3.10.27 \(\int \sqrt {x} (A+B x) (a+b x+c x^2)^3 \, dx\)

Optimal. Leaf size=182 \[ \frac {2}{3} a^3 A x^{3/2}+\frac {2}{5} a^2 x^{5/2} (a B+3 A b)+\frac {6}{13} c x^{13/2} \left (a B c+A b c+b^2 B\right )+\frac {6}{7} a x^{7/2} \left (A \left (a c+b^2\right )+a b B\right )+\frac {2}{11} x^{11/2} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac {2}{9} x^{9/2} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac {2}{15} c^2 x^{15/2} (A c+3 b B)+\frac {2}{17} B c^3 x^{17/2} \]

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Rubi [A]  time = 0.11, antiderivative size = 182, 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*} \frac {2}{5} a^2 x^{5/2} (a B+3 A b)+\frac {2}{3} a^3 A x^{3/2}+\frac {2}{11} x^{11/2} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac {6}{13} c x^{13/2} \left (a B c+A b c+b^2 B\right )+\frac {2}{9} x^{9/2} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac {6}{7} a x^{7/2} \left (A \left (a c+b^2\right )+a b B\right )+\frac {2}{15} c^2 x^{15/2} (A c+3 b B)+\frac {2}{17} B c^3 x^{17/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^3*A*x^(3/2))/3 + (2*a^2*(3*A*b + a*B)*x^(5/2))/5 + (6*a*(a*b*B + A*(b^2 + a*c))*x^(7/2))/7 + (2*(3*a*B*(b
^2 + a*c) + A*(b^3 + 6*a*b*c))*x^(9/2))/9 + (2*(b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^(11/2))/11 + (6*c
*(b^2*B + A*b*c + a*B*c)*x^(13/2))/13 + (2*c^2*(3*b*B + A*c)*x^(15/2))/15 + (2*B*c^3*x^(17/2))/17

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 \sqrt {x} (A+B x) \left (a+b x+c x^2\right )^3 \, dx &=\int \left (a^3 A \sqrt {x}+a^2 (3 A b+a B) x^{3/2}+3 a \left (a b B+A \left (b^2+a c\right )\right ) x^{5/2}+\left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^{7/2}+\left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^{9/2}+3 c \left (b^2 B+A b c+a B c\right ) x^{11/2}+c^2 (3 b B+A c) x^{13/2}+B c^3 x^{15/2}\right ) \, dx\\ &=\frac {2}{3} a^3 A x^{3/2}+\frac {2}{5} a^2 (3 A b+a B) x^{5/2}+\frac {6}{7} a \left (a b B+A \left (b^2+a c\right )\right ) x^{7/2}+\frac {2}{9} \left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^{9/2}+\frac {2}{11} \left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^{11/2}+\frac {6}{13} c \left (b^2 B+A b c+a B c\right ) x^{13/2}+\frac {2}{15} c^2 (3 b B+A c) x^{15/2}+\frac {2}{17} B c^3 x^{17/2}\\ \end {align*}

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Mathematica [A]  time = 0.20, size = 178, normalized size = 0.98 \begin {gather*} \frac {2 x^{3/2} \left (51051 a^3 (5 A+3 B x)+7293 a^2 x (9 A (7 b+5 c x)+5 B x (9 b+7 c x))+255 a x^2 \left (13 A \left (99 b^2+154 b c x+63 c^2 x^2\right )+7 B x \left (143 b^2+234 b c x+99 c^2 x^2\right )\right )+7 x^3 \left (17 A \left (715 b^3+1755 b^2 c x+1485 b c^2 x^2+429 c^3 x^3\right )+9 B x \left (1105 b^3+2805 b^2 c x+2431 b c^2 x^2+715 c^3 x^3\right )\right )\right )}{765765} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(3/2)*(51051*a^3*(5*A + 3*B*x) + 7293*a^2*x*(9*A*(7*b + 5*c*x) + 5*B*x*(9*b + 7*c*x)) + 255*a*x^2*(13*A*(
99*b^2 + 154*b*c*x + 63*c^2*x^2) + 7*B*x*(143*b^2 + 234*b*c*x + 99*c^2*x^2)) + 7*x^3*(17*A*(715*b^3 + 1755*b^2
*c*x + 1485*b*c^2*x^2 + 429*c^3*x^3) + 9*B*x*(1105*b^3 + 2805*b^2*c*x + 2431*b*c^2*x^2 + 715*c^3*x^3))))/76576
5

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IntegrateAlgebraic [A]  time = 0.12, size = 237, normalized size = 1.30 \begin {gather*} \frac {2 \left (255255 a^3 A x^{3/2}+153153 a^3 B x^{5/2}+459459 a^2 A b x^{5/2}+328185 a^2 A c x^{7/2}+328185 a^2 b B x^{7/2}+255255 a^2 B c x^{9/2}+328185 a A b^2 x^{7/2}+510510 a A b c x^{9/2}+208845 a A c^2 x^{11/2}+255255 a b^2 B x^{9/2}+417690 a b B c x^{11/2}+176715 a B c^2 x^{13/2}+85085 A b^3 x^{9/2}+208845 A b^2 c x^{11/2}+176715 A b c^2 x^{13/2}+51051 A c^3 x^{15/2}+69615 b^3 B x^{11/2}+176715 b^2 B c x^{13/2}+153153 b B c^2 x^{15/2}+45045 B c^3 x^{17/2}\right )}{765765} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(255255*a^3*A*x^(3/2) + 459459*a^2*A*b*x^(5/2) + 153153*a^3*B*x^(5/2) + 328185*a*A*b^2*x^(7/2) + 328185*a^2
*b*B*x^(7/2) + 328185*a^2*A*c*x^(7/2) + 85085*A*b^3*x^(9/2) + 255255*a*b^2*B*x^(9/2) + 510510*a*A*b*c*x^(9/2)
+ 255255*a^2*B*c*x^(9/2) + 69615*b^3*B*x^(11/2) + 208845*A*b^2*c*x^(11/2) + 417690*a*b*B*c*x^(11/2) + 208845*a
*A*c^2*x^(11/2) + 176715*b^2*B*c*x^(13/2) + 176715*A*b*c^2*x^(13/2) + 176715*a*B*c^2*x^(13/2) + 153153*b*B*c^2
*x^(15/2) + 51051*A*c^3*x^(15/2) + 45045*B*c^3*x^(17/2)))/765765

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fricas [A]  time = 0.43, size = 169, normalized size = 0.93 \begin {gather*} \frac {2}{765765} \, {\left (45045 \, B c^{3} x^{8} + 51051 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{7} + 176715 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{6} + 69615 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{5} + 255255 \, A a^{3} x + 85085 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{4} + 328185 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{3} + 153153 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2}\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^(1/2),x, algorithm="fricas")

[Out]

2/765765*(45045*B*c^3*x^8 + 51051*(3*B*b*c^2 + A*c^3)*x^7 + 176715*(B*b^2*c + (B*a + A*b)*c^2)*x^6 + 69615*(B*
b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^5 + 255255*A*a^3*x + 85085*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)
*c)*x^4 + 328185*(B*a^2*b + A*a*b^2 + A*a^2*c)*x^3 + 153153*(B*a^3 + 3*A*a^2*b)*x^2)*sqrt(x)

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giac [A]  time = 0.16, size = 193, normalized size = 1.06 \begin {gather*} \frac {2}{17} \, B c^{3} x^{\frac {17}{2}} + \frac {2}{5} \, B b c^{2} x^{\frac {15}{2}} + \frac {2}{15} \, A c^{3} x^{\frac {15}{2}} + \frac {6}{13} \, B b^{2} c x^{\frac {13}{2}} + \frac {6}{13} \, B a c^{2} x^{\frac {13}{2}} + \frac {6}{13} \, A b c^{2} x^{\frac {13}{2}} + \frac {2}{11} \, B b^{3} x^{\frac {11}{2}} + \frac {12}{11} \, B a b c x^{\frac {11}{2}} + \frac {6}{11} \, A b^{2} c x^{\frac {11}{2}} + \frac {6}{11} \, A a c^{2} x^{\frac {11}{2}} + \frac {2}{3} \, B a b^{2} x^{\frac {9}{2}} + \frac {2}{9} \, A b^{3} x^{\frac {9}{2}} + \frac {2}{3} \, B a^{2} c x^{\frac {9}{2}} + \frac {4}{3} \, A a b c x^{\frac {9}{2}} + \frac {6}{7} \, B a^{2} b x^{\frac {7}{2}} + \frac {6}{7} \, A a b^{2} x^{\frac {7}{2}} + \frac {6}{7} \, A a^{2} c x^{\frac {7}{2}} + \frac {2}{5} \, B a^{3} x^{\frac {5}{2}} + \frac {6}{5} \, A a^{2} b x^{\frac {5}{2}} + \frac {2}{3} \, A a^{3} x^{\frac {3}{2}} \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^(1/2),x, algorithm="giac")

[Out]

2/17*B*c^3*x^(17/2) + 2/5*B*b*c^2*x^(15/2) + 2/15*A*c^3*x^(15/2) + 6/13*B*b^2*c*x^(13/2) + 6/13*B*a*c^2*x^(13/
2) + 6/13*A*b*c^2*x^(13/2) + 2/11*B*b^3*x^(11/2) + 12/11*B*a*b*c*x^(11/2) + 6/11*A*b^2*c*x^(11/2) + 6/11*A*a*c
^2*x^(11/2) + 2/3*B*a*b^2*x^(9/2) + 2/9*A*b^3*x^(9/2) + 2/3*B*a^2*c*x^(9/2) + 4/3*A*a*b*c*x^(9/2) + 6/7*B*a^2*
b*x^(7/2) + 6/7*A*a*b^2*x^(7/2) + 6/7*A*a^2*c*x^(7/2) + 2/5*B*a^3*x^(5/2) + 6/5*A*a^2*b*x^(5/2) + 2/3*A*a^3*x^
(3/2)

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maple [A]  time = 0.05, size = 192, normalized size = 1.05 \begin {gather*} \frac {2 \left (45045 B \,c^{3} x^{7}+51051 A \,c^{3} x^{6}+153153 x^{6} B b \,c^{2}+176715 x^{5} A b \,c^{2}+176715 B a \,c^{2} x^{5}+176715 x^{5} B \,b^{2} c +208845 A a \,c^{2} x^{4}+208845 x^{4} A \,b^{2} c +417690 x^{4} a b B c +69615 x^{4} b^{3} B +510510 x^{3} A a b c +85085 A \,b^{3} x^{3}+255255 B \,a^{2} c \,x^{3}+255255 x^{3} B a \,b^{2}+328185 A \,a^{2} c \,x^{2}+328185 x^{2} A a \,b^{2}+328185 B \,a^{2} b \,x^{2}+459459 x A \,a^{2} b +153153 B \,a^{3} x +255255 A \,a^{3}\right ) x^{\frac {3}{2}}}{765765} \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^(1/2),x)

[Out]

2/765765*x^(3/2)*(45045*B*c^3*x^7+51051*A*c^3*x^6+153153*B*b*c^2*x^6+176715*A*b*c^2*x^5+176715*B*a*c^2*x^5+176
715*B*b^2*c*x^5+208845*A*a*c^2*x^4+208845*A*b^2*c*x^4+417690*B*a*b*c*x^4+69615*B*b^3*x^4+510510*A*a*b*c*x^3+85
085*A*b^3*x^3+255255*B*a^2*c*x^3+255255*B*a*b^2*x^3+328185*A*a^2*c*x^2+328185*A*a*b^2*x^2+328185*B*a^2*b*x^2+4
59459*A*a^2*b*x+153153*B*a^3*x+255255*A*a^3)

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maxima [A]  time = 0.53, size = 166, normalized size = 0.91 \begin {gather*} \frac {2}{17} \, B c^{3} x^{\frac {17}{2}} + \frac {2}{15} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac {15}{2}} + \frac {6}{13} \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{\frac {13}{2}} + \frac {2}{11} \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{\frac {11}{2}} + \frac {2}{3} \, A a^{3} x^{\frac {3}{2}} + \frac {2}{9} \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{\frac {9}{2}} + \frac {6}{7} \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{\frac {7}{2}} + \frac {2}{5} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac {5}{2}} \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^(1/2),x, algorithm="maxima")

[Out]

2/17*B*c^3*x^(17/2) + 2/15*(3*B*b*c^2 + A*c^3)*x^(15/2) + 6/13*(B*b^2*c + (B*a + A*b)*c^2)*x^(13/2) + 2/11*(B*
b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^(11/2) + 2/3*A*a^3*x^(3/2) + 2/9*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*
A*a*b)*c)*x^(9/2) + 6/7*(B*a^2*b + A*a*b^2 + A*a^2*c)*x^(7/2) + 2/5*(B*a^3 + 3*A*a^2*b)*x^(5/2)

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mupad [B]  time = 0.05, size = 169, normalized size = 0.93 \begin {gather*} x^{9/2}\,\left (\frac {2\,B\,c\,a^2}{3}+\frac {2\,B\,a\,b^2}{3}+\frac {4\,A\,c\,a\,b}{3}+\frac {2\,A\,b^3}{9}\right )+x^{11/2}\,\left (\frac {2\,B\,b^3}{11}+\frac {6\,A\,b^2\,c}{11}+\frac {12\,B\,a\,b\,c}{11}+\frac {6\,A\,a\,c^2}{11}\right )+x^{5/2}\,\left (\frac {2\,B\,a^3}{5}+\frac {6\,A\,b\,a^2}{5}\right )+x^{15/2}\,\left (\frac {2\,A\,c^3}{15}+\frac {2\,B\,b\,c^2}{5}\right )+x^{7/2}\,\left (\frac {6\,B\,a^2\,b}{7}+\frac {6\,A\,c\,a^2}{7}+\frac {6\,A\,a\,b^2}{7}\right )+x^{13/2}\,\left (\frac {6\,B\,b^2\,c}{13}+\frac {6\,A\,b\,c^2}{13}+\frac {6\,B\,a\,c^2}{13}\right )+\frac {2\,A\,a^3\,x^{3/2}}{3}+\frac {2\,B\,c^3\,x^{17/2}}{17} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)*(A + B*x)*(a + b*x + c*x^2)^3,x)

[Out]

x^(9/2)*((2*A*b^3)/9 + (2*B*a*b^2)/3 + (2*B*a^2*c)/3 + (4*A*a*b*c)/3) + x^(11/2)*((2*B*b^3)/11 + (6*A*a*c^2)/1
1 + (6*A*b^2*c)/11 + (12*B*a*b*c)/11) + x^(5/2)*((2*B*a^3)/5 + (6*A*a^2*b)/5) + x^(15/2)*((2*A*c^3)/15 + (2*B*
b*c^2)/5) + x^(7/2)*((6*A*a*b^2)/7 + (6*A*a^2*c)/7 + (6*B*a^2*b)/7) + x^(13/2)*((6*A*b*c^2)/13 + (6*B*a*c^2)/1
3 + (6*B*b^2*c)/13) + (2*A*a^3*x^(3/2))/3 + (2*B*c^3*x^(17/2))/17

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sympy [A]  time = 9.17, size = 216, normalized size = 1.19 \begin {gather*} \frac {2 A a^{3} x^{\frac {3}{2}}}{3} + \frac {2 B c^{3} x^{\frac {17}{2}}}{17} + \frac {2 x^{\frac {15}{2}} \left (A c^{3} + 3 B b c^{2}\right )}{15} + \frac {2 x^{\frac {13}{2}} \left (3 A b c^{2} + 3 B a c^{2} + 3 B b^{2} c\right )}{13} + \frac {2 x^{\frac {11}{2}} \left (3 A a c^{2} + 3 A b^{2} c + 6 B a b c + B b^{3}\right )}{11} + \frac {2 x^{\frac {9}{2}} \left (6 A a b c + A b^{3} + 3 B a^{2} c + 3 B a b^{2}\right )}{9} + \frac {2 x^{\frac {7}{2}} \left (3 A a^{2} c + 3 A a b^{2} + 3 B a^{2} b\right )}{7} + \frac {2 x^{\frac {5}{2}} \left (3 A a^{2} b + B a^{3}\right )}{5} \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**(1/2),x)

[Out]

2*A*a**3*x**(3/2)/3 + 2*B*c**3*x**(17/2)/17 + 2*x**(15/2)*(A*c**3 + 3*B*b*c**2)/15 + 2*x**(13/2)*(3*A*b*c**2 +
3*B*a*c**2 + 3*B*b**2*c)/13 + 2*x**(11/2)*(3*A*a*c**2 + 3*A*b**2*c + 6*B*a*b*c + B*b**3)/11 + 2*x**(9/2)*(6*A
*a*b*c + A*b**3 + 3*B*a**2*c + 3*B*a*b**2)/9 + 2*x**(7/2)*(3*A*a**2*c + 3*A*a*b**2 + 3*B*a**2*b)/7 + 2*x**(5/2
)*(3*A*a**2*b + B*a**3)/5

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