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

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

[Out]

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

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

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.04, size = 83, normalized size = 0.76 \[ \frac {2}{15} a^3 x^{3/2} (5 A+3 B x)+\frac {2}{21} a^2 c x^{7/2} (9 A+7 B x)+\frac {6}{143} a c^2 x^{11/2} (13 A+11 B x)+\frac {2}{255} c^3 x^{15/2} (17 A+15 B x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^3*x^(3/2)*(5*A + 3*B*x))/15 + (2*a^2*c*x^(7/2)*(9*A + 7*B*x))/21 + (6*a*c^2*x^(11/2)*(13*A + 11*B*x))/143
+ (2*c^3*x^(15/2)*(17*A + 15*B*x))/255

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fricas [A]  time = 0.96, size = 80, normalized size = 0.73 \[ \frac {2}{255255} \, {\left (15015 \, B c^{3} x^{8} + 17017 \, A c^{3} x^{7} + 58905 \, B a c^{2} x^{6} + 69615 \, A a c^{2} x^{5} + 85085 \, B a^{2} c x^{4} + 109395 \, A a^{2} c x^{3} + 51051 \, B a^{3} x^{2} + 85085 \, A a^{3} x\right )} \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3*x^(1/2),x, algorithm="fricas")

[Out]

2/255255*(15015*B*c^3*x^8 + 17017*A*c^3*x^7 + 58905*B*a*c^2*x^6 + 69615*A*a*c^2*x^5 + 85085*B*a^2*c*x^4 + 1093
95*A*a^2*c*x^3 + 51051*B*a^3*x^2 + 85085*A*a^3*x)*sqrt(x)

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giac [A]  time = 0.16, size = 77, normalized size = 0.71 \[ \frac {2}{17} \, B c^{3} x^{\frac {17}{2}} + \frac {2}{15} \, A c^{3} x^{\frac {15}{2}} + \frac {6}{13} \, B a c^{2} x^{\frac {13}{2}} + \frac {6}{11} \, A a c^{2} x^{\frac {11}{2}} + \frac {2}{3} \, B a^{2} c x^{\frac {9}{2}} + \frac {6}{7} \, A a^{2} c x^{\frac {7}{2}} + \frac {2}{5} \, B a^{3} x^{\frac {5}{2}} + \frac {2}{3} \, A a^{3} x^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3*x^(1/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.05, size = 78, normalized size = 0.72 \[ \frac {2 \left (15015 B \,c^{3} x^{7}+17017 A \,c^{3} x^{6}+58905 B a \,c^{2} x^{5}+69615 A a \,c^{2} x^{4}+85085 B \,a^{2} c \,x^{3}+109395 A \,a^{2} c \,x^{2}+51051 B \,a^{3} x +85085 A \,a^{3}\right ) x^{\frac {3}{2}}}{255255} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/255255*x^(3/2)*(15015*B*c^3*x^7+17017*A*c^3*x^6+58905*B*a*c^2*x^5+69615*A*a*c^2*x^4+85085*B*a^2*c*x^3+109395
*A*a^2*c*x^2+51051*B*a^3*x+85085*A*a^3)

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maxima [A]  time = 0.52, size = 77, normalized size = 0.71 \[ \frac {2}{17} \, B c^{3} x^{\frac {17}{2}} + \frac {2}{15} \, A c^{3} x^{\frac {15}{2}} + \frac {6}{13} \, B a c^{2} x^{\frac {13}{2}} + \frac {6}{11} \, A a c^{2} x^{\frac {11}{2}} + \frac {2}{3} \, B a^{2} c x^{\frac {9}{2}} + \frac {6}{7} \, A a^{2} c x^{\frac {7}{2}} + \frac {2}{5} \, B a^{3} x^{\frac {5}{2}} + \frac {2}{3} \, A a^{3} x^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3*x^(1/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.04, size = 77, normalized size = 0.71 \[ \frac {2\,A\,a^3\,x^{3/2}}{3}+\frac {2\,B\,a^3\,x^{5/2}}{5}+\frac {2\,A\,c^3\,x^{15/2}}{15}+\frac {2\,B\,c^3\,x^{17/2}}{17}+\frac {6\,A\,a^2\,c\,x^{7/2}}{7}+\frac {6\,A\,a\,c^2\,x^{11/2}}{11}+\frac {2\,B\,a^2\,c\,x^{9/2}}{3}+\frac {6\,B\,a\,c^2\,x^{13/2}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 4.41, size = 114, normalized size = 1.05 \[ \frac {2 A a^{3} x^{\frac {3}{2}}}{3} + \frac {6 A a^{2} c x^{\frac {7}{2}}}{7} + \frac {6 A a c^{2} x^{\frac {11}{2}}}{11} + \frac {2 A c^{3} x^{\frac {15}{2}}}{15} + \frac {2 B a^{3} x^{\frac {5}{2}}}{5} + \frac {2 B a^{2} c x^{\frac {9}{2}}}{3} + \frac {6 B a c^{2} x^{\frac {13}{2}}}{13} + \frac {2 B c^{3} x^{\frac {17}{2}}}{17} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)**3*x**(1/2),x)

[Out]

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

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