3.830 \(\int \frac{(a+b x^2)^2 \sqrt{c+d x^2}}{x^{13/2}} \, dx\)

Optimal. Leaf size=217 \[ \frac{2 d^{3/4} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (5 a^2 d^2-22 a b c d+77 b^2 c^2\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right ),\frac{1}{2}\right )}{231 c^{9/4} \sqrt{c+d x^2}}-\frac{2 \sqrt{c+d x^2} \left (5 a^2 d^2-22 a b c d+77 b^2 c^2\right )}{231 c^2 x^{3/2}}-\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11/2}}-\frac{2 a \left (c+d x^2\right )^{3/2} (22 b c-5 a d)}{77 c^2 x^{7/2}} \]

[Out]

(-2*(77*b^2*c^2 - 22*a*b*c*d + 5*a^2*d^2)*Sqrt[c + d*x^2])/(231*c^2*x^(3/2)) - (2*a^2*(c + d*x^2)^(3/2))/(11*c
*x^(11/2)) - (2*a*(22*b*c - 5*a*d)*(c + d*x^2)^(3/2))/(77*c^2*x^(7/2)) + (2*d^(3/4)*(77*b^2*c^2 - 22*a*b*c*d +
5*a^2*d^2)*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*EllipticF[2*ArcTan[(d^(1/4)*Sqrt[x
])/c^(1/4)], 1/2])/(231*c^(9/4)*Sqrt[c + d*x^2])

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Rubi [A]  time = 0.189566, antiderivative size = 213, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {462, 453, 277, 329, 220} \[ \frac{2 d^{3/4} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (5 a^2 d^2-22 a b c d+77 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{231 c^{9/4} \sqrt{c+d x^2}}-\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11/2}}-\frac{2 \sqrt{c+d x^2} \left (77 b^2-\frac{a d (22 b c-5 a d)}{c^2}\right )}{231 x^{3/2}}-\frac{2 a \left (c+d x^2\right )^{3/2} (22 b c-5 a d)}{77 c^2 x^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^(13/2),x]

[Out]

(-2*(77*b^2 - (a*d*(22*b*c - 5*a*d))/c^2)*Sqrt[c + d*x^2])/(231*x^(3/2)) - (2*a^2*(c + d*x^2)^(3/2))/(11*c*x^(
11/2)) - (2*a*(22*b*c - 5*a*d)*(c + d*x^2)^(3/2))/(77*c^2*x^(7/2)) + (2*d^(3/4)*(77*b^2*c^2 - 22*a*b*c*d + 5*a
^2*d^2)*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*EllipticF[2*ArcTan[(d^(1/4)*Sqrt[x])/c
^(1/4)], 1/2])/(231*c^(9/4)*Sqrt[c + d*x^2])

Rule 462

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^2, x_Symbol] :> Simp[(c^2*(e*x)^(
m + 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] - Dist[1/(a*e^n*(m + 1)), Int[(e*x)^(m + n)*(a + b*x^n)^p*Simp[b
*c^2*n*(p + 1) + c*(b*c - 2*a*d)*(m + 1) - a*(m + 1)*d^2*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && Ne
Q[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[m, -1] && GtQ[n, 0]

Rule 453

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

Rule 277

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 220

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2\right )^2 \sqrt{c+d x^2}}{x^{13/2}} \, dx &=-\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11/2}}+\frac{2 \int \frac{\left (\frac{1}{2} a (22 b c-5 a d)+\frac{11}{2} b^2 c x^2\right ) \sqrt{c+d x^2}}{x^{9/2}} \, dx}{11 c}\\ &=-\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11/2}}-\frac{2 a (22 b c-5 a d) \left (c+d x^2\right )^{3/2}}{77 c^2 x^{7/2}}-\frac{1}{77} \left (-77 b^2+\frac{a d (22 b c-5 a d)}{c^2}\right ) \int \frac{\sqrt{c+d x^2}}{x^{5/2}} \, dx\\ &=-\frac{2 \left (77 b^2-\frac{a d (22 b c-5 a d)}{c^2}\right ) \sqrt{c+d x^2}}{231 x^{3/2}}-\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11/2}}-\frac{2 a (22 b c-5 a d) \left (c+d x^2\right )^{3/2}}{77 c^2 x^{7/2}}+\frac{1}{231} \left (2 d \left (77 b^2-\frac{a d (22 b c-5 a d)}{c^2}\right )\right ) \int \frac{1}{\sqrt{x} \sqrt{c+d x^2}} \, dx\\ &=-\frac{2 \left (77 b^2-\frac{a d (22 b c-5 a d)}{c^2}\right ) \sqrt{c+d x^2}}{231 x^{3/2}}-\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11/2}}-\frac{2 a (22 b c-5 a d) \left (c+d x^2\right )^{3/2}}{77 c^2 x^{7/2}}+\frac{1}{231} \left (4 d \left (77 b^2-\frac{a d (22 b c-5 a d)}{c^2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+d x^4}} \, dx,x,\sqrt{x}\right )\\ &=-\frac{2 \left (77 b^2-\frac{a d (22 b c-5 a d)}{c^2}\right ) \sqrt{c+d x^2}}{231 x^{3/2}}-\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11/2}}-\frac{2 a (22 b c-5 a d) \left (c+d x^2\right )^{3/2}}{77 c^2 x^{7/2}}+\frac{2 d^{3/4} \left (77 b^2 c^2-22 a b c d+5 a^2 d^2\right ) \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{231 c^{9/4} \sqrt{c+d x^2}}\\ \end{align*}

Mathematica [C]  time = 0.156693, size = 187, normalized size = 0.86 \[ -\frac{2 \sqrt{c+d x^2} \left (a^2 \left (21 c^2+6 c d x^2-10 d^2 x^4\right )+22 a b c x^2 \left (3 c+2 d x^2\right )+77 b^2 c^2 x^4\right )}{231 c^2 x^{11/2}}+\frac{4 i d x \sqrt{\frac{c}{d x^2}+1} \left (5 a^2 d^2-22 a b c d+77 b^2 c^2\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right ),-1\right )}{231 c^2 \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^(13/2),x]

[Out]

(-2*Sqrt[c + d*x^2]*(77*b^2*c^2*x^4 + 22*a*b*c*x^2*(3*c + 2*d*x^2) + a^2*(21*c^2 + 6*c*d*x^2 - 10*d^2*x^4)))/(
231*c^2*x^(11/2)) + (((4*I)/231)*d*(77*b^2*c^2 - 22*a*b*c*d + 5*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x*EllipticF[I*Arc
Sinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1])/(c^2*Sqrt[(I*Sqrt[c])/Sqrt[d]]*Sqrt[c + d*x^2])

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Maple [A]  time = 0.03, size = 403, normalized size = 1.9 \begin{align*}{\frac{2}{231\,{c}^{2}} \left ( 5\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) \sqrt{-cd}{x}^{5}{a}^{2}{d}^{2}-22\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) \sqrt{-cd}{x}^{5}abcd+77\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) \sqrt{-cd}{x}^{5}{b}^{2}{c}^{2}+10\,{x}^{6}{a}^{2}{d}^{3}-44\,{x}^{6}abc{d}^{2}-77\,{x}^{6}{b}^{2}{c}^{2}d+4\,{x}^{4}{a}^{2}c{d}^{2}-110\,{x}^{4}ab{c}^{2}d-77\,{x}^{4}{b}^{2}{c}^{3}-27\,{x}^{2}{a}^{2}{c}^{2}d-66\,{x}^{2}ab{c}^{3}-21\,{a}^{2}{c}^{3} \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}{x}^{-{\frac{11}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^(13/2),x)

[Out]

2/231/(d*x^2+c)^(1/2)*(5*((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1
/2)*(-x/(-c*d)^(1/2)*d)^(1/2)*EllipticF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*(-c*d)^(1/2)*x^5*
a^2*d^2-22*((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*(-x/(-c*d)
^(1/2)*d)^(1/2)*EllipticF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*(-c*d)^(1/2)*x^5*a*b*c*d+77*((d
*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*(-x/(-c*d)^(1/2)*d)^(1/2
)*EllipticF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*(-c*d)^(1/2)*x^5*b^2*c^2+10*x^6*a^2*d^3-44*x^
6*a*b*c*d^2-77*x^6*b^2*c^2*d+4*x^4*a^2*c*d^2-110*x^4*a*b*c^2*d-77*x^4*b^2*c^3-27*x^2*a^2*c^2*d-66*x^2*a*b*c^3-
21*a^2*c^3)/x^(11/2)/c^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{2} + a\right )}^{2} \sqrt{d x^{2} + c}}{x^{\frac{13}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^(13/2),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^(13/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt{d x^{2} + c}}{x^{\frac{13}{2}}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^(13/2),x, algorithm="fricas")

[Out]

integral((b^2*x^4 + 2*a*b*x^2 + a^2)*sqrt(d*x^2 + c)/x^(13/2), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2*(d*x**2+c)**(1/2)/x**(13/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{2} + a\right )}^{2} \sqrt{d x^{2} + c}}{x^{\frac{13}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^(13/2),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^(13/2), x)