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

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

[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

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Rubi [A]  time = 0.0403782, antiderivative size = 83, normalized size of antiderivative = 1., 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} \[ \frac{2}{5} a^2 x^{5/2} (a B+3 A b)+2 a^3 A \sqrt{x}+\frac{2}{13} b^2 x^{13/2} (3 a B+A b)+\frac{2}{3} a b x^{9/2} (a B+A b)+\frac{2}{17} b^3 B x^{17/2} \]

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 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^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*}

Mathematica [A]  time = 0.0406482, size = 71, normalized size = 0.86 \[ \frac{2 \sqrt{x} \left (663 a^2 x^2 (a B+3 A b)+3315 a^3 A+255 b^2 x^6 (3 a B+A b)+1105 a b x^4 (a B+A b)+195 b^3 B x^8\right )}{3315} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[x]*(3315*a^3*A + 663*a^2*(3*A*b + a*B)*x^2 + 1105*a*b*(A*b + a*B)*x^4 + 255*b^2*(A*b + 3*a*B)*x^6 + 19
5*b^3*B*x^8))/3315

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Maple [A]  time = 0.005, size = 80, normalized size = 1. \begin{align*}{\frac{390\,{b}^{3}B{x}^{8}+510\,{x}^{6}{b}^{3}A+1530\,{x}^{6}a{b}^{2}B+2210\,{x}^{4}a{b}^{2}A+2210\,{x}^{4}{a}^{2}bB+3978\,{x}^{2}A{a}^{2}b+1326\,{x}^{2}B{a}^{3}+6630\,{a}^{3}A}{3315}\sqrt{x}} \end{align*}

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)

[Out]

2/3315*x^(1/2)*(195*B*b^3*x^8+255*A*b^3*x^6+765*B*a*b^2*x^6+1105*A*a*b^2*x^4+1105*B*a^2*b*x^4+1989*A*a^2*b*x^2
+663*B*a^3*x^2+3315*A*a^3)

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Maxima [A]  time = 1.47263, size = 99, normalized size = 1.19 \begin{align*} \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{align*}

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 = 0.700195, size = 184, normalized size = 2.22 \begin{align*} \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{align*}

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 = 4.95304, size = 112, normalized size = 1.35 \begin{align*} 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{align*}

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 = 1.11773, size = 104, normalized size = 1.25 \begin{align*} \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{align*}

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)