3.1003 \(\int \frac{(A+B x) (a+b x+c x^2)^3}{\sqrt{x}} \, dx\)

Optimal. Leaf size=180 \[ \frac{2}{3} a^2 x^{3/2} (a B+3 A b)+2 a^3 A \sqrt{x}+\frac{2}{9} x^{9/2} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{6}{11} c x^{11/2} \left (a B c+A b c+b^2 B\right )+\frac{2}{7} x^{7/2} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{6}{5} a x^{5/2} \left (A \left (a c+b^2\right )+a b B\right )+\frac{2}{13} c^2 x^{13/2} (A c+3 b B)+\frac{2}{15} B c^3 x^{15/2} \]

[Out]

2*a^3*A*Sqrt[x] + (2*a^2*(3*A*b + a*B)*x^(3/2))/3 + (6*a*(a*b*B + A*(b^2 + a*c))*x^(5/2))/5 + (2*(3*a*B*(b^2 +
a*c) + A*(b^3 + 6*a*b*c))*x^(7/2))/7 + (2*(b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^(9/2))/9 + (6*c*(b^2*
B + A*b*c + a*B*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 = 0.111077, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {765} \[ \frac{2}{3} a^2 x^{3/2} (a B+3 A b)+2 a^3 A \sqrt{x}+\frac{2}{9} x^{9/2} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{6}{11} c x^{11/2} \left (a B c+A b c+b^2 B\right )+\frac{2}{7} x^{7/2} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{6}{5} a x^{5/2} \left (A \left (a c+b^2\right )+a b B\right )+\frac{2}{13} c^2 x^{13/2} (A c+3 b B)+\frac{2}{15} B c^3 x^{15/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*a^3*A*Sqrt[x] + (2*a^2*(3*A*b + a*B)*x^(3/2))/3 + (6*a*(a*b*B + A*(b^2 + a*c))*x^(5/2))/5 + (2*(3*a*B*(b^2 +
a*c) + A*(b^3 + 6*a*b*c))*x^(7/2))/7 + (2*(b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^(9/2))/9 + (6*c*(b^2*
B + A*b*c + a*B*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 (a+b x+c x^2\right )^3}{\sqrt{x}} \, dx &=\int \left (\frac{a^3 A}{\sqrt{x}}+a^2 (3 A b+a B) \sqrt{x}+3 a \left (a b B+A \left (b^2+a c\right )\right ) x^{3/2}+\left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^{5/2}+\left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^{7/2}+3 c \left (b^2 B+A b c+a B c\right ) x^{9/2}+c^2 (3 b B+A c) x^{11/2}+B c^3 x^{13/2}\right ) \, dx\\ &=2 a^3 A \sqrt{x}+\frac{2}{3} a^2 (3 A b+a B) x^{3/2}+\frac{6}{5} a \left (a b B+A \left (b^2+a c\right )\right ) x^{5/2}+\frac{2}{7} \left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^{7/2}+\frac{2}{9} \left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^{9/2}+\frac{6}{11} c \left (b^2 B+A b c+a B c\right ) 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 = 0.20156, size = 176, normalized size = 0.98 \[ \frac{2 \sqrt{x} \left (1287 a^2 x (7 A (5 b+3 c x)+3 B x (7 b+5 c x))+15015 a^3 (3 A+B x)+39 a x^2 \left (11 A \left (63 b^2+90 b c x+35 c^2 x^2\right )+5 B x \left (99 b^2+154 b c x+63 c^2 x^2\right )\right )+x^3 \left (15 A \left (1001 b^2 c x+429 b^3+819 b c^2 x^2+231 c^3 x^3\right )+7 B x \left (1755 b^2 c x+715 b^3+1485 b c^2 x^2+429 c^3 x^3\right )\right )\right )}{45045} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[x]*(15015*a^3*(3*A + B*x) + 1287*a^2*x*(7*A*(5*b + 3*c*x) + 3*B*x*(7*b + 5*c*x)) + 39*a*x^2*(11*A*(63*
b^2 + 90*b*c*x + 35*c^2*x^2) + 5*B*x*(99*b^2 + 154*b*c*x + 63*c^2*x^2)) + x^3*(15*A*(429*b^3 + 1001*b^2*c*x +
819*b*c^2*x^2 + 231*c^3*x^3) + 7*B*x*(715*b^3 + 1755*b^2*c*x + 1485*b*c^2*x^2 + 429*c^3*x^3))))/45045

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Maple [A]  time = 0.007, size = 192, normalized size = 1.1 \begin{align*}{\frac{6006\,B{c}^{3}{x}^{7}+6930\,A{c}^{3}{x}^{6}+20790\,B{x}^{6}b{c}^{2}+24570\,A{x}^{5}b{c}^{2}+24570\,aB{c}^{2}{x}^{5}+24570\,B{x}^{5}{b}^{2}c+30030\,aA{c}^{2}{x}^{4}+30030\,A{x}^{4}{b}^{2}c+60060\,B{x}^{4}abc+10010\,B{x}^{4}{b}^{3}+77220\,A{x}^{3}abc+12870\,A{b}^{3}{x}^{3}+38610\,{a}^{2}Bc{x}^{3}+38610\,B{x}^{3}a{b}^{2}+54054\,{a}^{2}Ac{x}^{2}+54054\,A{x}^{2}a{b}^{2}+54054\,B{x}^{2}{a}^{2}b+90090\,A{a}^{2}bx+30030\,{a}^{3}Bx+90090\,A{a}^{3}}{45045}\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*x^(1/2)*(3003*B*c^3*x^7+3465*A*c^3*x^6+10395*B*b*c^2*x^6+12285*A*b*c^2*x^5+12285*B*a*c^2*x^5+12285*B*b
^2*c*x^5+15015*A*a*c^2*x^4+15015*A*b^2*c*x^4+30030*B*a*b*c*x^4+5005*B*b^3*x^4+38610*A*a*b*c*x^3+6435*A*b^3*x^3
+19305*B*a^2*c*x^3+19305*B*a*b^2*x^3+27027*A*a^2*c*x^2+27027*A*a*b^2*x^2+27027*B*a^2*b*x^2+45045*A*a^2*b*x+150
15*B*a^3*x+45045*A*a^3)

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Maxima [A]  time = 1.1147, size = 224, normalized size = 1.24 \begin{align*} \frac{2}{15} \, B c^{3} x^{\frac{15}{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 +{\left (B a + A b\right )} c^{2}\right )} x^{\frac{11}{2}} + \frac{2}{9} \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{\frac{9}{2}} + 2 \, A a^{3} \sqrt{x} + \frac{2}{7} \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{\frac{7}{2}} + \frac{6}{5} \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{\frac{5}{2}} + \frac{2}{3} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*B*c^3*x^(15/2) + 2/13*(3*B*b*c^2 + A*c^3)*x^(13/2) + 6/11*(B*b^2*c + (B*a + A*b)*c^2)*x^(11/2) + 2/9*(B*b
^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^(9/2) + 2*A*a^3*sqrt(x) + 2/7*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*
b)*c)*x^(7/2) + 6/5*(B*a^2*b + A*a*b^2 + A*a^2*c)*x^(5/2) + 2/3*(B*a^3 + 3*A*a^2*b)*x^(3/2)

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Fricas [A]  time = 1.04603, size = 406, normalized size = 2.26 \begin{align*} \frac{2}{45045} \,{\left (3003 \, B c^{3} x^{7} + 3465 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 12285 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{5} + 5005 \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} + 45045 \, A a^{3} + 6435 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + 27027 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 15015 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x\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+a)^3/x^(1/2),x, algorithm="fricas")

[Out]

2/45045*(3003*B*c^3*x^7 + 3465*(3*B*b*c^2 + A*c^3)*x^6 + 12285*(B*b^2*c + (B*a + A*b)*c^2)*x^5 + 5005*(B*b^3 +
3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^4 + 45045*A*a^3 + 6435*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^3 +
27027*(B*a^2*b + A*a*b^2 + A*a^2*c)*x^2 + 15015*(B*a^3 + 3*A*a^2*b)*x)*sqrt(x)

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Sympy [A]  time = 6.4922, size = 291, normalized size = 1.62 \begin{align*} 2 A a^{3} \sqrt{x} + 2 A a^{2} b x^{\frac{3}{2}} + \frac{6 A a^{2} c x^{\frac{5}{2}}}{5} + \frac{6 A a b^{2} x^{\frac{5}{2}}}{5} + \frac{12 A a b c x^{\frac{7}{2}}}{7} + \frac{2 A a c^{2} x^{\frac{9}{2}}}{3} + \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 a^{3} x^{\frac{3}{2}}}{3} + \frac{6 B a^{2} b x^{\frac{5}{2}}}{5} + \frac{6 B a^{2} c x^{\frac{7}{2}}}{7} + \frac{6 B a b^{2} x^{\frac{7}{2}}}{7} + \frac{4 B a b c x^{\frac{9}{2}}}{3} + \frac{6 B a c^{2} x^{\frac{11}{2}}}{11} + \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+a)**3/x**(1/2),x)

[Out]

2*A*a**3*sqrt(x) + 2*A*a**2*b*x**(3/2) + 6*A*a**2*c*x**(5/2)/5 + 6*A*a*b**2*x**(5/2)/5 + 12*A*a*b*c*x**(7/2)/7
+ 2*A*a*c**2*x**(9/2)/3 + 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*a**3*x**(3/2)/3 + 6*B*a**2*b*x**(5/2)/5 + 6*B*a**2*c*x**(7/2)/7 + 6*B*a*b**2*x**(7/2)/7 + 4*B
*a*b*c*x**(9/2)/3 + 6*B*a*c**2*x**(11/2)/11 + 2*B*b**3*x**(9/2)/9 + 6*B*b**2*c*x**(11/2)/11 + 6*B*b*c**2*x**(1
3/2)/13 + 2*B*c**3*x**(15/2)/15

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Giac [A]  time = 1.27479, size = 261, normalized size = 1.45 \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} \, B a c^{2} 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{4}{3} \, B a b c x^{\frac{9}{2}} + \frac{2}{3} \, A b^{2} c x^{\frac{9}{2}} + \frac{2}{3} \, A a c^{2} x^{\frac{9}{2}} + \frac{6}{7} \, B a b^{2} x^{\frac{7}{2}} + \frac{2}{7} \, A b^{3} x^{\frac{7}{2}} + \frac{6}{7} \, B a^{2} c x^{\frac{7}{2}} + \frac{12}{7} \, A a b c x^{\frac{7}{2}} + \frac{6}{5} \, B a^{2} b x^{\frac{5}{2}} + \frac{6}{5} \, A a b^{2} x^{\frac{5}{2}} + \frac{6}{5} \, A a^{2} c x^{\frac{5}{2}} + \frac{2}{3} \, B a^{3} x^{\frac{3}{2}} + 2 \, A a^{2} b x^{\frac{3}{2}} + 2 \, A a^{3} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)^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*B*a*c^2*x^(11
/2) + 6/11*A*b*c^2*x^(11/2) + 2/9*B*b^3*x^(9/2) + 4/3*B*a*b*c*x^(9/2) + 2/3*A*b^2*c*x^(9/2) + 2/3*A*a*c^2*x^(9
/2) + 6/7*B*a*b^2*x^(7/2) + 2/7*A*b^3*x^(7/2) + 6/7*B*a^2*c*x^(7/2) + 12/7*A*a*b*c*x^(7/2) + 6/5*B*a^2*b*x^(5/
2) + 6/5*A*a*b^2*x^(5/2) + 6/5*A*a^2*c*x^(5/2) + 2/3*B*a^3*x^(3/2) + 2*A*a^2*b*x^(3/2) + 2*A*a^3*sqrt(x)