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

Optimal. Leaf size=107 \[ 2 a^3 A \sqrt {x}+\frac {2}{3} a^3 B x^{3/2}+\frac {6}{5} a^2 A c x^{5/2}+\frac {6}{7} a^2 B c x^{7/2}+\frac {2}{3} a A c^2 x^{9/2}+\frac {6}{11} a B c^2 x^{11/2}+\frac {2}{13} A c^3 x^{13/2}+\frac {2}{15} B c^3 x^{15/2} \]

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Rubi [A]  time = 0.04, antiderivative size = 107, 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}{5} a^2 A c x^{5/2}+2 a^3 A \sqrt {x}+\frac {6}{7} a^2 B c x^{7/2}+\frac {2}{3} a^3 B x^{3/2}+\frac {2}{3} a A c^2 x^{9/2}+\frac {6}{11} a B c^2 x^{11/2}+\frac {2}{13} A c^3 x^{13/2}+\frac {2}{15} B c^3 x^{15/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.03, size = 72, normalized size = 0.67 \begin {gather*} \frac {2 \sqrt {x} \left (5005 a^3 (3 A+B x)+1287 a^2 c x^2 (7 A+5 B x)+455 a c^2 x^4 (11 A+9 B x)+77 c^3 x^6 (15 A+13 B x)\right )}{15015} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[x]*(5005*a^3*(3*A + B*x) + 1287*a^2*c*x^2*(7*A + 5*B*x) + 455*a*c^2*x^4*(11*A + 9*B*x) + 77*c^3*x^6*(1
5*A + 13*B*x)))/15015

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IntegrateAlgebraic [A]  time = 0.04, size = 97, normalized size = 0.91 \begin {gather*} \frac {2 \left (15015 a^3 A \sqrt {x}+5005 a^3 B x^{3/2}+9009 a^2 A c x^{5/2}+6435 a^2 B c x^{7/2}+5005 a A c^2 x^{9/2}+4095 a B c^2 x^{11/2}+1155 A c^3 x^{13/2}+1001 B c^3 x^{15/2}\right )}{15015} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(15015*a^3*A*Sqrt[x] + 5005*a^3*B*x^(3/2) + 9009*a^2*A*c*x^(5/2) + 6435*a^2*B*c*x^(7/2) + 5005*a*A*c^2*x^(9
/2) + 4095*a*B*c^2*x^(11/2) + 1155*A*c^3*x^(13/2) + 1001*B*c^3*x^(15/2)))/15015

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fricas [A]  time = 0.41, size = 77, normalized size = 0.72 \begin {gather*} \frac {2}{15015} \, {\left (1001 \, B c^{3} x^{7} + 1155 \, A c^{3} x^{6} + 4095 \, B a c^{2} x^{5} + 5005 \, A a c^{2} x^{4} + 6435 \, B a^{2} c x^{3} + 9009 \, A a^{2} c x^{2} + 5005 \, B a^{3} x + 15015 \, A a^{3}\right )} \sqrt {x} \end {gather*}

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]

2/15015*(1001*B*c^3*x^7 + 1155*A*c^3*x^6 + 4095*B*a*c^2*x^5 + 5005*A*a*c^2*x^4 + 6435*B*a^2*c*x^3 + 9009*A*a^2
*c*x^2 + 5005*B*a^3*x + 15015*A*a^3)*sqrt(x)

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giac [A]  time = 0.15, size = 77, normalized size = 0.72 \begin {gather*} \frac {2}{15} \, B c^{3} x^{\frac {15}{2}} + \frac {2}{13} \, A c^{3} x^{\frac {13}{2}} + \frac {6}{11} \, B a c^{2} x^{\frac {11}{2}} + \frac {2}{3} \, A a c^{2} x^{\frac {9}{2}} + \frac {6}{7} \, B a^{2} c x^{\frac {7}{2}} + \frac {6}{5} \, A a^{2} c x^{\frac {5}{2}} + \frac {2}{3} \, B a^{3} x^{\frac {3}{2}} + 2 \, A a^{3} \sqrt {x} \end {gather*}

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]

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

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maple [A]  time = 0.05, size = 78, normalized size = 0.73 \begin {gather*} \frac {2 \left (1001 B \,c^{3} x^{7}+1155 A \,c^{3} x^{6}+4095 B a \,c^{2} x^{5}+5005 A a \,c^{2} x^{4}+6435 B \,a^{2} c \,x^{3}+9009 A \,a^{2} c \,x^{2}+5005 B \,a^{3} x +15015 A \,a^{3}\right ) \sqrt {x}}{15015} \end {gather*}

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]

2/15015*x^(1/2)*(1001*B*c^3*x^7+1155*A*c^3*x^6+4095*B*a*c^2*x^5+5005*A*a*c^2*x^4+6435*B*a^2*c*x^3+9009*A*a^2*c
*x^2+5005*B*a^3*x+15015*A*a^3)

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maxima [A]  time = 0.51, size = 77, normalized size = 0.72 \begin {gather*} \frac {2}{15} \, B c^{3} x^{\frac {15}{2}} + \frac {2}{13} \, A c^{3} x^{\frac {13}{2}} + \frac {6}{11} \, B a c^{2} x^{\frac {11}{2}} + \frac {2}{3} \, A a c^{2} x^{\frac {9}{2}} + \frac {6}{7} \, B a^{2} c x^{\frac {7}{2}} + \frac {6}{5} \, A a^{2} c x^{\frac {5}{2}} + \frac {2}{3} \, B a^{3} x^{\frac {3}{2}} + 2 \, A a^{3} \sqrt {x} \end {gather*}

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]

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

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mupad [B]  time = 0.04, size = 77, normalized size = 0.72 \begin {gather*} 2\,A\,a^3\,\sqrt {x}+\frac {2\,B\,a^3\,x^{3/2}}{3}+\frac {2\,A\,c^3\,x^{13/2}}{13}+\frac {2\,B\,c^3\,x^{15/2}}{15}+\frac {6\,A\,a^2\,c\,x^{5/2}}{5}+\frac {2\,A\,a\,c^2\,x^{9/2}}{3}+\frac {6\,B\,a^2\,c\,x^{7/2}}{7}+\frac {6\,B\,a\,c^2\,x^{11/2}}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*A*a^3*x^(1/2) + (2*B*a^3*x^(3/2))/3 + (2*A*c^3*x^(13/2))/13 + (2*B*c^3*x^(15/2))/15 + (6*A*a^2*c*x^(5/2))/5
+ (2*A*a*c^2*x^(9/2))/3 + (6*B*a^2*c*x^(7/2))/7 + (6*B*a*c^2*x^(11/2))/11

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sympy [A]  time = 3.87, size = 112, normalized size = 1.05 \begin {gather*} 2 A a^{3} \sqrt {x} + \frac {6 A a^{2} c x^{\frac {5}{2}}}{5} + \frac {2 A a c^{2} x^{\frac {9}{2}}}{3} + \frac {2 A c^{3} x^{\frac {13}{2}}}{13} + \frac {2 B a^{3} x^{\frac {3}{2}}}{3} + \frac {6 B a^{2} c x^{\frac {7}{2}}}{7} + \frac {6 B a c^{2} x^{\frac {11}{2}}}{11} + \frac {2 B c^{3} x^{\frac {15}{2}}}{15} \end {gather*}

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

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