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

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

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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} \begin {gather*} \frac {2}{3} a^2 A c x^{9/2}+\frac {2}{5} a^3 A x^{5/2}+\frac {6}{11} a^2 B c x^{11/2}+\frac {2}{7} a^3 B x^{7/2}+\frac {6}{13} a A c^2 x^{13/2}+\frac {2}{5} a B c^2 x^{15/2}+\frac {2}{17} A c^3 x^{17/2}+\frac {2}{19} B c^3 x^{19/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.04, size = 83, normalized size = 0.76 \begin {gather*} \frac {2}{35} a^3 x^{5/2} (7 A+5 B x)+\frac {2}{33} a^2 c x^{9/2} (11 A+9 B x)+\frac {2}{65} a c^2 x^{13/2} (15 A+13 B x)+\frac {2}{323} c^3 x^{17/2} (19 A+17 B x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^3*x^(5/2)*(7*A + 5*B*x))/35 + (2*a^2*c*x^(9/2)*(11*A + 9*B*x))/33 + (2*a*c^2*x^(13/2)*(15*A + 13*B*x))/65
+ (2*c^3*x^(17/2)*(19*A + 17*B*x))/323

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IntegrateAlgebraic [A]  time = 0.04, size = 97, normalized size = 0.89 \begin {gather*} \frac {2 \left (969969 a^3 A x^{5/2}+692835 a^3 B x^{7/2}+1616615 a^2 A c x^{9/2}+1322685 a^2 B c x^{11/2}+1119195 a A c^2 x^{13/2}+969969 a B c^2 x^{15/2}+285285 A c^3 x^{17/2}+255255 B c^3 x^{19/2}\right )}{4849845} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(969969*a^3*A*x^(5/2) + 692835*a^3*B*x^(7/2) + 1616615*a^2*A*c*x^(9/2) + 1322685*a^2*B*c*x^(11/2) + 1119195
*a*A*c^2*x^(13/2) + 969969*a*B*c^2*x^(15/2) + 285285*A*c^3*x^(17/2) + 255255*B*c^3*x^(19/2)))/4849845

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fricas [A]  time = 0.41, size = 82, normalized size = 0.75 \begin {gather*} \frac {2}{4849845} \, {\left (255255 \, B c^{3} x^{9} + 285285 \, A c^{3} x^{8} + 969969 \, B a c^{2} x^{7} + 1119195 \, A a c^{2} x^{6} + 1322685 \, B a^{2} c x^{5} + 1616615 \, A a^{2} c x^{4} + 692835 \, B a^{3} x^{3} + 969969 \, A a^{3} x^{2}\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/4849845*(255255*B*c^3*x^9 + 285285*A*c^3*x^8 + 969969*B*a*c^2*x^7 + 1119195*A*a*c^2*x^6 + 1322685*B*a^2*c*x^
5 + 1616615*A*a^2*c*x^4 + 692835*B*a^3*x^3 + 969969*A*a^3*x^2)*sqrt(x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 78, normalized size = 0.72 \begin {gather*} \frac {2 \left (255255 B \,c^{3} x^{7}+285285 A \,c^{3} x^{6}+969969 B a \,c^{2} x^{5}+1119195 A a \,c^{2} x^{4}+1322685 B \,a^{2} c \,x^{3}+1616615 A \,a^{2} c \,x^{2}+692835 B \,a^{3} x +969969 A \,a^{3}\right ) x^{\frac {5}{2}}}{4849845} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/4849845*x^(5/2)*(255255*B*c^3*x^7+285285*A*c^3*x^6+969969*B*a*c^2*x^5+1119195*A*a*c^2*x^4+1322685*B*a^2*c*x^
3+1616615*A*a^2*c*x^2+692835*B*a^3*x+969969*A*a^3)

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maxima [A]  time = 0.59, size = 77, normalized size = 0.71 \begin {gather*} \frac {2}{19} \, B c^{3} x^{\frac {19}{2}} + \frac {2}{17} \, A c^{3} x^{\frac {17}{2}} + \frac {2}{5} \, B a c^{2} x^{\frac {15}{2}} + \frac {6}{13} \, A a c^{2} x^{\frac {13}{2}} + \frac {6}{11} \, B a^{2} c x^{\frac {11}{2}} + \frac {2}{3} \, A a^{2} c x^{\frac {9}{2}} + \frac {2}{7} \, B a^{3} x^{\frac {7}{2}} + \frac {2}{5} \, A a^{3} x^{\frac {5}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 9.82, size = 114, normalized size = 1.05 \begin {gather*} \frac {2 A a^{3} x^{\frac {5}{2}}}{5} + \frac {2 A a^{2} c x^{\frac {9}{2}}}{3} + \frac {6 A a c^{2} x^{\frac {13}{2}}}{13} + \frac {2 A c^{3} x^{\frac {17}{2}}}{17} + \frac {2 B a^{3} x^{\frac {7}{2}}}{7} + \frac {6 B a^{2} c x^{\frac {11}{2}}}{11} + \frac {2 B a c^{2} x^{\frac {15}{2}}}{5} + \frac {2 B c^{3} x^{\frac {19}{2}}}{19} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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