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

Optimal. Leaf size=182 \[ \frac {2}{9} a^3 A x^{9/2}+\frac {2}{11} a^2 x^{11/2} (a B+3 A b)+\frac {6}{19} c x^{19/2} \left (a B c+A b c+b^2 B\right )+\frac {6}{13} a x^{13/2} \left (A \left (a c+b^2\right )+a b B\right )+\frac {2}{17} x^{17/2} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac {2}{15} x^{15/2} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac {2}{21} c^2 x^{21/2} (A c+3 b B)+\frac {2}{23} B c^3 x^{23/2} \]

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Rubi [A]  time = 0.12, 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}{11} a^2 x^{11/2} (a B+3 A b)+\frac {2}{9} a^3 A x^{9/2}+\frac {2}{17} x^{17/2} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac {6}{19} c x^{19/2} \left (a B c+A b c+b^2 B\right )+\frac {2}{15} x^{15/2} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac {6}{13} a x^{13/2} \left (A \left (a c+b^2\right )+a b B\right )+\frac {2}{21} c^2 x^{21/2} (A c+3 b B)+\frac {2}{23} B c^3 x^{23/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^3*A*x^(9/2))/9 + (2*a^2*(3*A*b + a*B)*x^(11/2))/11 + (6*a*(a*b*B + A*(b^2 + a*c))*x^(13/2))/13 + (2*(3*a*
B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))*x^(15/2))/15 + (2*(b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^(17/2))/17
+ (6*c*(b^2*B + A*b*c + a*B*c)*x^(19/2))/19 + (2*c^2*(3*b*B + A*c)*x^(21/2))/21 + (2*B*c^3*x^(23/2))/23

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 x^{7/2} (A+B x) \left (a+b x+c x^2\right )^3 \, dx &=\int \left (a^3 A x^{7/2}+a^2 (3 A b+a B) x^{9/2}+3 a \left (a b B+A \left (b^2+a c\right )\right ) x^{11/2}+\left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^{13/2}+\left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^{15/2}+3 c \left (b^2 B+A b c+a B c\right ) x^{17/2}+c^2 (3 b B+A c) x^{19/2}+B c^3 x^{21/2}\right ) \, dx\\ &=\frac {2}{9} a^3 A x^{9/2}+\frac {2}{11} a^2 (3 A b+a B) x^{11/2}+\frac {6}{13} a \left (a b B+A \left (b^2+a c\right )\right ) x^{13/2}+\frac {2}{15} \left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^{15/2}+\frac {2}{17} \left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^{17/2}+\frac {6}{19} c \left (b^2 B+A b c+a B c\right ) x^{19/2}+\frac {2}{21} c^2 (3 b B+A c) x^{21/2}+\frac {2}{23} B c^3 x^{23/2}\\ \end {align*}

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Mathematica [A]  time = 0.20, size = 178, normalized size = 0.98 \begin {gather*} \frac {2 x^{9/2} \left (3380195 a^3 (11 A+9 B x)+468027 a^2 x (15 A (13 b+11 c x)+11 B x (15 b+13 c x))+15939 a x^2 \left (19 A \left (255 b^2+442 b c x+195 c^2 x^2\right )+13 B x \left (323 b^2+570 b c x+255 c^2 x^2\right )\right )+429 x^3 \left (23 A \left (2261 b^3+5985 b^2 c x+5355 b c^2 x^2+1615 c^3 x^3\right )+15 B x \left (3059 b^3+8211 b^2 c x+7429 b c^2 x^2+2261 c^3 x^3\right )\right )\right )}{334639305} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(9/2)*(3380195*a^3*(11*A + 9*B*x) + 468027*a^2*x*(15*A*(13*b + 11*c*x) + 11*B*x*(15*b + 13*c*x)) + 15939*
a*x^2*(19*A*(255*b^2 + 442*b*c*x + 195*c^2*x^2) + 13*B*x*(323*b^2 + 570*b*c*x + 255*c^2*x^2)) + 429*x^3*(23*A*
(2261*b^3 + 5985*b^2*c*x + 5355*b*c^2*x^2 + 1615*c^3*x^3) + 15*B*x*(3059*b^3 + 8211*b^2*c*x + 7429*b*c^2*x^2 +
2261*c^3*x^3))))/334639305

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IntegrateAlgebraic [A]  time = 0.12, size = 237, normalized size = 1.30 \begin {gather*} \frac {2 \left (37182145 a^3 A x^{9/2}[phone] a^3 B x^{11/2}[phone] a^2 A b x^{11/2}[phone] a^2 A c x^{13/2}[phone] a^2 b B x^{13/2}[phone] a^2 B c x^{15/2}[phone] a A b^2 x^{13/2}[phone] a A b c x^{15/2}[phone] a A c^2 x^{17/2}[phone] a b^2 B x^{15/2}[phone] a b B c x^{17/2}[phone] a B c^2 x^{19/2}[phone] A b^3 x^{15/2}[phone] A b^2 c x^{17/2}[phone] A b c^2 x^{19/2}[phone] A c^3 x^{21/2}[phone] b^3 B x^{17/2}[phone] b^2 B c x^{19/2}[phone] b B c^2 x^{21/2}[phone] B c^3 x^{23/2}\right )}{334639305} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(37182145*a^3*A*x^(9/2) + 91265265*a^2*A*b*x^(11/2) + 30421755*a^3*B*x^(11/2) + 77224455*a*A*b^2*x^(13/2) +
77224455*a^2*b*B*x^(13/2) + 77224455*a^2*A*c*x^(13/2) + 22309287*A*b^3*x^(15/2) + 66927861*a*b^2*B*x^(15/2) +
133855722*a*A*b*c*x^(15/2) + 66927861*a^2*B*c*x^(15/2) + 19684665*b^3*B*x^(17/2) + 59053995*A*b^2*c*x^(17/2)
+ 118107990*a*b*B*c*x^(17/2) + 59053995*a*A*c^2*x^(17/2) + 52837785*b^2*B*c*x^(19/2) + 52837785*A*b*c^2*x^(19/
2) + 52837785*a*B*c^2*x^(19/2) + 47805615*b*B*c^2*x^(21/2) + 15935205*A*c^3*x^(21/2) + 14549535*B*c^3*x^(23/2)
))/334639305

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fricas [A]  time = 0.42, size = 171, normalized size = 0.94 \begin {gather*} \frac {2}{334639305} \, {\left (14549535 \, B c^{3} x^{11} + 15935205 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{10} + 52837785 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{9} + 19684665 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{8} + 37182145 \, A a^{3} x^{4} + 22309287 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{7} + 77224455 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{6} + 30421755 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{5}\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/334639305*(14549535*B*c^3*x^11 + 15935205*(3*B*b*c^2 + A*c^3)*x^10 + 52837785*(B*b^2*c + (B*a + A*b)*c^2)*x^
9 + 19684665*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^8 + 37182145*A*a^3*x^4 + 22309287*(3*B*a*b^2 + A*b^
3 + 3*(B*a^2 + 2*A*a*b)*c)*x^7 + 77224455*(B*a^2*b + A*a*b^2 + A*a^2*c)*x^6 + 30421755*(B*a^3 + 3*A*a^2*b)*x^5
)*sqrt(x)

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giac [A]  time = 0.23, size = 193, normalized size = 1.06 \begin {gather*} \frac {2}{23} \, B c^{3} x^{\frac {23}{2}} + \frac {2}{7} \, B b c^{2} x^{\frac {21}{2}} + \frac {2}{21} \, A c^{3} x^{\frac {21}{2}} + \frac {6}{19} \, B b^{2} c x^{\frac {19}{2}} + \frac {6}{19} \, B a c^{2} x^{\frac {19}{2}} + \frac {6}{19} \, A b c^{2} x^{\frac {19}{2}} + \frac {2}{17} \, B b^{3} x^{\frac {17}{2}} + \frac {12}{17} \, B a b c x^{\frac {17}{2}} + \frac {6}{17} \, A b^{2} c x^{\frac {17}{2}} + \frac {6}{17} \, A a c^{2} x^{\frac {17}{2}} + \frac {2}{5} \, B a b^{2} x^{\frac {15}{2}} + \frac {2}{15} \, A b^{3} x^{\frac {15}{2}} + \frac {2}{5} \, B a^{2} c x^{\frac {15}{2}} + \frac {4}{5} \, A a b c x^{\frac {15}{2}} + \frac {6}{13} \, B a^{2} b x^{\frac {13}{2}} + \frac {6}{13} \, A a b^{2} x^{\frac {13}{2}} + \frac {6}{13} \, A a^{2} c x^{\frac {13}{2}} + \frac {2}{11} \, B a^{3} x^{\frac {11}{2}} + \frac {6}{11} \, A a^{2} b x^{\frac {11}{2}} + \frac {2}{9} \, A a^{3} x^{\frac {9}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/23*B*c^3*x^(23/2) + 2/7*B*b*c^2*x^(21/2) + 2/21*A*c^3*x^(21/2) + 6/19*B*b^2*c*x^(19/2) + 6/19*B*a*c^2*x^(19/
2) + 6/19*A*b*c^2*x^(19/2) + 2/17*B*b^3*x^(17/2) + 12/17*B*a*b*c*x^(17/2) + 6/17*A*b^2*c*x^(17/2) + 6/17*A*a*c
^2*x^(17/2) + 2/5*B*a*b^2*x^(15/2) + 2/15*A*b^3*x^(15/2) + 2/5*B*a^2*c*x^(15/2) + 4/5*A*a*b*c*x^(15/2) + 6/13*
B*a^2*b*x^(13/2) + 6/13*A*a*b^2*x^(13/2) + 6/13*A*a^2*c*x^(13/2) + 2/11*B*a^3*x^(11/2) + 6/11*A*a^2*b*x^(11/2)
+ 2/9*A*a^3*x^(9/2)

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maple [A]  time = 0.05, size = 192, normalized size = 1.05 \begin {gather*} \frac {2 \left (14549535 B \,c^{3} x^{7}[phone] A \,c^{3} x^{6}[phone] x^{6} B b \,c^{2}[phone] x^{5} A b \,c^{2}[phone] B a \,c^{2} x^{5}[phone] x^{5} B \,b^{2} c [phone] A a \,c^{2} x^{4}[phone] x^{4} A \,b^{2} c [phone] x^{4} a b B c [phone] x^{4} b^{3} B [phone] x^{3} A a b c [phone] A \,b^{3} x^{3}[phone] B \,a^{2} c \,x^{3}[phone] x^{3} B a \,b^{2}[phone] A \,a^{2} c \,x^{2}[phone] x^{2} A a \,b^{2}[phone] B \,a^{2} b \,x^{2}[phone] x A \,a^{2} b [phone] B \,a^{3} x [phone] A \,a^{3}\right ) x^{\frac {9}{2}}}{334639305} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/334639305*x^(9/2)*(14549535*B*c^3*x^7+15935205*A*c^3*x^6+47805615*B*b*c^2*x^6+52837785*A*b*c^2*x^5+52837785*
B*a*c^2*x^5+52837785*B*b^2*c*x^5+59053995*A*a*c^2*x^4+59053995*A*b^2*c*x^4+118107990*B*a*b*c*x^4+19684665*B*b^
3*x^4+133855722*A*a*b*c*x^3+22309287*A*b^3*x^3+66927861*B*a^2*c*x^3+66927861*B*a*b^2*x^3+77224455*A*a^2*c*x^2+
77224455*A*a*b^2*x^2+77224455*B*a^2*b*x^2+91265265*A*a^2*b*x+30421755*B*a^3*x+37182145*A*a^3)

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maxima [A]  time = 0.56, size = 166, normalized size = 0.91 \begin {gather*} \frac {2}{23} \, B c^{3} x^{\frac {23}{2}} + \frac {2}{21} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac {21}{2}} + \frac {6}{19} \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{\frac {19}{2}} + \frac {2}{17} \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{\frac {17}{2}} + \frac {2}{9} \, A a^{3} x^{\frac {9}{2}} + \frac {2}{15} \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{\frac {15}{2}} + \frac {6}{13} \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{\frac {13}{2}} + \frac {2}{11} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac {11}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/23*B*c^3*x^(23/2) + 2/21*(3*B*b*c^2 + A*c^3)*x^(21/2) + 6/19*(B*b^2*c + (B*a + A*b)*c^2)*x^(19/2) + 2/17*(B*
b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^(17/2) + 2/9*A*a^3*x^(9/2) + 2/15*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2
*A*a*b)*c)*x^(15/2) + 6/13*(B*a^2*b + A*a*b^2 + A*a^2*c)*x^(13/2) + 2/11*(B*a^3 + 3*A*a^2*b)*x^(11/2)

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mupad [B]  time = 0.07, size = 169, normalized size = 0.93 \begin {gather*} x^{15/2}\,\left (\frac {2\,B\,c\,a^2}{5}+\frac {2\,B\,a\,b^2}{5}+\frac {4\,A\,c\,a\,b}{5}+\frac {2\,A\,b^3}{15}\right )+x^{17/2}\,\left (\frac {2\,B\,b^3}{17}+\frac {6\,A\,b^2\,c}{17}+\frac {12\,B\,a\,b\,c}{17}+\frac {6\,A\,a\,c^2}{17}\right )+x^{11/2}\,\left (\frac {2\,B\,a^3}{11}+\frac {6\,A\,b\,a^2}{11}\right )+x^{21/2}\,\left (\frac {2\,A\,c^3}{21}+\frac {2\,B\,b\,c^2}{7}\right )+x^{13/2}\,\left (\frac {6\,B\,a^2\,b}{13}+\frac {6\,A\,c\,a^2}{13}+\frac {6\,A\,a\,b^2}{13}\right )+x^{19/2}\,\left (\frac {6\,B\,b^2\,c}{19}+\frac {6\,A\,b\,c^2}{19}+\frac {6\,B\,a\,c^2}{19}\right )+\frac {2\,A\,a^3\,x^{9/2}}{9}+\frac {2\,B\,c^3\,x^{23/2}}{23} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^(15/2)*((2*A*b^3)/15 + (2*B*a*b^2)/5 + (2*B*a^2*c)/5 + (4*A*a*b*c)/5) + x^(17/2)*((2*B*b^3)/17 + (6*A*a*c^2)
/17 + (6*A*b^2*c)/17 + (12*B*a*b*c)/17) + x^(11/2)*((2*B*a^3)/11 + (6*A*a^2*b)/11) + x^(21/2)*((2*A*c^3)/21 +
(2*B*b*c^2)/7) + x^(13/2)*((6*A*a*b^2)/13 + (6*A*a^2*c)/13 + (6*B*a^2*b)/13) + x^(19/2)*((6*A*b*c^2)/19 + (6*B
*a*c^2)/19 + (6*B*b^2*c)/19) + (2*A*a^3*x^(9/2))/9 + (2*B*c^3*x^(23/2))/23

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sympy [A]  time = 36.59, size = 294, normalized size = 1.62 \begin {gather*} \frac {2 A a^{3} x^{\frac {9}{2}}}{9} + \frac {6 A a^{2} b x^{\frac {11}{2}}}{11} + \frac {6 A a^{2} c x^{\frac {13}{2}}}{13} + \frac {6 A a b^{2} x^{\frac {13}{2}}}{13} + \frac {4 A a b c x^{\frac {15}{2}}}{5} + \frac {6 A a c^{2} x^{\frac {17}{2}}}{17} + \frac {2 A b^{3} x^{\frac {15}{2}}}{15} + \frac {6 A b^{2} c x^{\frac {17}{2}}}{17} + \frac {6 A b c^{2} x^{\frac {19}{2}}}{19} + \frac {2 A c^{3} x^{\frac {21}{2}}}{21} + \frac {2 B a^{3} x^{\frac {11}{2}}}{11} + \frac {6 B a^{2} b x^{\frac {13}{2}}}{13} + \frac {2 B a^{2} c x^{\frac {15}{2}}}{5} + \frac {2 B a b^{2} x^{\frac {15}{2}}}{5} + \frac {12 B a b c x^{\frac {17}{2}}}{17} + \frac {6 B a c^{2} x^{\frac {19}{2}}}{19} + \frac {2 B b^{3} x^{\frac {17}{2}}}{17} + \frac {6 B b^{2} c x^{\frac {19}{2}}}{19} + \frac {2 B b c^{2} x^{\frac {21}{2}}}{7} + \frac {2 B c^{3} x^{\frac {23}{2}}}{23} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*A*a**3*x**(9/2)/9 + 6*A*a**2*b*x**(11/2)/11 + 6*A*a**2*c*x**(13/2)/13 + 6*A*a*b**2*x**(13/2)/13 + 4*A*a*b*c*
x**(15/2)/5 + 6*A*a*c**2*x**(17/2)/17 + 2*A*b**3*x**(15/2)/15 + 6*A*b**2*c*x**(17/2)/17 + 6*A*b*c**2*x**(19/2)
/19 + 2*A*c**3*x**(21/2)/21 + 2*B*a**3*x**(11/2)/11 + 6*B*a**2*b*x**(13/2)/13 + 2*B*a**2*c*x**(15/2)/5 + 2*B*a
*b**2*x**(15/2)/5 + 12*B*a*b*c*x**(17/2)/17 + 6*B*a*c**2*x**(19/2)/19 + 2*B*b**3*x**(17/2)/17 + 6*B*b**2*c*x**
(19/2)/19 + 2*B*b*c**2*x**(21/2)/7 + 2*B*c**3*x**(23/2)/23

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