^3+1848*B*b*c^3*x^3+1890*A*b*c^3*x
^2-1008*B*b^2*c^2*x^2-840*A*b^2*c^2*x+448*B*b^3*c*x+240*A*b^3*c-128*B*b^4)*(c*x^
2+b*x)^(5/2)/c^5/x^(5/2)

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Maxima [A]  time = 0.709187, size = 595, normalized size = 3.5 \[ \frac{2 \,{\left (5 \,{\left (693 \, c^{6} x^{6} + 63 \, b c^{5} x^{5} - 70 \, b^{2} c^{4} x^{4} + 80 \, b^{3} c^{3} x^{3} - 96 \, b^{4} c^{2} x^{2} + 128 \, b^{5} c x - 256 \, b^{6}\right )} x^{5} + 26 \,{\left (315 \, b c^{5} x^{6} + 35 \, b^{2} c^{4} x^{5} - 40 \, b^{3} c^{3} x^{4} + 48 \, b^{4} c^{2} x^{3} - 64 \, b^{5} c x^{2} + 128 \, b^{6} x\right )} x^{4} + 143 \,{\left (35 \, b^{2} c^{4} x^{6} + 5 \, b^{3} c^{3} x^{5} - 6 \, b^{4} c^{2} x^{4} + 8 \, b^{5} c x^{3} - 16 \, b^{6} x^{2}\right )} x^{3}\right )} \sqrt{c x + b} A}{45045 \, c^{4} x^{5}} + \frac{2 \,{\left ({\left (3003 \, c^{7} x^{7} + 231 \, b c^{6} x^{6} - 252 \, b^{2} c^{5} x^{5} + 280 \, b^{3} c^{4} x^{4} - 320 \, b^{4} c^{3} x^{3} + 384 \, b^{5} c^{2} x^{2} - 512 \, b^{6} c x + 1024 \, b^{7}\right )} x^{6} + 10 \,{\left (693 \, b c^{6} x^{7} + 63 \, b^{2} c^{5} x^{6} - 70 \, b^{3} c^{4} x^{5} + 80 \, b^{4} c^{3} x^{4} - 96 \, b^{5} c^{2} x^{3} + 128 \, b^{6} c x^{2} - 256 \, b^{7} x\right )} x^{5} + 13 \,{\left (315 \, b^{2} c^{5} x^{7} + 35 \, b^{3} c^{4} x^{6} - 40 \, b^{4} c^{3} x^{5} + 48 \, b^{5} c^{2} x^{4} - 64 \, b^{6} c x^{3} + 128 \, b^{7} x^{2}\right )} x^{4}\right )} \sqrt{c x + b} B}{45045 \, c^{5} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(B*x + A)*sqrt(x),x, algorithm="maxima")

[Out]

2/45045*(5*(693*c^6*x^6 + 63*b*c^5*x^5 - 70*b^2*c^4*x^4 + 80*b^3*c^3*x^3 - 96*b^
4*c^2*x^2 + 128*b^5*c*x - 256*b^6)*x^5 + 26*(315*b*c^5*x^6 + 35*b^2*c^4*x^5 - 40
*b^3*c^3*x^4 + 48*b^4*c^2*x^3 - 64*b^5*c*x^2 + 128*b^6*x)*x^4 + 143*(35*b^2*c^4*
x^6 + 5*b^3*c^3*x^5 - 6*b^4*c^2*x^4 + 8*b^5*c*x^3 - 16*b^6*x^2)*x^3)*sqrt(c*x +
b)*A/(c^4*x^5) + 2/45045*((3003*c^7*x^7 + 231*b*c^6*x^6 - 252*b^2*c^5*x^5 + 280*
b^3*c^4*x^4 - 320*b^4*c^3*x^3 + 384*b^5*c^2*x^2 - 512*b^6*c*x + 1024*b^7)*x^6 +
10*(693*b*c^6*x^7 + 63*b^2*c^5*x^6 - 70*b^3*c^4*x^5 + 80*b^4*c^3*x^4 - 96*b^5*c^
2*x^3 + 128*b^6*c*x^2 - 256*b^7*x)*x^5 + 13*(315*b^2*c^5*x^7 + 35*b^3*c^4*x^6 -
40*b^4*c^3*x^5 + 48*b^5*c^2*x^4 - 64*b^6*c*x^3 + 128*b^7*x^2)*x^4)*sqrt(c*x + b)
*B/(c^5*x^6)

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Fricas [A]  time = 0.302718, size = 274, normalized size = 1.61 \[ \frac{2 \,{\left (3003 \, B c^{8} x^{9} + 231 \,{\left (44 \, B b c^{7} + 15 \, A c^{8}\right )} x^{8} + 42 \,{\left (277 \, B b^{2} c^{6} + 285 \, A b c^{7}\right )} x^{7} + 14 \,{\left (322 \, B b^{3} c^{5} + 1005 \, A b^{2} c^{6}\right )} x^{6} - 5 \,{\left (B b^{4} c^{4} - 1128 \, A b^{3} c^{5}\right )} x^{5} +{\left (8 \, B b^{5} c^{3} - 15 \, A b^{4} c^{4}\right )} x^{4} - 2 \,{\left (8 \, B b^{6} c^{2} - 15 \, A b^{5} c^{3}\right )} x^{3} + 8 \,{\left (8 \, B b^{7} c - 15 \, A b^{6} c^{2}\right )} x^{2} + 16 \,{\left (8 \, B b^{8} - 15 \, A b^{7} c\right )} x\right )}}{45045 \, \sqrt{c x^{2} + b x} c^{5} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(B*x + A)*sqrt(x),x, algorithm="fricas")

[Out]

2/45045*(3003*B*c^8*x^9 + 231*(44*B*b*c^7 + 15*A*c^8)*x^8 + 42*(277*B*b^2*c^6 +
285*A*b*c^7)*x^7 + 14*(322*B*b^3*c^5 + 1005*A*b^2*c^6)*x^6 - 5*(B*b^4*c^4 - 1128
*A*b^3*c^5)*x^5 + (8*B*b^5*c^3 - 15*A*b^4*c^4)*x^4 - 2*(8*B*b^6*c^2 - 15*A*b^5*c
^3)*x^3 + 8*(8*B*b^7*c - 15*A*b^6*c^2)*x^2 + 16*(8*B*b^8 - 15*A*b^7*c)*x)/(sqrt(
c*x^2 + b*x)*c^5*sqrt(x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{x} \left (x \left (b + c x\right )\right )^{\frac{5}{2}} \left (A + B x\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)**(5/2)*x**(1/2),x)

[Out]

Integral(sqrt(x)*(x*(b + c*x))**(5/2)*(A + B*x), x)

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GIAC/XCAS [A]  time = 0.297063, size = 659, normalized size = 3.88 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(B*x + A)*sqrt(x),x, algorithm="giac")

[Out]

-2/45045*B*c^2*(1024*b^(15/2)/c^7 - (3003*(c*x + b)^(15/2) - 20790*(c*x + b)^(13
/2)*b + 61425*(c*x + b)^(11/2)*b^2 - 100100*(c*x + b)^(9/2)*b^3 + 96525*(c*x + b
)^(7/2)*b^4 - 54054*(c*x + b)^(5/2)*b^5 + 15015*(c*x + b)^(3/2)*b^6)/c^7) + 4/90
09*B*b*c*(256*b^(13/2)/c^6 + (693*(c*x + b)^(13/2) - 4095*(c*x + b)^(11/2)*b + 1
0010*(c*x + b)^(9/2)*b^2 - 12870*(c*x + b)^(7/2)*b^3 + 9009*(c*x + b)^(5/2)*b^4
- 3003*(c*x + b)^(3/2)*b^5)/c^6) + 2/9009*A*c^2*(256*b^(13/2)/c^6 + (693*(c*x +
b)^(13/2) - 4095*(c*x + b)^(11/2)*b + 10010*(c*x + b)^(9/2)*b^2 - 12870*(c*x + b
)^(7/2)*b^3 + 9009*(c*x + b)^(5/2)*b^4 - 3003*(c*x + b)^(3/2)*b^5)/c^6) - 2/3465
*B*b^2*(128*b^(11/2)/c^5 - (315*(c*x + b)^(11/2) - 1540*(c*x + b)^(9/2)*b + 2970
*(c*x + b)^(7/2)*b^2 - 2772*(c*x + b)^(5/2)*b^3 + 1155*(c*x + b)^(3/2)*b^4)/c^5)
- 4/3465*A*b*c*(128*b^(11/2)/c^5 - (315*(c*x + b)^(11/2) - 1540*(c*x + b)^(9/2)
*b + 2970*(c*x + b)^(7/2)*b^2 - 2772*(c*x + b)^(5/2)*b^3 + 1155*(c*x + b)^(3/2)*
b^4)/c^5) + 2/315*A*b^2*(16*b^(9/2)/c^4 + (35*(c*x + b)^(9/2) - 135*(c*x + b)^(7
/2)*b + 189*(c*x + b)^(5/2)*b^2 - 105*(c*x + b)^(3/2)*b^3)/c^4)