3.828 \(\int \frac{(a+b x^2)^2 \sqrt{c+d x^2}}{x^{9/2}} \, dx\)

Optimal. Leaf size=213 \[ \frac{2 \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (a d (14 b c-a d)+7 b^2 c^2\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right ),\frac{1}{2}\right )}{21 c^{5/4} \sqrt [4]{d} \sqrt{c+d x^2}}-\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{7 c x^{7/2}}+\frac{2 \sqrt{x} \sqrt{c+d x^2} \left (a d (14 b c-a d)+7 b^2 c^2\right )}{21 c^2}-\frac{2 a \left (c+d x^2\right )^{3/2} (14 b c-a d)}{21 c^2 x^{3/2}} \]

[Out]

(2*(7*b^2*c^2 + a*d*(14*b*c - a*d))*Sqrt[x]*Sqrt[c + d*x^2])/(21*c^2) - (2*a^2*(c + d*x^2)^(3/2))/(7*c*x^(7/2)
) - (2*a*(14*b*c - a*d)*(c + d*x^2)^(3/2))/(21*c^2*x^(3/2)) + (2*(7*b^2*c^2 + a*d*(14*b*c - a*d))*(Sqrt[c] + S
qrt[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])/(21*c
^(5/4)*d^(1/4)*Sqrt[c + d*x^2])

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Rubi [A]  time = 0.170769, antiderivative size = 210, normalized size of antiderivative = 0.99, 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, 279, 329, 220} \[ -\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{7 c x^{7/2}}+\frac{2}{21} \sqrt{x} \sqrt{c+d x^2} \left (\frac{a d (14 b c-a d)}{c^2}+7 b^2\right )+\frac{2 \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (a d (14 b c-a d)+7 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{21 c^{5/4} \sqrt [4]{d} \sqrt{c+d x^2}}-\frac{2 a \left (c+d x^2\right )^{3/2} (14 b c-a d)}{21 c^2 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^(9/2),x]

[Out]

(2*(7*b^2 + (a*d*(14*b*c - a*d))/c^2)*Sqrt[x]*Sqrt[c + d*x^2])/21 - (2*a^2*(c + d*x^2)^(3/2))/(7*c*x^(7/2)) -
(2*a*(14*b*c - a*d)*(c + d*x^2)^(3/2))/(21*c^2*x^(3/2)) + (2*(7*b^2*c^2 + a*d*(14*b*c - a*d))*(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])/(21*c^(5/
4)*d^(1/4)*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 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
n*p + 1)), x] + Dist[(a*n*p)/(m + n*p + 1), Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x]
&& IGtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 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^{9/2}} \, dx &=-\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{7 c x^{7/2}}+\frac{2 \int \frac{\left (\frac{1}{2} a (14 b c-a d)+\frac{7}{2} b^2 c x^2\right ) \sqrt{c+d x^2}}{x^{5/2}} \, dx}{7 c}\\ &=-\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{7 c x^{7/2}}-\frac{2 a (14 b c-a d) \left (c+d x^2\right )^{3/2}}{21 c^2 x^{3/2}}+\frac{1}{7} \left (7 b^2+\frac{a d (14 b c-a d)}{c^2}\right ) \int \frac{\sqrt{c+d x^2}}{\sqrt{x}} \, dx\\ &=\frac{2}{21} \left (7 b^2+\frac{a d (14 b c-a d)}{c^2}\right ) \sqrt{x} \sqrt{c+d x^2}-\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{7 c x^{7/2}}-\frac{2 a (14 b c-a d) \left (c+d x^2\right )^{3/2}}{21 c^2 x^{3/2}}+\frac{1}{21} \left (2 c \left (7 b^2+\frac{a d (14 b c-a d)}{c^2}\right )\right ) \int \frac{1}{\sqrt{x} \sqrt{c+d x^2}} \, dx\\ &=\frac{2}{21} \left (7 b^2+\frac{a d (14 b c-a d)}{c^2}\right ) \sqrt{x} \sqrt{c+d x^2}-\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{7 c x^{7/2}}-\frac{2 a (14 b c-a d) \left (c+d x^2\right )^{3/2}}{21 c^2 x^{3/2}}+\frac{1}{21} \left (4 c \left (7 b^2+\frac{a d (14 b c-a d)}{c^2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+d x^4}} \, dx,x,\sqrt{x}\right )\\ &=\frac{2}{21} \left (7 b^2+\frac{a d (14 b c-a d)}{c^2}\right ) \sqrt{x} \sqrt{c+d x^2}-\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{7 c x^{7/2}}-\frac{2 a (14 b c-a d) \left (c+d x^2\right )^{3/2}}{21 c^2 x^{3/2}}+\frac{2 c^{3/4} \left (7 b^2+\frac{a d (14 b c-a d)}{c^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 )}{21 \sqrt [4]{d} \sqrt{c+d x^2}}\\ \end{align*}

Mathematica [C]  time = 0.204546, size = 160, normalized size = 0.75 \[ \frac{2 \left (\left (c+d x^2\right ) \left (-a^2 \left (3 c+2 d x^2\right )-14 a b c x^2+7 b^2 c x^4\right )+\frac{2 i x^{9/2} \sqrt{\frac{c}{d x^2}+1} \left (-a^2 d^2+14 a b c d+7 b^2 c^2\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right ),-1\right )}{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}\right )}{21 c x^{7/2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^(9/2),x]

[Out]

(2*((c + d*x^2)*(-14*a*b*c*x^2 + 7*b^2*c*x^4 - a^2*(3*c + 2*d*x^2)) + ((2*I)*(7*b^2*c^2 + 14*a*b*c*d - a^2*d^2
)*Sqrt[1 + c/(d*x^2)]*x^(9/2)*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1])/Sqrt[(I*Sqrt[c])/Sq
rt[d]]))/(21*c*x^(7/2)*Sqrt[c + d*x^2])

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Maple [A]  time = 0.046, size = 385, normalized size = 1.8 \begin{align*} -{\frac{2}{21\,cd} \left ( \sqrt{{ \left ( dx+\sqrt{-cd} \right ){\frac{1}{\sqrt{-cd}}}}}\sqrt{2}\sqrt{{ \left ( -dx+\sqrt{-cd} \right ){\frac{1}{\sqrt{-cd}}}}}\sqrt{-{dx{\frac{1}{\sqrt{-cd}}}}}{\it EllipticF} \left ( \sqrt{{ \left ( dx+\sqrt{-cd} \right ){\frac{1}{\sqrt{-cd}}}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{-cd}{x}^{3}{a}^{2}{d}^{2}-14\,\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}^{3}abcd-7\,\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}^{3}{b}^{2}{c}^{2}-7\,{x}^{6}{b}^{2}c{d}^{2}+2\,{x}^{4}{a}^{2}{d}^{3}+14\,{x}^{4}abc{d}^{2}-7\,{x}^{4}{b}^{2}{c}^{2}d+5\,{x}^{2}{a}^{2}c{d}^{2}+14\,{x}^{2}ab{c}^{2}d+3\,{a}^{2}{c}^{2}d \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}{x}^{-{\frac{7}{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^(9/2),x)

[Out]

-2/21/(d*x^2+c)^(1/2)*(((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^3*a^
2*d^2-14*((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^3*a*b*c*d-7*((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)*E
llipticF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*(-c*d)^(1/2)*x^3*b^2*c^2-7*x^6*b^2*c*d^2+2*x^4*a
^2*d^3+14*x^4*a*b*c*d^2-7*x^4*b^2*c^2*d+5*x^2*a^2*c*d^2+14*x^2*a*b*c^2*d+3*a^2*c^2*d)/x^(7/2)/d/c

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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{9}{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^(9/2),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^(9/2), 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{9}{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^(9/2),x, algorithm="fricas")

[Out]

integral((b^2*x^4 + 2*a*b*x^2 + a^2)*sqrt(d*x^2 + c)/x^(9/2), x)

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Sympy [C]  time = 140.602, size = 144, normalized size = 0.68 \begin{align*} \frac{a^{2} \sqrt{c} \Gamma \left (- \frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{7}{4}, - \frac{1}{2} \\ - \frac{3}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{2 x^{\frac{7}{2}} \Gamma \left (- \frac{3}{4}\right )} + \frac{a b \sqrt{c} \Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, - \frac{1}{2} \\ \frac{1}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{x^{\frac{3}{2}} \Gamma \left (\frac{1}{4}\right )} + \frac{b^{2} \sqrt{c} \sqrt{x} \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{2 \Gamma \left (\frac{5}{4}\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**(9/2),x)

[Out]

a**2*sqrt(c)*gamma(-7/4)*hyper((-7/4, -1/2), (-3/4,), d*x**2*exp_polar(I*pi)/c)/(2*x**(7/2)*gamma(-3/4)) + a*b
*sqrt(c)*gamma(-3/4)*hyper((-3/4, -1/2), (1/4,), d*x**2*exp_polar(I*pi)/c)/(x**(3/2)*gamma(1/4)) + b**2*sqrt(c
)*sqrt(x)*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), d*x**2*exp_polar(I*pi)/c)/(2*gamma(5/4))

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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{9}{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^(9/2),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^(9/2), x)