3.162 \(\int \frac{(A+B x) (b x+c x^2)^3}{\sqrt{x}} \, dx\)

Optimal. Leaf size=85 \[ \frac{2}{9} b^2 x^{9/2} (3 A c+b B)+\frac{2}{7} A b^3 x^{7/2}+\frac{2}{13} c^2 x^{13/2} (A c+3 b B)+\frac{6}{11} b c x^{11/2} (A c+b B)+\frac{2}{15} B c^3 x^{15/2} \]

[Out]

(2*A*b^3*x^(7/2))/7 + (2*b^2*(b*B + 3*A*c)*x^(9/2))/9 + (6*b*c*(b*B + A*c)*x^(11/2))/11 + (2*c^2*(3*b*B + A*c)
*x^(13/2))/13 + (2*B*c^3*x^(15/2))/15

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Rubi [A]  time = 0.0405242, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {765} \[ \frac{2}{9} b^2 x^{9/2} (3 A c+b B)+\frac{2}{7} A b^3 x^{7/2}+\frac{2}{13} c^2 x^{13/2} (A c+3 b B)+\frac{6}{11} b c x^{11/2} (A c+b B)+\frac{2}{15} B c^3 x^{15/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*A*b^3*x^(7/2))/7 + (2*b^2*(b*B + 3*A*c)*x^(9/2))/9 + (6*b*c*(b*B + A*c)*x^(11/2))/11 + (2*c^2*(3*b*B + A*c)
*x^(13/2))/13 + (2*B*c^3*x^(15/2))/15

Rule 765

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand
Integrand[(e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, e, f, g, m}, x] && IntegerQ[p] && (
GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int \frac{(A+B x) \left (b x+c x^2\right )^3}{\sqrt{x}} \, dx &=\int \left (A b^3 x^{5/2}+b^2 (b B+3 A c) x^{7/2}+3 b c (b B+A c) x^{9/2}+c^2 (3 b B+A c) x^{11/2}+B c^3 x^{13/2}\right ) \, dx\\ &=\frac{2}{7} A b^3 x^{7/2}+\frac{2}{9} b^2 (b B+3 A c) x^{9/2}+\frac{6}{11} b c (b B+A c) x^{11/2}+\frac{2}{13} c^2 (3 b B+A c) x^{13/2}+\frac{2}{15} B c^3 x^{15/2}\\ \end{align*}

Mathematica [A]  time = 0.0466555, size = 70, normalized size = 0.82 \[ \frac{2 \left (B x^{7/2} (b+c x)^4-\frac{x^{7/2} \left (1001 b^2 c x+429 b^3+819 b c^2 x^2+231 c^3 x^3\right ) (7 b B-15 A c)}{3003}\right )}{15 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(B*x^(7/2)*(b + c*x)^4 - ((7*b*B - 15*A*c)*x^(7/2)*(429*b^3 + 1001*b^2*c*x + 819*b*c^2*x^2 + 231*c^3*x^3))/
3003))/(15*c)

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Maple [A]  time = 0.006, size = 76, normalized size = 0.9 \begin{align*}{\frac{6006\,B{c}^{3}{x}^{4}+6930\,A{x}^{3}{c}^{3}+20790\,B{x}^{3}b{c}^{2}+24570\,A{x}^{2}b{c}^{2}+24570\,B{x}^{2}{b}^{2}c+30030\,A{b}^{2}cx+10010\,{b}^{3}Bx+12870\,A{b}^{3}}{45045}{x}^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)^3/x^(1/2),x)

[Out]

2/45045*x^(7/2)*(3003*B*c^3*x^4+3465*A*c^3*x^3+10395*B*b*c^2*x^3+12285*A*b*c^2*x^2+12285*B*b^2*c*x^2+15015*A*b
^2*c*x+5005*B*b^3*x+6435*A*b^3)

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Maxima [A]  time = 1.06587, size = 99, normalized size = 1.16 \begin{align*} \frac{2}{15} \, B c^{3} x^{\frac{15}{2}} + \frac{2}{7} \, A b^{3} x^{\frac{7}{2}} + \frac{2}{13} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac{13}{2}} + \frac{6}{11} \,{\left (B b^{2} c + A b c^{2}\right )} x^{\frac{11}{2}} + \frac{2}{9} \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{\frac{9}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^3/x^(1/2),x, algorithm="maxima")

[Out]

2/15*B*c^3*x^(15/2) + 2/7*A*b^3*x^(7/2) + 2/13*(3*B*b*c^2 + A*c^3)*x^(13/2) + 6/11*(B*b^2*c + A*b*c^2)*x^(11/2
) + 2/9*(B*b^3 + 3*A*b^2*c)*x^(9/2)

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Fricas [A]  time = 1.72616, size = 196, normalized size = 2.31 \begin{align*} \frac{2}{45045} \,{\left (3003 \, B c^{3} x^{7} + 6435 \, A b^{3} x^{3} + 3465 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 12285 \,{\left (B b^{2} c + A b c^{2}\right )} x^{5} + 5005 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{4}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^3/x^(1/2),x, algorithm="fricas")

[Out]

2/45045*(3003*B*c^3*x^7 + 6435*A*b^3*x^3 + 3465*(3*B*b*c^2 + A*c^3)*x^6 + 12285*(B*b^2*c + A*b*c^2)*x^5 + 5005
*(B*b^3 + 3*A*b^2*c)*x^4)*sqrt(x)

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Sympy [A]  time = 3.57524, size = 114, normalized size = 1.34 \begin{align*} \frac{2 A b^{3} x^{\frac{7}{2}}}{7} + \frac{2 A b^{2} c x^{\frac{9}{2}}}{3} + \frac{6 A b c^{2} x^{\frac{11}{2}}}{11} + \frac{2 A c^{3} x^{\frac{13}{2}}}{13} + \frac{2 B b^{3} x^{\frac{9}{2}}}{9} + \frac{6 B b^{2} c x^{\frac{11}{2}}}{11} + \frac{6 B b c^{2} x^{\frac{13}{2}}}{13} + \frac{2 B c^{3} x^{\frac{15}{2}}}{15} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)**3/x**(1/2),x)

[Out]

2*A*b**3*x**(7/2)/7 + 2*A*b**2*c*x**(9/2)/3 + 6*A*b*c**2*x**(11/2)/11 + 2*A*c**3*x**(13/2)/13 + 2*B*b**3*x**(9
/2)/9 + 6*B*b**2*c*x**(11/2)/11 + 6*B*b*c**2*x**(13/2)/13 + 2*B*c**3*x**(15/2)/15

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Giac [A]  time = 1.12682, size = 104, normalized size = 1.22 \begin{align*} \frac{2}{15} \, B c^{3} x^{\frac{15}{2}} + \frac{6}{13} \, B b c^{2} x^{\frac{13}{2}} + \frac{2}{13} \, A c^{3} x^{\frac{13}{2}} + \frac{6}{11} \, B b^{2} c x^{\frac{11}{2}} + \frac{6}{11} \, A b c^{2} x^{\frac{11}{2}} + \frac{2}{9} \, B b^{3} x^{\frac{9}{2}} + \frac{2}{3} \, A b^{2} c x^{\frac{9}{2}} + \frac{2}{7} \, A b^{3} x^{\frac{7}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^3/x^(1/2),x, algorithm="giac")

[Out]

2/15*B*c^3*x^(15/2) + 6/13*B*b*c^2*x^(13/2) + 2/13*A*c^3*x^(13/2) + 6/11*B*b^2*c*x^(11/2) + 6/11*A*b*c^2*x^(11
/2) + 2/9*B*b^3*x^(9/2) + 2/3*A*b^2*c*x^(9/2) + 2/7*A*b^3*x^(7/2)