3.1008 \(\int \frac{(A+B x) (a+b x+c x^2)^3}{x^{11/2}} \, dx\)

Optimal. Leaf size=178 \[ -\frac{2 a^2 (a B+3 A b)}{7 x^{7/2}}-\frac{2 a^3 A}{9 x^{9/2}}-\frac{2 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )}{\sqrt{x}}-\frac{6 a \left (A \left (a c+b^2\right )+a b B\right )}{5 x^{5/2}}-\frac{2 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )}{3 x^{3/2}}+6 c \sqrt{x} \left (a B c+A b c+b^2 B\right )+\frac{2}{3} c^2 x^{3/2} (A c+3 b B)+\frac{2}{5} B c^3 x^{5/2} \]

[Out]

(-2*a^3*A)/(9*x^(9/2)) - (2*a^2*(3*A*b + a*B))/(7*x^(7/2)) - (6*a*(a*b*B + A*(b^2 + a*c)))/(5*x^(5/2)) - (2*(3
*a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c)))/(3*x^(3/2)) - (2*(b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2))/Sqrt[x] +
6*c*(b^2*B + A*b*c + a*B*c)*Sqrt[x] + (2*c^2*(3*b*B + A*c)*x^(3/2))/3 + (2*B*c^3*x^(5/2))/5

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Rubi [A]  time = 0.112042, antiderivative size = 178, 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} \[ -\frac{2 a^2 (a B+3 A b)}{7 x^{7/2}}-\frac{2 a^3 A}{9 x^{9/2}}-\frac{2 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )}{\sqrt{x}}-\frac{6 a \left (A \left (a c+b^2\right )+a b B\right )}{5 x^{5/2}}-\frac{2 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )}{3 x^{3/2}}+6 c \sqrt{x} \left (a B c+A b c+b^2 B\right )+\frac{2}{3} c^2 x^{3/2} (A c+3 b B)+\frac{2}{5} B c^3 x^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + b*x + c*x^2)^3)/x^(11/2),x]

[Out]

(-2*a^3*A)/(9*x^(9/2)) - (2*a^2*(3*A*b + a*B))/(7*x^(7/2)) - (6*a*(a*b*B + A*(b^2 + a*c)))/(5*x^(5/2)) - (2*(3
*a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c)))/(3*x^(3/2)) - (2*(b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2))/Sqrt[x] +
6*c*(b^2*B + A*b*c + a*B*c)*Sqrt[x] + (2*c^2*(3*b*B + A*c)*x^(3/2))/3 + (2*B*c^3*x^(5/2))/5

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

Mathematica [A]  time = 0.1999, size = 172, normalized size = 0.97 \[ -\frac{2 \left (9 a^2 x (3 A (5 b+7 c x)+7 B x (3 b+5 c x))+5 a^3 (7 A+9 B x)+63 a x^2 \left (A \left (3 b^2+10 b c x+15 c^2 x^2\right )+5 B x \left (b^2+6 b c x-3 c^2 x^2\right )\right )+21 x^3 \left (5 A \left (9 b^2 c x+b^3-9 b c^2 x^2-c^3 x^3\right )-3 B x \left (15 b^2 c x-5 b^3+5 b c^2 x^2+c^3 x^3\right )\right )\right )}{315 x^{9/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + b*x + c*x^2)^3)/x^(11/2),x]

[Out]

(-2*(5*a^3*(7*A + 9*B*x) + 9*a^2*x*(7*B*x*(3*b + 5*c*x) + 3*A*(5*b + 7*c*x)) + 63*a*x^2*(5*B*x*(b^2 + 6*b*c*x
- 3*c^2*x^2) + A*(3*b^2 + 10*b*c*x + 15*c^2*x^2)) + 21*x^3*(5*A*(b^3 + 9*b^2*c*x - 9*b*c^2*x^2 - c^3*x^3) - 3*
B*x*(-5*b^3 + 15*b^2*c*x + 5*b*c^2*x^2 + c^3*x^3))))/(315*x^(9/2))

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Maple [A]  time = 0.007, size = 192, normalized size = 1.1 \begin{align*} -{\frac{-126\,B{c}^{3}{x}^{7}-210\,A{c}^{3}{x}^{6}-630\,B{x}^{6}b{c}^{2}-1890\,A{x}^{5}b{c}^{2}-1890\,aB{c}^{2}{x}^{5}-1890\,B{x}^{5}{b}^{2}c+1890\,aA{c}^{2}{x}^{4}+1890\,A{x}^{4}{b}^{2}c+3780\,B{x}^{4}abc+630\,B{x}^{4}{b}^{3}+1260\,A{x}^{3}abc+210\,A{b}^{3}{x}^{3}+630\,{a}^{2}Bc{x}^{3}+630\,B{x}^{3}a{b}^{2}+378\,{a}^{2}Ac{x}^{2}+378\,A{x}^{2}a{b}^{2}+378\,B{x}^{2}{a}^{2}b+270\,A{a}^{2}bx+90\,{a}^{3}Bx+70\,A{a}^{3}}{315}{x}^{-{\frac{9}{2}}}} \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^(11/2),x)

[Out]

-2/315*(-63*B*c^3*x^7-105*A*c^3*x^6-315*B*b*c^2*x^6-945*A*b*c^2*x^5-945*B*a*c^2*x^5-945*B*b^2*c*x^5+945*A*a*c^
2*x^4+945*A*b^2*c*x^4+1890*B*a*b*c*x^4+315*B*b^3*x^4+630*A*a*b*c*x^3+105*A*b^3*x^3+315*B*a^2*c*x^3+315*B*a*b^2
*x^3+189*A*a^2*c*x^2+189*A*a*b^2*x^2+189*B*a^2*b*x^2+135*A*a^2*b*x+45*B*a^3*x+35*A*a^3)/x^(9/2)

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Maxima [A]  time = 1.06637, size = 225, normalized size = 1.26 \begin{align*} \frac{2}{5} \, B c^{3} x^{\frac{5}{2}} + \frac{2}{3} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac{3}{2}} + 6 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} \sqrt{x} - \frac{2 \,{\left (315 \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} + 35 \, A a^{3} + 105 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + 189 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 45 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )}}{315 \, 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+a)^3/x^(11/2),x, algorithm="maxima")

[Out]

2/5*B*c^3*x^(5/2) + 2/3*(3*B*b*c^2 + A*c^3)*x^(3/2) + 6*(B*b^2*c + (B*a + A*b)*c^2)*sqrt(x) - 2/315*(315*(B*b^
3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^4 + 35*A*a^3 + 105*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^3 +
189*(B*a^2*b + A*a*b^2 + A*a^2*c)*x^2 + 45*(B*a^3 + 3*A*a^2*b)*x)/x^(9/2)

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Fricas [A]  time = 1.05541, size = 383, normalized size = 2.15 \begin{align*} \frac{2 \,{\left (63 \, B c^{3} x^{7} + 105 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 945 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{5} - 315 \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} - 35 \, A a^{3} - 105 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} - 189 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} - 45 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )}}{315 \, 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+a)^3/x^(11/2),x, algorithm="fricas")

[Out]

2/315*(63*B*c^3*x^7 + 105*(3*B*b*c^2 + A*c^3)*x^6 + 945*(B*b^2*c + (B*a + A*b)*c^2)*x^5 - 315*(B*b^3 + 3*A*a*c
^2 + 3*(2*B*a*b + A*b^2)*c)*x^4 - 35*A*a^3 - 105*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^3 - 189*(B*a^2*
b + A*a*b^2 + A*a^2*c)*x^2 - 45*(B*a^3 + 3*A*a^2*b)*x)/x^(9/2)

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Sympy [A]  time = 22.0206, size = 275, normalized size = 1.54 \begin{align*} - \frac{2 A a^{3}}{9 x^{\frac{9}{2}}} - \frac{6 A a^{2} b}{7 x^{\frac{7}{2}}} - \frac{6 A a^{2} c}{5 x^{\frac{5}{2}}} - \frac{6 A a b^{2}}{5 x^{\frac{5}{2}}} - \frac{4 A a b c}{x^{\frac{3}{2}}} - \frac{6 A a c^{2}}{\sqrt{x}} - \frac{2 A b^{3}}{3 x^{\frac{3}{2}}} - \frac{6 A b^{2} c}{\sqrt{x}} + 6 A b c^{2} \sqrt{x} + \frac{2 A c^{3} x^{\frac{3}{2}}}{3} - \frac{2 B a^{3}}{7 x^{\frac{7}{2}}} - \frac{6 B a^{2} b}{5 x^{\frac{5}{2}}} - \frac{2 B a^{2} c}{x^{\frac{3}{2}}} - \frac{2 B a b^{2}}{x^{\frac{3}{2}}} - \frac{12 B a b c}{\sqrt{x}} + 6 B a c^{2} \sqrt{x} - \frac{2 B b^{3}}{\sqrt{x}} + 6 B b^{2} c \sqrt{x} + 2 B b c^{2} x^{\frac{3}{2}} + \frac{2 B c^{3} x^{\frac{5}{2}}}{5} \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**(11/2),x)

[Out]

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

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Giac [A]  time = 1.25947, size = 259, normalized size = 1.46 \begin{align*} \frac{2}{5} \, B c^{3} x^{\frac{5}{2}} + 2 \, B b c^{2} x^{\frac{3}{2}} + \frac{2}{3} \, A c^{3} x^{\frac{3}{2}} + 6 \, B b^{2} c \sqrt{x} + 6 \, B a c^{2} \sqrt{x} + 6 \, A b c^{2} \sqrt{x} - \frac{2 \,{\left (315 \, B b^{3} x^{4} + 1890 \, B a b c x^{4} + 945 \, A b^{2} c x^{4} + 945 \, A a c^{2} x^{4} + 315 \, B a b^{2} x^{3} + 105 \, A b^{3} x^{3} + 315 \, B a^{2} c x^{3} + 630 \, A a b c x^{3} + 189 \, B a^{2} b x^{2} + 189 \, A a b^{2} x^{2} + 189 \, A a^{2} c x^{2} + 45 \, B a^{3} x + 135 \, A a^{2} b x + 35 \, A a^{3}\right )}}{315 \, 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+a)^3/x^(11/2),x, algorithm="giac")

[Out]

2/5*B*c^3*x^(5/2) + 2*B*b*c^2*x^(3/2) + 2/3*A*c^3*x^(3/2) + 6*B*b^2*c*sqrt(x) + 6*B*a*c^2*sqrt(x) + 6*A*b*c^2*
sqrt(x) - 2/315*(315*B*b^3*x^4 + 1890*B*a*b*c*x^4 + 945*A*b^2*c*x^4 + 945*A*a*c^2*x^4 + 315*B*a*b^2*x^3 + 105*
A*b^3*x^3 + 315*B*a^2*c*x^3 + 630*A*a*b*c*x^3 + 189*B*a^2*b*x^2 + 189*A*a*b^2*x^2 + 189*A*a^2*c*x^2 + 45*B*a^3
*x + 135*A*a^2*b*x + 35*A*a^3)/x^(9/2)