3.1006 \(\int \frac{(A+B x) \left (a+b x+c x^2\right )^3}{x^{7/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.277916, 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 \[ -\frac{2 a^3 A}{5 x^{5/2}}-\frac{2 a^2 (a B+3 A b)}{3 x^{3/2}}+\frac{6}{5} c x^{5/2} \left (a B c+A b c+b^2 B\right )-\frac{6 a \left (A \left (a c+b^2\right )+a b B\right )}{\sqrt{x}}+\frac{2}{3} x^{3/2} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+2 \sqrt{x} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{2}{7} c^2 x^{7/2} (A c+3 b B)+\frac{2}{9} B c^3 x^{9/2} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a + b*x + c*x^2)^3)/x^(7/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 39.019, size = 190, normalized size = 1.07 \[ - \frac{2 A a^{3}}{5 x^{\frac{5}{2}}} + \frac{2 B c^{3} x^{\frac{9}{2}}}{9} - \frac{2 a^{2} \left (A b + \frac{B a}{3}\right )}{x^{\frac{3}{2}}} - \frac{6 a \left (A a c + A b^{2} + B a b\right )}{\sqrt{x}} + \frac{2 c^{2} x^{\frac{7}{2}} \left (A c + 3 B b\right )}{7} + \frac{6 c x^{\frac{5}{2}} \left (A b c + B a c + B b^{2}\right )}{5} + x^{\frac{3}{2}} \left (2 A a c^{2} + 2 A b^{2} c + 4 B a b c + \frac{2 B b^{3}}{3}\right ) + \sqrt{x} \left (12 A a b c + 2 A b^{3} + 6 B a^{2} c + 6 B a b^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x+a)**3/x**(7/2),x)

[Out]

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

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Mathematica [A]  time = 0.125615, size = 169, normalized size = 0.95 \[ \frac{2 \left (-21 a^3 (3 A+5 B x)-315 a^2 x (A (b+3 c x)+3 B x (b-c x))+63 a x^2 \left (5 A \left (-3 b^2+6 b c x+c^2 x^2\right )+B x \left (15 b^2+10 b c x+3 c^2 x^2\right )\right )+x^3 \left (9 A \left (35 b^3+35 b^2 c x+21 b c^2 x^2+5 c^3 x^3\right )+B x \left (105 b^3+189 b^2 c x+135 b c^2 x^2+35 c^3 x^3\right )\right )\right )}{315 x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a + b*x + c*x^2)^3)/x^(7/2),x]

[Out]

(2*(-21*a^3*(3*A + 5*B*x) - 315*a^2*x*(3*B*x*(b - c*x) + A*(b + 3*c*x)) + 63*a*x
^2*(5*A*(-3*b^2 + 6*b*c*x + c^2*x^2) + B*x*(15*b^2 + 10*b*c*x + 3*c^2*x^2)) + x^
3*(9*A*(35*b^3 + 35*b^2*c*x + 21*b*c^2*x^2 + 5*c^3*x^3) + B*x*(105*b^3 + 189*b^2
*c*x + 135*b*c^2*x^2 + 35*c^3*x^3))))/(315*x^(5/2))

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Maple [A]  time = 0.01, size = 192, normalized size = 1.1 \[ -{\frac{-70\,B{c}^{3}{x}^{7}-90\,A{c}^{3}{x}^{6}-270\,B{x}^{6}b{c}^{2}-378\,A{x}^{5}b{c}^{2}-378\,aB{c}^{2}{x}^{5}-378\,B{x}^{5}{b}^{2}c-630\,aA{c}^{2}{x}^{4}-630\,A{x}^{4}{b}^{2}c-1260\,B{x}^{4}abc-210\,B{x}^{4}{b}^{3}-3780\,A{x}^{3}abc-630\,A{b}^{3}{x}^{3}-1890\,{a}^{2}Bc{x}^{3}-1890\,B{x}^{3}a{b}^{2}+1890\,{a}^{2}Ac{x}^{2}+1890\,A{x}^{2}a{b}^{2}+1890\,B{x}^{2}{a}^{2}b+630\,A{a}^{2}bx+210\,{a}^{3}Bx+126\,A{a}^{3}}{315}{x}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x+a)^3/x^(7/2),x)

[Out]

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

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Maxima [A]  time = 0.725859, size = 225, normalized size = 1.26 \[ \frac{2}{9} \, B c^{3} x^{\frac{9}{2}} + \frac{2}{7} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac{7}{2}} + \frac{6}{5} \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{\frac{5}{2}} + \frac{2}{3} \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{\frac{3}{2}} + 2 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} \sqrt{x} - \frac{2 \,{\left (3 \, A a^{3} + 45 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 5 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )}}{15 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^3*(B*x + A)/x^(7/2),x, algorithm="maxima")

[Out]

2/9*B*c^3*x^(9/2) + 2/7*(3*B*b*c^2 + A*c^3)*x^(7/2) + 6/5*(B*b^2*c + (B*a + A*b)
*c^2)*x^(5/2) + 2/3*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^(3/2) + 2*(3*B
*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*sqrt(x) - 2/15*(3*A*a^3 + 45*(B*a^2*b +
A*a*b^2 + A*a^2*c)*x^2 + 5*(B*a^3 + 3*A*a^2*b)*x)/x^(5/2)

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Fricas [A]  time = 0.279724, size = 224, normalized size = 1.26 \[ \frac{2 \,{\left (35 \, B c^{3} x^{7} + 45 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 189 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{5} + 105 \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} - 63 \, A a^{3} + 315 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} - 945 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} - 105 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )}}{315 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^3*(B*x + A)/x^(7/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 22.1244, size = 275, normalized size = 1.54 \[ - \frac{2 A a^{3}}{5 x^{\frac{5}{2}}} - \frac{2 A a^{2} b}{x^{\frac{3}{2}}} - \frac{6 A a^{2} c}{\sqrt{x}} - \frac{6 A a b^{2}}{\sqrt{x}} + 12 A a b c \sqrt{x} + 2 A a c^{2} x^{\frac{3}{2}} + 2 A b^{3} \sqrt{x} + 2 A b^{2} c x^{\frac{3}{2}} + \frac{6 A b c^{2} x^{\frac{5}{2}}}{5} + \frac{2 A c^{3} x^{\frac{7}{2}}}{7} - \frac{2 B a^{3}}{3 x^{\frac{3}{2}}} - \frac{6 B a^{2} b}{\sqrt{x}} + 6 B a^{2} c \sqrt{x} + 6 B a b^{2} \sqrt{x} + 4 B a b c x^{\frac{3}{2}} + \frac{6 B a c^{2} x^{\frac{5}{2}}}{5} + \frac{2 B b^{3} x^{\frac{3}{2}}}{3} + \frac{6 B b^{2} c x^{\frac{5}{2}}}{5} + \frac{6 B b c^{2} x^{\frac{7}{2}}}{7} + \frac{2 B c^{3} x^{\frac{9}{2}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x+a)**3/x**(7/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.271993, size = 259, normalized size = 1.46 \[ \frac{2}{9} \, B c^{3} x^{\frac{9}{2}} + \frac{6}{7} \, B b c^{2} x^{\frac{7}{2}} + \frac{2}{7} \, A c^{3} x^{\frac{7}{2}} + \frac{6}{5} \, B b^{2} c x^{\frac{5}{2}} + \frac{6}{5} \, B a c^{2} x^{\frac{5}{2}} + \frac{6}{5} \, A b c^{2} x^{\frac{5}{2}} + \frac{2}{3} \, B b^{3} x^{\frac{3}{2}} + 4 \, B a b c x^{\frac{3}{2}} + 2 \, A b^{2} c x^{\frac{3}{2}} + 2 \, A a c^{2} x^{\frac{3}{2}} + 6 \, B a b^{2} \sqrt{x} + 2 \, A b^{3} \sqrt{x} + 6 \, B a^{2} c \sqrt{x} + 12 \, A a b c \sqrt{x} - \frac{2 \,{\left (45 \, B a^{2} b x^{2} + 45 \, A a b^{2} x^{2} + 45 \, A a^{2} c x^{2} + 5 \, B a^{3} x + 15 \, A a^{2} b x + 3 \, A a^{3}\right )}}{15 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^3*(B*x + A)/x^(7/2),x, algorithm="giac")

[Out]

2/9*B*c^3*x^(9/2) + 6/7*B*b*c^2*x^(7/2) + 2/7*A*c^3*x^(7/2) + 6/5*B*b^2*c*x^(5/2
) + 6/5*B*a*c^2*x^(5/2) + 6/5*A*b*c^2*x^(5/2) + 2/3*B*b^3*x^(3/2) + 4*B*a*b*c*x^
(3/2) + 2*A*b^2*c*x^(3/2) + 2*A*a*c^2*x^(3/2) + 6*B*a*b^2*sqrt(x) + 2*A*b^3*sqrt
(x) + 6*B*a^2*c*sqrt(x) + 12*A*a*b*c*sqrt(x) - 2/15*(45*B*a^2*b*x^2 + 45*A*a*b^2
*x^2 + 45*A*a^2*c*x^2 + 5*B*a^3*x + 15*A*a^2*b*x + 3*A*a^3)/x^(5/2)