3.403 \(\int x^{5/2} (A+B x) (a+c x^2)^3 \, dx\)

Optimal. Leaf size=109 \[ \frac {2}{7} a^3 A x^{7/2}+\frac {2}{9} a^3 B x^{9/2}+\frac {6}{11} a^2 A c x^{11/2}+\frac {6}{13} a^2 B c x^{13/2}+\frac {2}{5} a A c^2 x^{15/2}+\frac {6}{17} a B c^2 x^{17/2}+\frac {2}{19} A c^3 x^{19/2}+\frac {2}{21} B c^3 x^{21/2} \]

[Out]

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

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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} \[ \frac {6}{11} a^2 A c x^{11/2}+\frac {2}{7} a^3 A x^{7/2}+\frac {6}{13} a^2 B c x^{13/2}+\frac {2}{9} a^3 B x^{9/2}+\frac {2}{5} a A c^2 x^{15/2}+\frac {6}{17} a B c^2 x^{17/2}+\frac {2}{19} A c^3 x^{19/2}+\frac {2}{21} B c^3 x^{21/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(5/2)*(A + B*x)*(a + c*x^2)^3,x]

[Out]

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

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

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Mathematica [A]  time = 0.04, size = 83, normalized size = 0.76 \[ \frac {2}{63} a^3 x^{7/2} (9 A+7 B x)+\frac {6}{143} a^2 c x^{11/2} (13 A+11 B x)+\frac {2}{85} a c^2 x^{15/2} (17 A+15 B x)+\frac {2}{399} c^3 x^{19/2} (21 A+19 B x) \]

Antiderivative was successfully verified.

[In]

Integrate[x^(5/2)*(A + B*x)*(a + c*x^2)^3,x]

[Out]

(2*a^3*x^(7/2)*(9*A + 7*B*x))/63 + (6*a^2*c*x^(11/2)*(13*A + 11*B*x))/143 + (2*a*c^2*x^(15/2)*(17*A + 15*B*x))
/85 + (2*c^3*x^(19/2)*(21*A + 19*B*x))/399

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fricas [A]  time = 0.99, size = 82, normalized size = 0.75 \[ \frac {2}{14549535} \, {\left (692835 \, B c^{3} x^{10} + 765765 \, A c^{3} x^{9} + 2567565 \, B a c^{2} x^{8} + 2909907 \, A a c^{2} x^{7} + 3357585 \, B a^{2} c x^{6} + 3968055 \, A a^{2} c x^{5} + 1616615 \, B a^{3} x^{4} + 2078505 \, A a^{3} x^{3}\right )} \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(B*x+A)*(c*x^2+a)^3,x, algorithm="fricas")

[Out]

2/14549535*(692835*B*c^3*x^10 + 765765*A*c^3*x^9 + 2567565*B*a*c^2*x^8 + 2909907*A*a*c^2*x^7 + 3357585*B*a^2*c
*x^6 + 3968055*A*a^2*c*x^5 + 1616615*B*a^3*x^4 + 2078505*A*a^3*x^3)*sqrt(x)

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giac [A]  time = 0.15, size = 77, normalized size = 0.71 \[ \frac {2}{21} \, B c^{3} x^{\frac {21}{2}} + \frac {2}{19} \, A c^{3} x^{\frac {19}{2}} + \frac {6}{17} \, B a c^{2} x^{\frac {17}{2}} + \frac {2}{5} \, A a c^{2} x^{\frac {15}{2}} + \frac {6}{13} \, B a^{2} c x^{\frac {13}{2}} + \frac {6}{11} \, A a^{2} c x^{\frac {11}{2}} + \frac {2}{9} \, B a^{3} x^{\frac {9}{2}} + \frac {2}{7} \, A a^{3} x^{\frac {7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(B*x+A)*(c*x^2+a)^3,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.05, size = 78, normalized size = 0.72 \[ \frac {2 \left (692835 B \,c^{3} x^{7}+765765 A \,c^{3} x^{6}+2567565 B a \,c^{2} x^{5}+2909907 A a \,c^{2} x^{4}+3357585 B \,a^{2} c \,x^{3}+3968055 A \,a^{2} c \,x^{2}+1616615 B \,a^{3} x +2078505 A \,a^{3}\right ) x^{\frac {7}{2}}}{14549535} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(B*x+A)*(c*x^2+a)^3,x)

[Out]

2/14549535*x^(7/2)*(692835*B*c^3*x^7+765765*A*c^3*x^6+2567565*B*a*c^2*x^5+2909907*A*a*c^2*x^4+3357585*B*a^2*c*
x^3+3968055*A*a^2*c*x^2+1616615*B*a^3*x+2078505*A*a^3)

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maxima [A]  time = 0.60, size = 77, normalized size = 0.71 \[ \frac {2}{21} \, B c^{3} x^{\frac {21}{2}} + \frac {2}{19} \, A c^{3} x^{\frac {19}{2}} + \frac {6}{17} \, B a c^{2} x^{\frac {17}{2}} + \frac {2}{5} \, A a c^{2} x^{\frac {15}{2}} + \frac {6}{13} \, B a^{2} c x^{\frac {13}{2}} + \frac {6}{11} \, A a^{2} c x^{\frac {11}{2}} + \frac {2}{9} \, B a^{3} x^{\frac {9}{2}} + \frac {2}{7} \, A a^{3} x^{\frac {7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(B*x+A)*(c*x^2+a)^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.03, size = 77, normalized size = 0.71 \[ \frac {2\,A\,a^3\,x^{7/2}}{7}+\frac {2\,B\,a^3\,x^{9/2}}{9}+\frac {2\,A\,c^3\,x^{19/2}}{19}+\frac {2\,B\,c^3\,x^{21/2}}{21}+\frac {6\,A\,a^2\,c\,x^{11/2}}{11}+\frac {2\,A\,a\,c^2\,x^{15/2}}{5}+\frac {6\,B\,a^2\,c\,x^{13/2}}{13}+\frac {6\,B\,a\,c^2\,x^{17/2}}{17} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(a + c*x^2)^3*(A + B*x),x)

[Out]

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

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sympy [A]  time = 16.45, size = 114, normalized size = 1.05 \[ \frac {2 A a^{3} x^{\frac {7}{2}}}{7} + \frac {6 A a^{2} c x^{\frac {11}{2}}}{11} + \frac {2 A a c^{2} x^{\frac {15}{2}}}{5} + \frac {2 A c^{3} x^{\frac {19}{2}}}{19} + \frac {2 B a^{3} x^{\frac {9}{2}}}{9} + \frac {6 B a^{2} c x^{\frac {13}{2}}}{13} + \frac {6 B a c^{2} x^{\frac {17}{2}}}{17} + \frac {2 B c^{3} x^{\frac {21}{2}}}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(5/2)*(B*x+A)*(c*x**2+a)**3,x)

[Out]

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

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