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

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

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Rubi [A]  time = 0.12, antiderivative size = 176, normalized size of antiderivative = 1.00, 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} \begin {gather*} 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} \end {gather*}

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*}

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Mathematica [A]  time = 0.19, size = 173, normalized size = 0.98 \begin {gather*} \frac {-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 \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 (231 b^3+495 b^2 c x+385 b c^2 x^2+105 c^3 x^3\right )+5 B x \left (429 b^3+1001 b^2 c x+819 b c^2 x^2+231 c^3 x^3\right )\right )}{15015 \sqrt {x}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.16, size = 195, normalized size = 1.11 \begin {gather*} \frac {2 \left (-15015 a^3 A+15015 a^3 B x+45045 a^2 A b x+15015 a^2 A c x^2+15015 a^2 b B x^2+9009 a^2 B c x^3+15015 a A b^2 x^2+18018 a A b c x^3+6435 a A c^2 x^4+9009 a b^2 B x^3+12870 a b B c x^4+5005 a B c^2 x^5+3003 A b^3 x^3+6435 A b^2 c x^4+5005 A b c^2 x^5+1365 A c^3 x^6+2145 b^3 B x^4+5005 b^2 B c x^5+4095 b B c^2 x^6+1155 B c^3 x^7\right )}{15015 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.43, size = 166, normalized size = 0.94 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.17, size = 193, normalized size = 1.10 \begin {gather*} \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 {gather*}

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)

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maple [A]  time = 0.05, size = 192, normalized size = 1.09 \begin {gather*} -\frac {2 \left (-1155 B \,c^{3} x^{7}-1365 A \,c^{3} x^{6}-4095 x^{6} B b \,c^{2}-5005 x^{5} A b \,c^{2}-5005 B a \,c^{2} x^{5}-5005 x^{5} B \,b^{2} c -6435 A a \,c^{2} x^{4}-6435 x^{4} A \,b^{2} c -12870 x^{4} a b B c -2145 x^{4} b^{3} B -18018 x^{3} A a b c -3003 A \,b^{3} x^{3}-9009 B \,a^{2} c \,x^{3}-9009 x^{3} B a \,b^{2}-15015 A \,a^{2} c \,x^{2}-15015 x^{2} A a \,b^{2}-15015 B \,a^{2} b \,x^{2}-45045 x A \,a^{2} b -15015 B \,a^{3} x +15015 A \,a^{3}\right )}{15015 \sqrt {x}} \end {gather*}

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 = 0.59, size = 166, normalized size = 0.94 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 0.06, size = 169, normalized size = 0.96 \begin {gather*} x^{5/2}\,\left (\frac {6\,B\,c\,a^2}{5}+\frac {6\,B\,a\,b^2}{5}+\frac {12\,A\,c\,a\,b}{5}+\frac {2\,A\,b^3}{5}\right )+x^{7/2}\,\left (\frac {2\,B\,b^3}{7}+\frac {6\,A\,b^2\,c}{7}+\frac {12\,B\,a\,b\,c}{7}+\frac {6\,A\,a\,c^2}{7}\right )+\sqrt {x}\,\left (2\,B\,a^3+6\,A\,b\,a^2\right )+x^{11/2}\,\left (\frac {2\,A\,c^3}{11}+\frac {6\,B\,b\,c^2}{11}\right )+x^{3/2}\,\left (2\,B\,a^2\,b+2\,A\,c\,a^2+2\,A\,a\,b^2\right )+x^{9/2}\,\left (\frac {2\,B\,b^2\,c}{3}+\frac {2\,A\,b\,c^2}{3}+\frac {2\,B\,a\,c^2}{3}\right )-\frac {2\,A\,a^3}{\sqrt {x}}+\frac {2\,B\,c^3\,x^{13/2}}{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 5.81, size = 284, normalized size = 1.61 \begin {gather*} - \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 {gather*}

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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