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

Optimal. Leaf size=61 \[ 2 a^2 A \sqrt {x}+\frac {2}{13} b x^{13/2} (2 a B+A b)+\frac {2}{7} a x^{7/2} (a B+2 A b)+\frac {2}{19} b^2 B x^{19/2} \]

[Out]

2/7*a*(2*A*b+B*a)*x^(7/2)+2/13*b*(A*b+2*B*a)*x^(13/2)+2/19*b^2*B*x^(19/2)+2*a^2*A*x^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {448} \[ 2 a^2 A \sqrt {x}+\frac {2}{13} b x^{13/2} (2 a B+A b)+\frac {2}{7} a x^{7/2} (a B+2 A b)+\frac {2}{19} b^2 B x^{19/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*a^2*A*Sqrt[x] + (2*a*(2*A*b + a*B)*x^(7/2))/7 + (2*b*(A*b + 2*a*B)*x^(13/2))/13 + (2*b^2*B*x^(19/2))/19

Rule 448

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.07, size = 53, normalized size = 0.87 \[ \frac {2 \sqrt {x} \left (1729 a^2 A+133 b x^6 (2 a B+A b)+247 a x^3 (a B+2 A b)+91 b^2 B x^9\right )}{1729} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[x]*(1729*a^2*A + 247*a*(2*A*b + a*B)*x^3 + 133*b*(A*b + 2*a*B)*x^6 + 91*b^2*B*x^9))/1729

________________________________________________________________________________________

fricas [A]  time = 0.65, size = 53, normalized size = 0.87 \[ \frac {2}{1729} \, {\left (91 \, B b^{2} x^{9} + 133 \, {\left (2 \, B a b + A b^{2}\right )} x^{6} + 247 \, {\left (B a^{2} + 2 \, A a b\right )} x^{3} + 1729 \, A a^{2}\right )} \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1729*(91*B*b^2*x^9 + 133*(2*B*a*b + A*b^2)*x^6 + 247*(B*a^2 + 2*A*a*b)*x^3 + 1729*A*a^2)*sqrt(x)

________________________________________________________________________________________

giac [A]  time = 0.23, size = 53, normalized size = 0.87 \[ \frac {2}{19} \, B b^{2} x^{\frac {19}{2}} + \frac {4}{13} \, B a b x^{\frac {13}{2}} + \frac {2}{13} \, A b^{2} x^{\frac {13}{2}} + \frac {2}{7} \, B a^{2} x^{\frac {7}{2}} + \frac {4}{7} \, A a b x^{\frac {7}{2}} + 2 \, A a^{2} \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/19*B*b^2*x^(19/2) + 4/13*B*a*b*x^(13/2) + 2/13*A*b^2*x^(13/2) + 2/7*B*a^2*x^(7/2) + 4/7*A*a*b*x^(7/2) + 2*A*
a^2*sqrt(x)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 56, normalized size = 0.92 \[ \frac {2 \left (91 b^{2} B \,x^{9}+133 A \,b^{2} x^{6}+266 B a b \,x^{6}+494 A a b \,x^{3}+247 B \,a^{2} x^{3}+1729 a^{2} A \right ) \sqrt {x}}{1729} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1729*x^(1/2)*(91*B*b^2*x^9+133*A*b^2*x^6+266*B*a*b*x^6+494*A*a*b*x^3+247*B*a^2*x^3+1729*A*a^2)

________________________________________________________________________________________

maxima [A]  time = 0.51, size = 51, normalized size = 0.84 \[ \frac {2}{19} \, B b^{2} x^{\frac {19}{2}} + \frac {2}{13} \, {\left (2 \, B a b + A b^{2}\right )} x^{\frac {13}{2}} + \frac {2}{7} \, {\left (B a^{2} + 2 \, A a b\right )} x^{\frac {7}{2}} + 2 \, A a^{2} \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/19*B*b^2*x^(19/2) + 2/13*(2*B*a*b + A*b^2)*x^(13/2) + 2/7*(B*a^2 + 2*A*a*b)*x^(7/2) + 2*A*a^2*sqrt(x)

________________________________________________________________________________________

mupad [B]  time = 0.05, size = 51, normalized size = 0.84 \[ x^{7/2}\,\left (\frac {2\,B\,a^2}{7}+\frac {4\,A\,b\,a}{7}\right )+x^{13/2}\,\left (\frac {2\,A\,b^2}{13}+\frac {4\,B\,a\,b}{13}\right )+2\,A\,a^2\,\sqrt {x}+\frac {2\,B\,b^2\,x^{19/2}}{19} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^(7/2)*((2*B*a^2)/7 + (4*A*a*b)/7) + x^(13/2)*((2*A*b^2)/13 + (4*B*a*b)/13) + 2*A*a^2*x^(1/2) + (2*B*b^2*x^(1
9/2))/19

________________________________________________________________________________________

sympy [A]  time = 9.22, size = 78, normalized size = 1.28 \[ 2 A a^{2} \sqrt {x} + \frac {4 A a b x^{\frac {7}{2}}}{7} + \frac {2 A b^{2} x^{\frac {13}{2}}}{13} + \frac {2 B a^{2} x^{\frac {7}{2}}}{7} + \frac {4 B a b x^{\frac {13}{2}}}{13} + \frac {2 B b^{2} x^{\frac {19}{2}}}{19} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*A*a**2*sqrt(x) + 4*A*a*b*x**(7/2)/7 + 2*A*b**2*x**(13/2)/13 + 2*B*a**2*x**(7/2)/7 + 4*B*a*b*x**(13/2)/13 + 2
*B*b**2*x**(19/2)/19

________________________________________________________________________________________