3.217 \(\int \frac{(A+B x) \left (b x+c x^2\right )^{5/2}}{x^{3/2}} \, dx\)

Optimal. Leaf size=96 \[ \frac{4 b \left (b x+c x^2\right )^{7/2} (4 b B-11 A c)}{693 c^3 x^{7/2}}-\frac{2 \left (b x+c x^2\right )^{7/2} (4 b B-11 A c)}{99 c^2 x^{5/2}}+\frac{2 B \left (b x+c x^2\right )^{7/2}}{11 c x^{3/2}} \]

[Out]

(4*b*(4*b*B - 11*A*c)*(b*x + c*x^2)^(7/2))/(693*c^3*x^(7/2)) - (2*(4*b*B - 11*A*
c)*(b*x + c*x^2)^(7/2))/(99*c^2*x^(5/2)) + (2*B*(b*x + c*x^2)^(7/2))/(11*c*x^(3/
2))

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Rubi [A]  time = 0.213848, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{4 b \left (b x+c x^2\right )^{7/2} (4 b B-11 A c)}{693 c^3 x^{7/2}}-\frac{2 \left (b x+c x^2\right )^{7/2} (4 b B-11 A c)}{99 c^2 x^{5/2}}+\frac{2 B \left (b x+c x^2\right )^{7/2}}{11 c x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(b*x + c*x^2)^(5/2))/x^(3/2),x]

[Out]

(4*b*(4*b*B - 11*A*c)*(b*x + c*x^2)^(7/2))/(693*c^3*x^(7/2)) - (2*(4*b*B - 11*A*
c)*(b*x + c*x^2)^(7/2))/(99*c^2*x^(5/2)) + (2*B*(b*x + c*x^2)^(7/2))/(11*c*x^(3/
2))

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Rubi in Sympy [A]  time = 11.5036, size = 92, normalized size = 0.96 \[ \frac{2 B \left (b x + c x^{2}\right )^{\frac{7}{2}}}{11 c x^{\frac{3}{2}}} - \frac{4 b \left (11 A c - 4 B b\right ) \left (b x + c x^{2}\right )^{\frac{7}{2}}}{693 c^{3} x^{\frac{7}{2}}} + \frac{2 \left (11 A c - 4 B b\right ) \left (b x + c x^{2}\right )^{\frac{7}{2}}}{99 c^{2} x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**(5/2)/x**(3/2),x)

[Out]

2*B*(b*x + c*x**2)**(7/2)/(11*c*x**(3/2)) - 4*b*(11*A*c - 4*B*b)*(b*x + c*x**2)*
*(7/2)/(693*c**3*x**(7/2)) + 2*(11*A*c - 4*B*b)*(b*x + c*x**2)**(7/2)/(99*c**2*x
**(5/2))

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Mathematica [A]  time = 0.0787158, size = 63, normalized size = 0.66 \[ \frac{2 (b+c x)^3 \sqrt{x (b+c x)} \left (-2 b c (11 A+14 B x)+7 c^2 x (11 A+9 B x)+8 b^2 B\right )}{693 c^3 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(b*x + c*x^2)^(5/2))/x^(3/2),x]

[Out]

(2*(b + c*x)^3*Sqrt[x*(b + c*x)]*(8*b^2*B + 7*c^2*x*(11*A + 9*B*x) - 2*b*c*(11*A
+ 14*B*x)))/(693*c^3*Sqrt[x])

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Maple [A]  time = 0.005, size = 59, normalized size = 0.6 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -63\,B{c}^{2}{x}^{2}-77\,Ax{c}^{2}+28\,Bxbc+22\,Abc-8\,{b}^{2}B \right ) }{693\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}{x}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^(5/2)/x^(3/2),x)

[Out]

-2/693*(c*x+b)*(-63*B*c^2*x^2-77*A*c^2*x+28*B*b*c*x+22*A*b*c-8*B*b^2)*(c*x^2+b*x
)^(5/2)/c^3/x^(5/2)

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Maxima [A]  time = 0.714499, size = 412, normalized size = 4.29 \[ \frac{2 \,{\left ({\left (35 \, c^{4} x^{4} + 5 \, b c^{3} x^{3} - 6 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x - 16 \, b^{4}\right )} x^{3} + 6 \,{\left (15 \, b c^{3} x^{4} + 3 \, b^{2} c^{2} x^{3} - 4 \, b^{3} c x^{2} + 8 \, b^{4} x\right )} x^{2} + 21 \,{\left (3 \, b^{2} c^{2} x^{4} + b^{3} c x^{3} - 2 \, b^{4} x^{2}\right )} x\right )} \sqrt{c x + b} A}{315 \, c^{2} x^{3}} + \frac{2 \,{\left ({\left (315 \, c^{5} x^{5} + 35 \, b c^{4} x^{4} - 40 \, b^{2} c^{3} x^{3} + 48 \, b^{3} c^{2} x^{2} - 64 \, b^{4} c x + 128 \, b^{5}\right )} x^{4} + 22 \,{\left (35 \, b c^{4} x^{5} + 5 \, b^{2} c^{3} x^{4} - 6 \, b^{3} c^{2} x^{3} + 8 \, b^{4} c x^{2} - 16 \, b^{5} x\right )} x^{3} + 33 \,{\left (15 \, b^{2} c^{3} x^{5} + 3 \, b^{3} c^{2} x^{4} - 4 \, b^{4} c x^{3} + 8 \, b^{5} x^{2}\right )} x^{2}\right )} \sqrt{c x + b} B}{3465 \, c^{3} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(B*x + A)/x^(3/2),x, algorithm="maxima")

[Out]

2/315*((35*c^4*x^4 + 5*b*c^3*x^3 - 6*b^2*c^2*x^2 + 8*b^3*c*x - 16*b^4)*x^3 + 6*(
15*b*c^3*x^4 + 3*b^2*c^2*x^3 - 4*b^3*c*x^2 + 8*b^4*x)*x^2 + 21*(3*b^2*c^2*x^4 +
b^3*c*x^3 - 2*b^4*x^2)*x)*sqrt(c*x + b)*A/(c^2*x^3) + 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*(1
5*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)*B/(c
^3*x^4)

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Fricas [A]  time = 0.290856, size = 209, normalized size = 2.18 \[ \frac{2 \,{\left (63 \, B c^{6} x^{7} + 7 \,{\left (32 \, B b c^{5} + 11 \, A c^{6}\right )} x^{6} + 2 \,{\left (137 \, B b^{2} c^{4} + 143 \, A b c^{5}\right )} x^{5} + 2 \,{\left (58 \, B b^{3} c^{3} + 187 \, A b^{2} c^{4}\right )} x^{4} -{\left (B b^{4} c^{2} - 176 \, A b^{3} c^{3}\right )} x^{3} +{\left (4 \, B b^{5} c - 11 \, A b^{4} c^{2}\right )} x^{2} + 2 \,{\left (4 \, B b^{6} - 11 \, A b^{5} c\right )} x\right )}}{693 \, \sqrt{c x^{2} + b x} c^{3} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(B*x + A)/x^(3/2),x, algorithm="fricas")

[Out]

2/693*(63*B*c^6*x^7 + 7*(32*B*b*c^5 + 11*A*c^6)*x^6 + 2*(137*B*b^2*c^4 + 143*A*b
*c^5)*x^5 + 2*(58*B*b^3*c^3 + 187*A*b^2*c^4)*x^4 - (B*b^4*c^2 - 176*A*b^3*c^3)*x
^3 + (4*B*b^5*c - 11*A*b^4*c^2)*x^2 + 2*(4*B*b^6 - 11*A*b^5*c)*x)/(sqrt(c*x^2 +
b*x)*c^3*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**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.28986, size = 464, normalized size = 4.83 \[ -\frac{2}{3465} \, B c^{2}{\left (\frac{128 \, b^{\frac{11}{2}}}{c^{5}} - \frac{315 \,{\left (c x + b\right )}^{\frac{11}{2}} - 1540 \,{\left (c x + b\right )}^{\frac{9}{2}} b + 2970 \,{\left (c x + b\right )}^{\frac{7}{2}} b^{2} - 2772 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{3} + 1155 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{4}}{c^{5}}\right )} + \frac{4}{315} \, B b c{\left (\frac{16 \, b^{\frac{9}{2}}}{c^{4}} + \frac{35 \,{\left (c x + b\right )}^{\frac{9}{2}} - 135 \,{\left (c x + b\right )}^{\frac{7}{2}} b + 189 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{2} - 105 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{3}}{c^{4}}\right )} + \frac{2}{315} \, A c^{2}{\left (\frac{16 \, b^{\frac{9}{2}}}{c^{4}} + \frac{35 \,{\left (c x + b\right )}^{\frac{9}{2}} - 135 \,{\left (c x + b\right )}^{\frac{7}{2}} b + 189 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{2} - 105 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{3}}{c^{4}}\right )} - \frac{2}{105} \, B b^{2}{\left (\frac{8 \, b^{\frac{7}{2}}}{c^{3}} - \frac{15 \,{\left (c x + b\right )}^{\frac{7}{2}} - 42 \,{\left (c x + b\right )}^{\frac{5}{2}} b + 35 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{2}}{c^{3}}\right )} - \frac{4}{105} \, A b c{\left (\frac{8 \, b^{\frac{7}{2}}}{c^{3}} - \frac{15 \,{\left (c x + b\right )}^{\frac{7}{2}} - 42 \,{\left (c x + b\right )}^{\frac{5}{2}} b + 35 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{2}}{c^{3}}\right )} + \frac{2}{15} \, A b^{2}{\left (\frac{2 \, b^{\frac{5}{2}}}{c^{2}} + \frac{3 \,{\left (c x + b\right )}^{\frac{5}{2}} - 5 \,{\left (c x + b\right )}^{\frac{3}{2}} b}{c^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(B*x + A)/x^(3/2),x, algorithm="giac")

[Out]

-2/3465*B*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 - 2772*(c*x + b)^(5/2)*b^3 + 1155*(c*x + b)^(3/2)*b^
4)/c^5) + 4/315*B*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) + 2/315*A*c^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) - 2/105*B*b^2*(8*b^(7/2)/c^3 - (15*(c*x + b)
^(7/2) - 42*(c*x + b)^(5/2)*b + 35*(c*x + b)^(3/2)*b^2)/c^3) - 4/105*A*b*c*(8*b^
(7/2)/c^3 - (15*(c*x + b)^(7/2) - 42*(c*x + b)^(5/2)*b + 35*(c*x + b)^(3/2)*b^2)
/c^3) + 2/15*A*b^2*(2*b^(5/2)/c^2 + (3*(c*x + b)^(5/2) - 5*(c*x + b)^(3/2)*b)/c^
2)