3.1005 \(\int \frac{(A+B x) (a+b x+c x^2)^3}{x^{5/2}} \, dx\)

Optimal. Leaf size=178 \[ -\frac{2 a^2 (a B+3 A b)}{\sqrt{x}}-\frac{2 a^3 A}{3 x^{3/2}}+\frac{2}{5} x^{5/2} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{6}{7} c x^{7/2} \left (a B c+A b c+b^2 B\right )+\frac{2}{3} x^{3/2} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+6 a \sqrt{x} \left (A \left (a c+b^2\right )+a b B\right )+\frac{2}{9} c^2 x^{9/2} (A c+3 b B)+\frac{2}{11} B c^3 x^{11/2} \]

[Out]

(-2*a^3*A)/(3*x^(3/2)) - (2*a^2*(3*A*b + a*B))/Sqrt[x] + 6*a*(a*b*B + A*(b^2 + a*c))*Sqrt[x] + (2*(3*a*B*(b^2
+ a*c) + A*(b^3 + 6*a*b*c))*x^(3/2))/3 + (2*(b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^(5/2))/5 + (6*c*(b^2
*B + A*b*c + a*B*c)*x^(7/2))/7 + (2*c^2*(3*b*B + A*c)*x^(9/2))/9 + (2*B*c^3*x^(11/2))/11

________________________________________________________________________________________

Rubi [A]  time = 0.111978, 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} \[ -\frac{2 a^2 (a B+3 A b)}{\sqrt{x}}-\frac{2 a^3 A}{3 x^{3/2}}+\frac{2}{5} x^{5/2} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{6}{7} c x^{7/2} \left (a B c+A b c+b^2 B\right )+\frac{2}{3} x^{3/2} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+6 a \sqrt{x} \left (A \left (a c+b^2\right )+a b B\right )+\frac{2}{9} c^2 x^{9/2} (A c+3 b B)+\frac{2}{11} B c^3 x^{11/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^3*A)/(3*x^(3/2)) - (2*a^2*(3*A*b + a*B))/Sqrt[x] + 6*a*(a*b*B + A*(b^2 + a*c))*Sqrt[x] + (2*(3*a*B*(b^2
+ a*c) + A*(b^3 + 6*a*b*c))*x^(3/2))/3 + (2*(b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^(5/2))/5 + (6*c*(b^2
*B + A*b*c + a*B*c)*x^(7/2))/7 + (2*c^2*(3*b*B + A*c)*x^(9/2))/9 + (2*B*c^3*x^(11/2))/11

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

Mathematica [A]  time = 0.163685, size = 170, normalized size = 0.96 \[ \frac{2 \left (3465 a^2 x (B x (3 b+c x)-3 A (b-c x))-1155 a^3 (A+3 B x)+99 a x^2 \left (7 A \left (15 b^2+10 b c x+3 c^2 x^2\right )+B x \left (35 b^2+42 b c x+15 c^2 x^2\right )\right )+x^3 \left (11 A \left (189 b^2 c x+105 b^3+135 b c^2 x^2+35 c^3 x^3\right )+3 B x \left (495 b^2 c x+231 b^3+385 b c^2 x^2+105 c^3 x^3\right )\right )\right )}{3465 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-1155*a^3*(A + 3*B*x) + 3465*a^2*x*(-3*A*(b - c*x) + B*x*(3*b + c*x)) + 99*a*x^2*(7*A*(15*b^2 + 10*b*c*x +
3*c^2*x^2) + B*x*(35*b^2 + 42*b*c*x + 15*c^2*x^2)) + x^3*(11*A*(105*b^3 + 189*b^2*c*x + 135*b*c^2*x^2 + 35*c^
3*x^3) + 3*B*x*(231*b^3 + 495*b^2*c*x + 385*b*c^2*x^2 + 105*c^3*x^3))))/(3465*x^(3/2))

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 192, normalized size = 1.1 \begin{align*} -{\frac{-630\,B{c}^{3}{x}^{7}-770\,A{c}^{3}{x}^{6}-2310\,B{x}^{6}b{c}^{2}-2970\,A{x}^{5}b{c}^{2}-2970\,aB{c}^{2}{x}^{5}-2970\,B{x}^{5}{b}^{2}c-4158\,aA{c}^{2}{x}^{4}-4158\,A{x}^{4}{b}^{2}c-8316\,B{x}^{4}abc-1386\,B{x}^{4}{b}^{3}-13860\,A{x}^{3}abc-2310\,A{b}^{3}{x}^{3}-6930\,{a}^{2}Bc{x}^{3}-6930\,B{x}^{3}a{b}^{2}-20790\,{a}^{2}Ac{x}^{2}-20790\,A{x}^{2}a{b}^{2}-20790\,B{x}^{2}{a}^{2}b+20790\,A{a}^{2}bx+6930\,{a}^{3}Bx+2310\,A{a}^{3}}{3465}{x}^{-{\frac{3}{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^(5/2),x)

[Out]

-2/3465*(-315*B*c^3*x^7-385*A*c^3*x^6-1155*B*b*c^2*x^6-1485*A*b*c^2*x^5-1485*B*a*c^2*x^5-1485*B*b^2*c*x^5-2079
*A*a*c^2*x^4-2079*A*b^2*c*x^4-4158*B*a*b*c*x^4-693*B*b^3*x^4-6930*A*a*b*c*x^3-1155*A*b^3*x^3-3465*B*a^2*c*x^3-
3465*B*a*b^2*x^3-10395*A*a^2*c*x^2-10395*A*a*b^2*x^2-10395*B*a^2*b*x^2+10395*A*a^2*b*x+3465*B*a^3*x+1155*A*a^3
)/x^(3/2)

________________________________________________________________________________________

Maxima [A]  time = 1.01985, size = 224, normalized size = 1.26 \begin{align*} \frac{2}{11} \, B c^{3} x^{\frac{11}{2}} + \frac{2}{9} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac{9}{2}} + \frac{6}{7} \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{\frac{7}{2}} + \frac{2}{5} \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{\frac{5}{2}} + \frac{2}{3} \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{\frac{3}{2}} + 6 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} \sqrt{x} - \frac{2 \,{\left (A a^{3} + 3 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )}}{3 \, 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^(5/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.03988, size = 397, normalized size = 2.23 \begin{align*} \frac{2 \,{\left (315 \, B c^{3} x^{7} + 385 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 1485 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{5} + 693 \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} - 1155 \, A a^{3} + 1155 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + 10395 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} - 3465 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )}}{3465 \, 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^(5/2),x, algorithm="fricas")

[Out]

2/3465*(315*B*c^3*x^7 + 385*(3*B*b*c^2 + A*c^3)*x^6 + 1485*(B*b^2*c + (B*a + A*b)*c^2)*x^5 + 693*(B*b^3 + 3*A*
a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^4 - 1155*A*a^3 + 1155*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^3 + 10395
*(B*a^2*b + A*a*b^2 + A*a^2*c)*x^2 - 3465*(B*a^3 + 3*A*a^2*b)*x)/x^(3/2)

________________________________________________________________________________________

Sympy [A]  time = 8.55106, size = 280, normalized size = 1.57 \begin{align*} - \frac{2 A a^{3}}{3 x^{\frac{3}{2}}} - \frac{6 A a^{2} b}{\sqrt{x}} + 6 A a^{2} c \sqrt{x} + 6 A a b^{2} \sqrt{x} + 4 A a b c x^{\frac{3}{2}} + \frac{6 A a c^{2} x^{\frac{5}{2}}}{5} + \frac{2 A b^{3} x^{\frac{3}{2}}}{3} + \frac{6 A b^{2} c x^{\frac{5}{2}}}{5} + \frac{6 A b c^{2} x^{\frac{7}{2}}}{7} + \frac{2 A c^{3} x^{\frac{9}{2}}}{9} - \frac{2 B a^{3}}{\sqrt{x}} + 6 B a^{2} b \sqrt{x} + 2 B a^{2} c x^{\frac{3}{2}} + 2 B a b^{2} x^{\frac{3}{2}} + \frac{12 B a b c x^{\frac{5}{2}}}{5} + \frac{6 B a c^{2} x^{\frac{7}{2}}}{7} + \frac{2 B b^{3} x^{\frac{5}{2}}}{5} + \frac{6 B b^{2} c x^{\frac{7}{2}}}{7} + \frac{2 B b c^{2} x^{\frac{9}{2}}}{3} + \frac{2 B c^{3} x^{\frac{11}{2}}}{11} \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**(5/2),x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.21634, size = 258, normalized size = 1.45 \begin{align*} \frac{2}{11} \, B c^{3} x^{\frac{11}{2}} + \frac{2}{3} \, B b c^{2} x^{\frac{9}{2}} + \frac{2}{9} \, A c^{3} x^{\frac{9}{2}} + \frac{6}{7} \, B b^{2} c x^{\frac{7}{2}} + \frac{6}{7} \, B a c^{2} x^{\frac{7}{2}} + \frac{6}{7} \, A b c^{2} x^{\frac{7}{2}} + \frac{2}{5} \, B b^{3} x^{\frac{5}{2}} + \frac{12}{5} \, B a b c x^{\frac{5}{2}} + \frac{6}{5} \, A b^{2} c x^{\frac{5}{2}} + \frac{6}{5} \, A a c^{2} x^{\frac{5}{2}} + 2 \, B a b^{2} x^{\frac{3}{2}} + \frac{2}{3} \, A b^{3} x^{\frac{3}{2}} + 2 \, B a^{2} c x^{\frac{3}{2}} + 4 \, A a b c x^{\frac{3}{2}} + 6 \, B a^{2} b \sqrt{x} + 6 \, A a b^{2} \sqrt{x} + 6 \, A a^{2} c \sqrt{x} - \frac{2 \,{\left (3 \, B a^{3} x + 9 \, A a^{2} b x + A a^{3}\right )}}{3 \, 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^(5/2),x, algorithm="giac")

[Out]

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