3.1004 \(\int \frac{(A+B x) (a+b x+c x^2)^3}{x^{3/2}} \, dx\)

Optimal. Leaf size=176 \[ 2 a^2 \sqrt{x} (a B+3 A b)-\frac{2 a^3 A}{\sqrt{x}}+\frac{2}{7} x^{7/2} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{2}{3} c x^{9/2} \left (a B c+A b c+b^2 B\right )+\frac{2}{5} x^{5/2} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+2 a x^{3/2} \left (A \left (a c+b^2\right )+a b B\right )+\frac{2}{11} c^2 x^{11/2} (A c+3 b B)+\frac{2}{13} B c^3 x^{13/2} \]

[Out]

(-2*a^3*A)/Sqrt[x] + 2*a^2*(3*A*b + a*B)*Sqrt[x] + 2*a*(a*b*B + A*(b^2 + a*c))*x^(3/2) + (2*(3*a*B*(b^2 + a*c)
+ A*(b^3 + 6*a*b*c))*x^(5/2))/5 + (2*(b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^(7/2))/7 + (2*c*(b^2*B + A
*b*c + a*B*c)*x^(9/2))/3 + (2*c^2*(3*b*B + A*c)*x^(11/2))/11 + (2*B*c^3*x^(13/2))/13

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Rubi [A]  time = 0.117435, antiderivative size = 176, 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} \[ 2 a^2 \sqrt{x} (a B+3 A b)-\frac{2 a^3 A}{\sqrt{x}}+\frac{2}{7} x^{7/2} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{2}{3} c x^{9/2} \left (a B c+A b c+b^2 B\right )+\frac{2}{5} x^{5/2} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+2 a x^{3/2} \left (A \left (a c+b^2\right )+a b B\right )+\frac{2}{11} c^2 x^{11/2} (A c+3 b B)+\frac{2}{13} B c^3 x^{13/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^3*A)/Sqrt[x] + 2*a^2*(3*A*b + a*B)*Sqrt[x] + 2*a*(a*b*B + A*(b^2 + a*c))*x^(3/2) + (2*(3*a*B*(b^2 + a*c)
+ A*(b^3 + 6*a*b*c))*x^(5/2))/5 + (2*(b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^(7/2))/7 + (2*c*(b^2*B + A
*b*c + a*B*c)*x^(9/2))/3 + (2*c^2*(3*b*B + A*c)*x^(11/2))/11 + (2*B*c^3*x^(13/2))/13

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

Mathematica [A]  time = 0.200093, size = 173, normalized size = 0.98 \[ \frac{6006 a^2 x (5 A (3 b+c x)+B x (5 b+3 c x))-30030 a^3 (A-B x)+286 a x^2 \left (3 A \left (35 b^2+42 b c x+15 c^2 x^2\right )+B x \left (63 b^2+90 b c x+35 c^2 x^2\right )\right )+2 x^3 \left (13 A \left (495 b^2 c x+231 b^3+385 b c^2 x^2+105 c^3 x^3\right )+5 B x \left (1001 b^2 c x+429 b^3+819 b c^2 x^2+231 c^3 x^3\right )\right )}{15015 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-30030*a^3*(A - B*x) + 6006*a^2*x*(5*A*(3*b + c*x) + B*x*(5*b + 3*c*x)) + 286*a*x^2*(3*A*(35*b^2 + 42*b*c*x +
15*c^2*x^2) + B*x*(63*b^2 + 90*b*c*x + 35*c^2*x^2)) + 2*x^3*(13*A*(231*b^3 + 495*b^2*c*x + 385*b*c^2*x^2 + 10
5*c^3*x^3) + 5*B*x*(429*b^3 + 1001*b^2*c*x + 819*b*c^2*x^2 + 231*c^3*x^3)))/(15015*Sqrt[x])

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Maple [A]  time = 0.006, size = 192, normalized size = 1.1 \begin{align*} -{\frac{-2310\,B{c}^{3}{x}^{7}-2730\,A{c}^{3}{x}^{6}-8190\,B{x}^{6}b{c}^{2}-10010\,A{x}^{5}b{c}^{2}-10010\,aB{c}^{2}{x}^{5}-10010\,B{x}^{5}{b}^{2}c-12870\,aA{c}^{2}{x}^{4}-12870\,A{x}^{4}{b}^{2}c-25740\,B{x}^{4}abc-4290\,B{x}^{4}{b}^{3}-36036\,A{x}^{3}abc-6006\,A{b}^{3}{x}^{3}-18018\,{a}^{2}Bc{x}^{3}-18018\,B{x}^{3}a{b}^{2}-30030\,{a}^{2}Ac{x}^{2}-30030\,A{x}^{2}a{b}^{2}-30030\,B{x}^{2}{a}^{2}b-90090\,A{a}^{2}bx-30030\,{a}^{3}Bx+30030\,A{a}^{3}}{15015}{\frac{1}{\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^(3/2),x)

[Out]

-2/15015*(-1155*B*c^3*x^7-1365*A*c^3*x^6-4095*B*b*c^2*x^6-5005*A*b*c^2*x^5-5005*B*a*c^2*x^5-5005*B*b^2*c*x^5-6
435*A*a*c^2*x^4-6435*A*b^2*c*x^4-12870*B*a*b*c*x^4-2145*B*b^3*x^4-18018*A*a*b*c*x^3-3003*A*b^3*x^3-9009*B*a^2*
c*x^3-9009*B*a*b^2*x^3-15015*A*a^2*c*x^2-15015*A*a*b^2*x^2-15015*B*a^2*b*x^2-45045*A*a^2*b*x-15015*B*a^3*x+150
15*A*a^3)/x^(1/2)

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Maxima [A]  time = 1.12468, size = 224, normalized size = 1.27 \begin{align*} \frac{2}{13} \, B c^{3} x^{\frac{13}{2}} + \frac{2}{11} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac{11}{2}} + \frac{2}{3} \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{\frac{9}{2}} + \frac{2}{7} \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{\frac{7}{2}} - \frac{2 \, A a^{3}}{\sqrt{x}} + \frac{2}{5} \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{\frac{5}{2}} + 2 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{\frac{3}{2}} + 2 \,{\left (B a^{3} + 3 \, A a^{2} b\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^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.0403, size = 405, normalized size = 2.3 \begin{align*} \frac{2 \,{\left (1155 \, B c^{3} x^{7} + 1365 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 5005 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{5} + 2145 \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} - 15015 \, A a^{3} + 3003 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + 15015 \,{\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 )}}{15015 \, \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^(3/2),x, algorithm="fricas")

[Out]

2/15015*(1155*B*c^3*x^7 + 1365*(3*B*b*c^2 + A*c^3)*x^6 + 5005*(B*b^2*c + (B*a + A*b)*c^2)*x^5 + 2145*(B*b^3 +
3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^4 - 15015*A*a^3 + 3003*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^3 +
15015*(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 = 7.19779, size = 284, normalized size = 1.61 \begin{align*} - \frac{2 A a^{3}}{\sqrt{x}} + 6 A a^{2} b \sqrt{x} + 2 A a^{2} c x^{\frac{3}{2}} + 2 A a b^{2} x^{\frac{3}{2}} + \frac{12 A a b c x^{\frac{5}{2}}}{5} + \frac{6 A a c^{2} x^{\frac{7}{2}}}{7} + \frac{2 A b^{3} x^{\frac{5}{2}}}{5} + \frac{6 A b^{2} c x^{\frac{7}{2}}}{7} + \frac{2 A b c^{2} x^{\frac{9}{2}}}{3} + \frac{2 A c^{3} x^{\frac{11}{2}}}{11} + 2 B a^{3} \sqrt{x} + 2 B a^{2} b x^{\frac{3}{2}} + \frac{6 B a^{2} c x^{\frac{5}{2}}}{5} + \frac{6 B a b^{2} x^{\frac{5}{2}}}{5} + \frac{12 B a b c x^{\frac{7}{2}}}{7} + \frac{2 B a c^{2} x^{\frac{9}{2}}}{3} + \frac{2 B b^{3} x^{\frac{7}{2}}}{7} + \frac{2 B b^{2} c x^{\frac{9}{2}}}{3} + \frac{6 B b c^{2} x^{\frac{11}{2}}}{11} + \frac{2 B c^{3} x^{\frac{13}{2}}}{13} \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**(3/2),x)

[Out]

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

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Giac [A]  time = 1.28029, size = 261, normalized size = 1.48 \begin{align*} \frac{2}{13} \, B c^{3} x^{\frac{13}{2}} + \frac{6}{11} \, B b c^{2} x^{\frac{11}{2}} + \frac{2}{11} \, A c^{3} x^{\frac{11}{2}} + \frac{2}{3} \, B b^{2} c x^{\frac{9}{2}} + \frac{2}{3} \, B a c^{2} x^{\frac{9}{2}} + \frac{2}{3} \, A b c^{2} x^{\frac{9}{2}} + \frac{2}{7} \, B b^{3} x^{\frac{7}{2}} + \frac{12}{7} \, B a b c x^{\frac{7}{2}} + \frac{6}{7} \, A b^{2} c x^{\frac{7}{2}} + \frac{6}{7} \, A a c^{2} x^{\frac{7}{2}} + \frac{6}{5} \, B a b^{2} x^{\frac{5}{2}} + \frac{2}{5} \, A b^{3} x^{\frac{5}{2}} + \frac{6}{5} \, B a^{2} c x^{\frac{5}{2}} + \frac{12}{5} \, A a b c x^{\frac{5}{2}} + 2 \, B a^{2} b x^{\frac{3}{2}} + 2 \, A a b^{2} x^{\frac{3}{2}} + 2 \, A a^{2} c x^{\frac{3}{2}} + 2 \, B a^{3} \sqrt{x} + 6 \, A a^{2} b \sqrt{x} - \frac{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^(3/2),x, algorithm="giac")

[Out]

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