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

Optimal. Leaf size=109 \[ \frac {2}{9} a^3 A x^{9/2}+\frac {2}{11} a^3 B x^{11/2}+\frac {6}{13} a^2 A c x^{13/2}+\frac {2}{5} a^2 B c x^{15/2}+\frac {6}{17} a A c^2 x^{17/2}+\frac {6}{19} a B c^2 x^{19/2}+\frac {2}{21} A c^3 x^{21/2}+\frac {2}{23} B c^3 x^{23/2} \]

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {766} \begin {gather*} \frac {6}{13} a^2 A c x^{13/2}+\frac {2}{9} a^3 A x^{9/2}+\frac {2}{5} a^2 B c x^{15/2}+\frac {2}{11} a^3 B x^{11/2}+\frac {6}{17} a A c^2 x^{17/2}+\frac {6}{19} a B c^2 x^{19/2}+\frac {2}{21} A c^3 x^{21/2}+\frac {2}{23} B c^3 x^{23/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 766

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(e*x
)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int x^{7/2} (A+B x) \left (a+c x^2\right )^3 \, dx &=\int \left (a^3 A x^{7/2}+a^3 B x^{9/2}+3 a^2 A c x^{11/2}+3 a^2 B c x^{13/2}+3 a A c^2 x^{15/2}+3 a B c^2 x^{17/2}+A c^3 x^{19/2}+B c^3 x^{21/2}\right ) \, dx\\ &=\frac {2}{9} a^3 A x^{9/2}+\frac {2}{11} a^3 B x^{11/2}+\frac {6}{13} a^2 A c x^{13/2}+\frac {2}{5} a^2 B c x^{15/2}+\frac {6}{17} a A c^2 x^{17/2}+\frac {6}{19} a B c^2 x^{19/2}+\frac {2}{21} A c^3 x^{21/2}+\frac {2}{23} B c^3 x^{23/2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 83, normalized size = 0.76 \begin {gather*} \frac {2}{99} a^3 x^{9/2} (11 A+9 B x)+\frac {2}{65} a^2 c x^{13/2} (15 A+13 B x)+\frac {6}{323} a c^2 x^{17/2} (19 A+17 B x)+\frac {2}{483} c^3 x^{21/2} (23 A+21 B x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^3*x^(9/2)*(11*A + 9*B*x))/99 + (2*a^2*c*x^(13/2)*(15*A + 13*B*x))/65 + (6*a*c^2*x^(17/2)*(19*A + 17*B*x))
/323 + (2*c^3*x^(21/2)*(23*A + 21*B*x))/483

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.04, size = 97, normalized size = 0.89 \begin {gather*} \frac {2 \left (37182145 a^3 A x^{9/2}[phone] a^3 B x^{11/2}[phone] a^2 A c x^{13/2}[phone] a^2 B c x^{15/2}[phone] a A c^2 x^{17/2}[phone] a B c^2 x^{19/2}[phone] A c^3 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 + c*x^2)^3,x]

[Out]

(2*(37182145*a^3*A*x^(9/2) + 30421755*a^3*B*x^(11/2) + 77224455*a^2*A*c*x^(13/2) + 66927861*a^2*B*c*x^(15/2) +
59053995*a*A*c^2*x^(17/2) + 52837785*a*B*c^2*x^(19/2) + 15935205*A*c^3*x^(21/2) + 14549535*B*c^3*x^(23/2)))/3
34639305

________________________________________________________________________________________

fricas [A]  time = 0.40, size = 82, normalized size = 0.75 \begin {gather*} \frac {2}{334639305} \, {\left (14549535 \, B c^{3} x^{11} + 15935205 \, A c^{3} x^{10} + 52837785 \, B a c^{2} x^{9} + 59053995 \, A a c^{2} x^{8} + 66927861 \, B a^{2} c x^{7} + 77224455 \, A a^{2} c x^{6} + 30421755 \, B a^{3} x^{5} + 37182145 \, A a^{3} x^{4}\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+a)^3,x, algorithm="fricas")

[Out]

2/334639305*(14549535*B*c^3*x^11 + 15935205*A*c^3*x^10 + 52837785*B*a*c^2*x^9 + 59053995*A*a*c^2*x^8 + 6692786
1*B*a^2*c*x^7 + 77224455*A*a^2*c*x^6 + 30421755*B*a^3*x^5 + 37182145*A*a^3*x^4)*sqrt(x)

________________________________________________________________________________________

giac [A]  time = 0.15, size = 77, normalized size = 0.71 \begin {gather*} \frac {2}{23} \, B c^{3} x^{\frac {23}{2}} + \frac {2}{21} \, A c^{3} x^{\frac {21}{2}} + \frac {6}{19} \, B a c^{2} x^{\frac {19}{2}} + \frac {6}{17} \, A a c^{2} x^{\frac {17}{2}} + \frac {2}{5} \, B a^{2} c x^{\frac {15}{2}} + \frac {6}{13} \, A a^{2} c x^{\frac {13}{2}} + \frac {2}{11} \, B a^{3} 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+a)^3,x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.05, size = 78, normalized size = 0.72 \begin {gather*} \frac {2 \left (14549535 B \,c^{3} x^{7}[phone] A \,c^{3} x^{6}[phone] B a \,c^{2} x^{5}[phone] A a \,c^{2} x^{4}[phone] B \,a^{2} c \,x^{3}[phone] A \,a^{2} c \,x^{2}[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+a)^3,x)

[Out]

2/334639305*x^(9/2)*(14549535*B*c^3*x^7+15935205*A*c^3*x^6+52837785*B*a*c^2*x^5+59053995*A*a*c^2*x^4+66927861*
B*a^2*c*x^3+77224455*A*a^2*c*x^2+30421755*B*a^3*x+37182145*A*a^3)

________________________________________________________________________________________

maxima [A]  time = 0.52, size = 77, normalized size = 0.71 \begin {gather*} \frac {2}{23} \, B c^{3} x^{\frac {23}{2}} + \frac {2}{21} \, A c^{3} x^{\frac {21}{2}} + \frac {6}{19} \, B a c^{2} x^{\frac {19}{2}} + \frac {6}{17} \, A a c^{2} x^{\frac {17}{2}} + \frac {2}{5} \, B a^{2} c x^{\frac {15}{2}} + \frac {6}{13} \, A a^{2} c x^{\frac {13}{2}} + \frac {2}{11} \, B a^{3} 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+a)^3,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 77, normalized size = 0.71 \begin {gather*} \frac {2\,A\,a^3\,x^{9/2}}{9}+\frac {2\,B\,a^3\,x^{11/2}}{11}+\frac {2\,A\,c^3\,x^{21/2}}{21}+\frac {2\,B\,c^3\,x^{23/2}}{23}+\frac {6\,A\,a^2\,c\,x^{13/2}}{13}+\frac {6\,A\,a\,c^2\,x^{17/2}}{17}+\frac {2\,B\,a^2\,c\,x^{15/2}}{5}+\frac {6\,B\,a\,c^2\,x^{19/2}}{19} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 27.94, size = 114, normalized size = 1.05 \begin {gather*} \frac {2 A a^{3} x^{\frac {9}{2}}}{9} + \frac {6 A a^{2} c x^{\frac {13}{2}}}{13} + \frac {6 A a c^{2} x^{\frac {17}{2}}}{17} + \frac {2 A c^{3} x^{\frac {21}{2}}}{21} + \frac {2 B a^{3} x^{\frac {11}{2}}}{11} + \frac {2 B a^{2} c x^{\frac {15}{2}}}{5} + \frac {6 B a c^{2} x^{\frac {19}{2}}}{19} + \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+a)**3,x)

[Out]

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

________________________________________________________________________________________