3.4.63 \(\int \frac {(a+b x^2)^3 (A+B x^2)}{\sqrt {x}} \, dx\) [363]

Optimal. Leaf size=83 \[ 2 a^3 A \sqrt {x}+\frac {2}{5} a^2 (3 A b+a B) x^{5/2}+\frac {2}{3} a b (A b+a B) x^{9/2}+\frac {2}{13} b^2 (A b+3 a B) x^{13/2}+\frac {2}{17} b^3 B x^{17/2} \]

[Out]

2/5*a^2*(3*A*b+B*a)*x^(5/2)+2/3*a*b*(A*b+B*a)*x^(9/2)+2/13*b^2*(A*b+3*B*a)*x^(13/2)+2/17*b^3*B*x^(17/2)+2*a^3*
A*x^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 83, 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 = {459} \begin {gather*} 2 a^3 A \sqrt {x}+\frac {2}{5} a^2 x^{5/2} (a B+3 A b)+\frac {2}{13} b^2 x^{13/2} (3 a B+A b)+\frac {2}{3} a b x^{9/2} (a B+A b)+\frac {2}{17} b^3 B x^{17/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*a^3*A*Sqrt[x] + (2*a^2*(3*A*b + a*B)*x^(5/2))/5 + (2*a*b*(A*b + a*B)*x^(9/2))/3 + (2*b^2*(A*b + 3*a*B)*x^(13
/2))/13 + (2*b^3*B*x^(17/2))/17

Rule 459

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

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Mathematica [A]
time = 0.04, size = 80, normalized size = 0.96 \begin {gather*} \frac {2 \sqrt {x} \left (663 a^3 \left (5 A+B x^2\right )+221 a^2 b x^2 \left (9 A+5 B x^2\right )+85 a b^2 x^4 \left (13 A+9 B x^2\right )+15 b^3 x^6 \left (17 A+13 B x^2\right )\right )}{3315} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[x]*(663*a^3*(5*A + B*x^2) + 221*a^2*b*x^2*(9*A + 5*B*x^2) + 85*a*b^2*x^4*(13*A + 9*B*x^2) + 15*b^3*x^6
*(17*A + 13*B*x^2)))/3315

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Maple [A]
time = 0.10, size = 76, normalized size = 0.92

method	result	size
derivativedivides	\(\frac {2 b^{3} B \,x^{\frac {17}{2}}}{17}+\frac {2 \left (A \,b^{3}+3 B a \,b^{2}\right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (3 A a \,b^{2}+3 B \,a^{2} b \right ) x^{\frac {9}{2}}}{9}+\frac {2 \left (3 A \,a^{2} b +B \,a^{3}\right ) x^{\frac {5}{2}}}{5}+2 a^{3} A \sqrt {x}\)	\(76\)
default	\(\frac {2 b^{3} B \,x^{\frac {17}{2}}}{17}+\frac {2 \left (A \,b^{3}+3 B a \,b^{2}\right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (3 A a \,b^{2}+3 B \,a^{2} b \right ) x^{\frac {9}{2}}}{9}+\frac {2 \left (3 A \,a^{2} b +B \,a^{3}\right ) x^{\frac {5}{2}}}{5}+2 a^{3} A \sqrt {x}\)	\(76\)
trager	\(\left (\frac {2}{17} B \,b^{3} x^{8}+\frac {2}{13} x^{6} A \,b^{3}+\frac {6}{13} x^{6} B a \,b^{2}+\frac {2}{3} A a \,b^{2} x^{4}+\frac {2}{3} x^{4} B \,a^{2} b +\frac {6}{5} x^{2} A \,a^{2} b +\frac {2}{5} B \,a^{3} x^{2}+2 A \,a^{3}\right ) \sqrt {x}\)	\(79\)
gosper	\(\frac {2 \sqrt {x}\, \left (195 B \,b^{3} x^{8}+255 x^{6} A \,b^{3}+765 x^{6} B a \,b^{2}+1105 A a \,b^{2} x^{4}+1105 x^{4} B \,a^{2} b +1989 x^{2} A \,a^{2} b +663 B \,a^{3} x^{2}+3315 A \,a^{3}\right )}{3315}\)	\(80\)
risch	\(\frac {2 \sqrt {x}\, \left (195 B \,b^{3} x^{8}+255 x^{6} A \,b^{3}+765 x^{6} B a \,b^{2}+1105 A a \,b^{2} x^{4}+1105 x^{4} B \,a^{2} b +1989 x^{2} A \,a^{2} b +663 B \,a^{3} x^{2}+3315 A \,a^{3}\right )}{3315}\)	\(80\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/17*b^3*B*x^(17/2)+2/13*(A*b^3+3*B*a*b^2)*x^(13/2)+2/9*(3*A*a*b^2+3*B*a^2*b)*x^(9/2)+2/5*(3*A*a^2*b+B*a^3)*x^
(5/2)+2*a^3*A*x^(1/2)

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Maxima [A]
time = 0.28, size = 73, normalized size = 0.88 \begin {gather*} \frac {2}{17} \, B b^{3} x^{\frac {17}{2}} + \frac {2}{13} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac {13}{2}} + \frac {2}{3} \, {\left (B a^{2} b + A a b^{2}\right )} x^{\frac {9}{2}} + 2 \, A a^{3} \sqrt {x} + \frac {2}{5} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac {5}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/17*B*b^3*x^(17/2) + 2/13*(3*B*a*b^2 + A*b^3)*x^(13/2) + 2/3*(B*a^2*b + A*a*b^2)*x^(9/2) + 2*A*a^3*sqrt(x) +
2/5*(B*a^3 + 3*A*a^2*b)*x^(5/2)

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Fricas [A]
time = 1.70, size = 75, normalized size = 0.90 \begin {gather*} \frac {2}{3315} \, {\left (195 \, B b^{3} x^{8} + 255 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{6} + 1105 \, {\left (B a^{2} b + A a b^{2}\right )} x^{4} + 3315 \, A a^{3} + 663 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2}\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3315*(195*B*b^3*x^8 + 255*(3*B*a*b^2 + A*b^3)*x^6 + 1105*(B*a^2*b + A*a*b^2)*x^4 + 3315*A*a^3 + 663*(B*a^3 +
3*A*a^2*b)*x^2)*sqrt(x)

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Sympy [A]
time = 0.48, size = 112, normalized size = 1.35 \begin {gather*} 2 A a^{3} \sqrt {x} + \frac {6 A a^{2} b x^{\frac {5}{2}}}{5} + \frac {2 A a b^{2} x^{\frac {9}{2}}}{3} + \frac {2 A b^{3} x^{\frac {13}{2}}}{13} + \frac {2 B a^{3} x^{\frac {5}{2}}}{5} + \frac {2 B a^{2} b x^{\frac {9}{2}}}{3} + \frac {6 B a b^{2} x^{\frac {13}{2}}}{13} + \frac {2 B b^{3} x^{\frac {17}{2}}}{17} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*A*a**3*sqrt(x) + 6*A*a**2*b*x**(5/2)/5 + 2*A*a*b**2*x**(9/2)/3 + 2*A*b**3*x**(13/2)/13 + 2*B*a**3*x**(5/2)/5
+ 2*B*a**2*b*x**(9/2)/3 + 6*B*a*b**2*x**(13/2)/13 + 2*B*b**3*x**(17/2)/17

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Giac [A]
time = 0.64, size = 77, normalized size = 0.93 \begin {gather*} \frac {2}{17} \, B b^{3} x^{\frac {17}{2}} + \frac {6}{13} \, B a b^{2} x^{\frac {13}{2}} + \frac {2}{13} \, A b^{3} x^{\frac {13}{2}} + \frac {2}{3} \, B a^{2} b x^{\frac {9}{2}} + \frac {2}{3} \, A a b^{2} x^{\frac {9}{2}} + \frac {2}{5} \, B a^{3} x^{\frac {5}{2}} + \frac {6}{5} \, A a^{2} b x^{\frac {5}{2}} + 2 \, A a^{3} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/17*B*b^3*x^(17/2) + 6/13*B*a*b^2*x^(13/2) + 2/13*A*b^3*x^(13/2) + 2/3*B*a^2*b*x^(9/2) + 2/3*A*a*b^2*x^(9/2)
+ 2/5*B*a^3*x^(5/2) + 6/5*A*a^2*b*x^(5/2) + 2*A*a^3*sqrt(x)

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Mupad [B]
time = 0.02, size = 69, normalized size = 0.83 \begin {gather*} x^{5/2}\,\left (\frac {2\,B\,a^3}{5}+\frac {6\,A\,b\,a^2}{5}\right )+x^{13/2}\,\left (\frac {2\,A\,b^3}{13}+\frac {6\,B\,a\,b^2}{13}\right )+2\,A\,a^3\,\sqrt {x}+\frac {2\,B\,b^3\,x^{17/2}}{17}+\frac {2\,a\,b\,x^{9/2}\,\left (A\,b+B\,a\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^(5/2)*((2*B*a^3)/5 + (6*A*a^2*b)/5) + x^(13/2)*((2*A*b^3)/13 + (6*B*a*b^2)/13) + 2*A*a^3*x^(1/2) + (2*B*b^3*
x^(17/2))/17 + (2*a*b*x^(9/2)*(A*b + B*a))/3

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