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

Optimal. Leaf size=133 \[ -\frac{16 b^2 \left (b x+c x^2\right )^{7/2} (6 b B-13 A c)}{9009 c^4 x^{7/2}}+\frac{8 b \left (b x+c x^2\right )^{7/2} (6 b B-13 A c)}{1287 c^3 x^{5/2}}-\frac{2 \left (b x+c x^2\right )^{7/2} (6 b B-13 A c)}{143 c^2 x^{3/2}}+\frac{2 B \left (b x+c x^2\right )^{7/2}}{13 c \sqrt{x}} \]

[Out]

(-16*b^2*(6*b*B - 13*A*c)*(b*x + c*x^2)^(7/2))/(9009*c^4*x^(7/2)) + (8*b*(6*b*B
- 13*A*c)*(b*x + c*x^2)^(7/2))/(1287*c^3*x^(5/2)) - (2*(6*b*B - 13*A*c)*(b*x + c
*x^2)^(7/2))/(143*c^2*x^(3/2)) + (2*B*(b*x + c*x^2)^(7/2))/(13*c*Sqrt[x])

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Rubi [A]  time = 0.265117, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{16 b^2 \left (b x+c x^2\right )^{7/2} (6 b B-13 A c)}{9009 c^4 x^{7/2}}+\frac{8 b \left (b x+c x^2\right )^{7/2} (6 b B-13 A c)}{1287 c^3 x^{5/2}}-\frac{2 \left (b x+c x^2\right )^{7/2} (6 b B-13 A c)}{143 c^2 x^{3/2}}+\frac{2 B \left (b x+c x^2\right )^{7/2}}{13 c \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

(-16*b^2*(6*b*B - 13*A*c)*(b*x + c*x^2)^(7/2))/(9009*c^4*x^(7/2)) + (8*b*(6*b*B
- 13*A*c)*(b*x + c*x^2)^(7/2))/(1287*c^3*x^(5/2)) - (2*(6*b*B - 13*A*c)*(b*x + c
*x^2)^(7/2))/(143*c^2*x^(3/2)) + (2*B*(b*x + c*x^2)^(7/2))/(13*c*Sqrt[x])

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Rubi in Sympy [A]  time = 16.1643, size = 129, normalized size = 0.97 \[ \frac{2 B \left (b x + c x^{2}\right )^{\frac{7}{2}}}{13 c \sqrt{x}} + \frac{16 b^{2} \left (13 A c - 6 B b\right ) \left (b x + c x^{2}\right )^{\frac{7}{2}}}{9009 c^{4} x^{\frac{7}{2}}} - \frac{8 b \left (13 A c - 6 B b\right ) \left (b x + c x^{2}\right )^{\frac{7}{2}}}{1287 c^{3} x^{\frac{5}{2}}} + \frac{2 \left (13 A c - 6 B b\right ) \left (b x + c x^{2}\right )^{\frac{7}{2}}}{143 c^{2} x^{\frac{3}{2}}} \]

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*(b*x + c*x**2)**(7/2)/(13*c*sqrt(x)) + 16*b**2*(13*A*c - 6*B*b)*(b*x + c*x**
2)**(7/2)/(9009*c**4*x**(7/2)) - 8*b*(13*A*c - 6*B*b)*(b*x + c*x**2)**(7/2)/(128
7*c**3*x**(5/2)) + 2*(13*A*c - 6*B*b)*(b*x + c*x**2)**(7/2)/(143*c**2*x**(3/2))

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Mathematica [A]  time = 0.0962016, size = 82, normalized size = 0.62 \[ \frac{2 (b+c x)^3 \sqrt{x (b+c x)} \left (8 b^2 c (13 A+21 B x)-14 b c^2 x (26 A+27 B x)+63 c^3 x^2 (13 A+11 B x)-48 b^3 B\right )}{9009 c^4 \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b + c*x)^3*Sqrt[x*(b + c*x)]*(-48*b^3*B + 63*c^3*x^2*(13*A + 11*B*x) + 8*b^2
*c*(13*A + 21*B*x) - 14*b*c^2*x*(26*A + 27*B*x)))/(9009*c^4*Sqrt[x])

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Maple [A]  time = 0.009, size = 83, normalized size = 0.6 \[{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 693\,B{c}^{3}{x}^{3}+819\,A{c}^{3}{x}^{2}-378\,Bb{c}^{2}{x}^{2}-364\,Ab{c}^{2}x+168\,B{b}^{2}cx+104\,A{b}^{2}c-48\,B{b}^{3} \right ) }{9009\,{c}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}{x}^{-{\frac{5}{2}}}} \]

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]

2/9009*(c*x+b)*(693*B*c^3*x^3+819*A*c^3*x^2-378*B*b*c^2*x^2-364*A*b*c^2*x+168*B*
b^2*c*x+104*A*b^2*c-48*B*b^3)*(c*x^2+b*x)^(5/2)/c^4/x^(5/2)

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Maxima [A]  time = 0.712251, size = 506, normalized size = 3.8 \[ \frac{2 \,{\left ({\left (315 \, c^{5} x^{5} + 35 \, b c^{4} x^{4} - 40 \, b^{2} c^{3} x^{3} + 48 \, b^{3} c^{2} x^{2} - 64 \, b^{4} c x + 128 \, b^{5}\right )} x^{4} + 22 \,{\left (35 \, b c^{4} x^{5} + 5 \, b^{2} c^{3} x^{4} - 6 \, b^{3} c^{2} x^{3} + 8 \, b^{4} c x^{2} - 16 \, b^{5} x\right )} x^{3} + 33 \,{\left (15 \, b^{2} c^{3} x^{5} + 3 \, b^{3} c^{2} x^{4} - 4 \, b^{4} c x^{3} + 8 \, b^{5} x^{2}\right )} x^{2}\right )} \sqrt{c x + b} A}{3465 \, c^{3} x^{4}} + \frac{2 \,{\left (5 \,{\left (693 \, c^{6} x^{6} + 63 \, b c^{5} x^{5} - 70 \, b^{2} c^{4} x^{4} + 80 \, b^{3} c^{3} x^{3} - 96 \, b^{4} c^{2} x^{2} + 128 \, b^{5} c x - 256 \, b^{6}\right )} x^{5} + 26 \,{\left (315 \, b c^{5} x^{6} + 35 \, b^{2} c^{4} x^{5} - 40 \, b^{3} c^{3} x^{4} + 48 \, b^{4} c^{2} x^{3} - 64 \, b^{5} c x^{2} + 128 \, b^{6} x\right )} x^{4} + 143 \,{\left (35 \, b^{2} c^{4} x^{6} + 5 \, b^{3} c^{3} x^{5} - 6 \, b^{4} c^{2} x^{4} + 8 \, b^{5} c x^{3} - 16 \, b^{6} x^{2}\right )} x^{3}\right )} \sqrt{c x + b} B}{45045 \, c^{4} x^{5}} \]

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]

2/3465*((315*c^5*x^5 + 35*b*c^4*x^4 - 40*b^2*c^3*x^3 + 48*b^3*c^2*x^2 - 64*b^4*c
*x + 128*b^5)*x^4 + 22*(35*b*c^4*x^5 + 5*b^2*c^3*x^4 - 6*b^3*c^2*x^3 + 8*b^4*c*x
^2 - 16*b^5*x)*x^3 + 33*(15*b^2*c^3*x^5 + 3*b^3*c^2*x^4 - 4*b^4*c*x^3 + 8*b^5*x^
2)*x^2)*sqrt(c*x + b)*A/(c^3*x^4) + 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)*B/(c^4*x^5)

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Fricas [A]  time = 0.299152, size = 243, normalized size = 1.83 \[ \frac{2 \,{\left (693 \, B c^{7} x^{8} + 63 \,{\left (38 \, B b c^{6} + 13 \, A c^{7}\right )} x^{7} + 14 \,{\left (201 \, B b^{2} c^{5} + 208 \, A b c^{6}\right )} x^{6} + 2 \,{\left (564 \, B b^{3} c^{4} + 1781 \, A b^{2} c^{5}\right )} x^{5} -{\left (3 \, B b^{4} c^{3} - 1508 \, A b^{3} c^{4}\right )} x^{4} +{\left (6 \, B b^{5} c^{2} - 13 \, A b^{4} c^{3}\right )} x^{3} - 4 \,{\left (6 \, B b^{6} c - 13 \, A b^{5} c^{2}\right )} x^{2} - 8 \,{\left (6 \, B b^{7} - 13 \, A b^{6} c\right )} x\right )}}{9009 \, \sqrt{c x^{2} + b x} c^{4} \sqrt{x}} \]

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]

2/9009*(693*B*c^7*x^8 + 63*(38*B*b*c^6 + 13*A*c^7)*x^7 + 14*(201*B*b^2*c^5 + 208
*A*b*c^6)*x^6 + 2*(564*B*b^3*c^4 + 1781*A*b^2*c^5)*x^5 - (3*B*b^4*c^3 - 1508*A*b
^3*c^4)*x^4 + (6*B*b^5*c^2 - 13*A*b^4*c^3)*x^3 - 4*(6*B*b^6*c - 13*A*b^5*c^2)*x^
2 - 8*(6*B*b^7 - 13*A*b^6*c)*x)/(sqrt(c*x^2 + b*x)*c^4*sqrt(x))

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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)*(c*x**2+b*x)**(5/2)/x**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.292854, size = 562, normalized size = 4.23 \[ \frac{2}{9009} \, B c^{2}{\left (\frac{256 \, b^{\frac{13}{2}}}{c^{6}} + \frac{693 \,{\left (c x + b\right )}^{\frac{13}{2}} - 4095 \,{\left (c x + b\right )}^{\frac{11}{2}} b + 10010 \,{\left (c x + b\right )}^{\frac{9}{2}} b^{2} - 12870 \,{\left (c x + b\right )}^{\frac{7}{2}} b^{3} + 9009 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{4} - 3003 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{5}}{c^{6}}\right )} - \frac{4}{3465} \, B b c{\left (\frac{128 \, b^{\frac{11}{2}}}{c^{5}} - \frac{315 \,{\left (c x + b\right )}^{\frac{11}{2}} - 1540 \,{\left (c x + b\right )}^{\frac{9}{2}} b + 2970 \,{\left (c x + b\right )}^{\frac{7}{2}} b^{2} - 2772 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{3} + 1155 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{4}}{c^{5}}\right )} - \frac{2}{3465} \, A c^{2}{\left (\frac{128 \, b^{\frac{11}{2}}}{c^{5}} - \frac{315 \,{\left (c x + b\right )}^{\frac{11}{2}} - 1540 \,{\left (c x + b\right )}^{\frac{9}{2}} b + 2970 \,{\left (c x + b\right )}^{\frac{7}{2}} b^{2} - 2772 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{3} + 1155 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{4}}{c^{5}}\right )} + \frac{2}{315} \, B b^{2}{\left (\frac{16 \, b^{\frac{9}{2}}}{c^{4}} + \frac{35 \,{\left (c x + b\right )}^{\frac{9}{2}} - 135 \,{\left (c x + b\right )}^{\frac{7}{2}} b + 189 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{2} - 105 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{3}}{c^{4}}\right )} + \frac{4}{315} \, A b c{\left (\frac{16 \, b^{\frac{9}{2}}}{c^{4}} + \frac{35 \,{\left (c x + b\right )}^{\frac{9}{2}} - 135 \,{\left (c x + b\right )}^{\frac{7}{2}} b + 189 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{2} - 105 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{3}}{c^{4}}\right )} - \frac{2}{105} \, A b^{2}{\left (\frac{8 \, b^{\frac{7}{2}}}{c^{3}} - \frac{15 \,{\left (c x + b\right )}^{\frac{7}{2}} - 42 \,{\left (c x + b\right )}^{\frac{5}{2}} b + 35 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{2}}{c^{3}}\right )} \]

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]

2/9009*B*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) - 4/3465*B*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) - 2/3465*A*c^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 - 277
2*(c*x + b)^(5/2)*b^3 + 1155*(c*x + b)^(3/2)*b^4)/c^5) + 2/315*B*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 - 1
05*(c*x + b)^(3/2)*b^3)/c^4) + 4/315*A*b*c*(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) - 2/105*A*b^2*(8*b^(7/2)/c^3 - (15*(c*x + b)^(7/2) - 42*(c*x + b)^(5/2)*b + 3
5*(c*x + b)^(3/2)*b^2)/c^3)