3.963 \(\int \frac{(a+b x^2)^{5/2}}{x^4 \sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=336 \[ \frac{b \sqrt{a+b x^2} (9 b c-a d) \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right ),1-\frac{b c}{a d}\right )}{3 \sqrt{c} \sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{x \sqrt{a+b x^2} \left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right )}{3 c^2 \sqrt{c+d x^2}}-\frac{\sqrt{a+b x^2} \left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 c^{3/2} \sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{2 a \sqrt{a+b x^2} \sqrt{c+d x^2} (3 b c-a d)}{3 c^2 x}-\frac{a \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{3 c x^3} \]

[Out]

((3*b^2*c^2 + 7*a*b*c*d - 2*a^2*d^2)*x*Sqrt[a + b*x^2])/(3*c^2*Sqrt[c + d*x^2]) - (2*a*(3*b*c - a*d)*Sqrt[a +
b*x^2]*Sqrt[c + d*x^2])/(3*c^2*x) - (a*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(3*c*x^3) - ((3*b^2*c^2 + 7*a*b*c*d
- 2*a^2*d^2)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*c^(3/2)*Sqrt[d]*Sqrt[
(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (b*(9*b*c - a*d)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]
*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*Sqrt[c]*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])

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Rubi [A]  time = 0.291161, antiderivative size = 336, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {474, 580, 531, 418, 492, 411} \[ \frac{x \sqrt{a+b x^2} \left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right )}{3 c^2 \sqrt{c+d x^2}}-\frac{\sqrt{a+b x^2} \left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 c^{3/2} \sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{2 a \sqrt{a+b x^2} \sqrt{c+d x^2} (3 b c-a d)}{3 c^2 x}-\frac{a \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{3 c x^3}+\frac{b \sqrt{a+b x^2} (9 b c-a d) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 \sqrt{c} \sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(5/2)/(x^4*Sqrt[c + d*x^2]),x]

[Out]

((3*b^2*c^2 + 7*a*b*c*d - 2*a^2*d^2)*x*Sqrt[a + b*x^2])/(3*c^2*Sqrt[c + d*x^2]) - (2*a*(3*b*c - a*d)*Sqrt[a +
b*x^2]*Sqrt[c + d*x^2])/(3*c^2*x) - (a*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(3*c*x^3) - ((3*b^2*c^2 + 7*a*b*c*d
- 2*a^2*d^2)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*c^(3/2)*Sqrt[d]*Sqrt[
(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (b*(9*b*c - a*d)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]
*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*Sqrt[c]*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])

Rule 474

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

Rule 580

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(a*g*(m + 1)), x] - Dist[1/(a*g^n*(m + 1
)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c*(p + 1) + a*d*q)
+ d*((b*e - a*f)*(m + 1) + b*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && IGtQ[n
, 0] && GtQ[q, 0] && LtQ[m, -1] &&  !(EqQ[q, 1] && SimplerQ[e + f*x^n, c + d*x^n])

Rule 531

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Dist[
e, Int[(a + b*x^n)^p*(c + d*x^n)^q, x], x] + Dist[f, Int[x^n*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, n, p, q}, x]

Rule 418

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(Sqrt[a + b*x^2]*EllipticF[ArcT
an[Rt[d/c, 2]*x], 1 - (b*c)/(a*d)])/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]), x] /
; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]

Rule 492

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(x*Sqrt[a + b*x^2])/(b*Sqr
t[c + d*x^2]), x] - Dist[c/b, Int[Sqrt[a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b
*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]

Rule 411

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(Sqrt[a + b*x^2]*EllipticE[ArcTan
[Rt[d/c, 2]*x], 1 - (b*c)/(a*d)])/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]), x] /;
FreeQ[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2\right )^{5/2}}{x^4 \sqrt{c+d x^2}} \, dx &=-\frac{a \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{3 c x^3}+\frac{\int \frac{\sqrt{a+b x^2} \left (2 a (3 b c-a d)+b (3 b c+a d) x^2\right )}{x^2 \sqrt{c+d x^2}} \, dx}{3 c}\\ &=-\frac{2 a (3 b c-a d) \sqrt{a+b x^2} \sqrt{c+d x^2}}{3 c^2 x}-\frac{a \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{3 c x^3}+\frac{\int \frac{a b c (9 b c-a d)+b \left (3 b^2 c^2+7 a b c d-2 a^2 d^2\right ) x^2}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{3 c^2}\\ &=-\frac{2 a (3 b c-a d) \sqrt{a+b x^2} \sqrt{c+d x^2}}{3 c^2 x}-\frac{a \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{3 c x^3}+\frac{(a b (9 b c-a d)) \int \frac{1}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{3 c}+\frac{\left (b \left (3 b^2 c^2+7 a b c d-2 a^2 d^2\right )\right ) \int \frac{x^2}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{3 c^2}\\ &=\frac{\left (3 b^2 c^2+7 a b c d-2 a^2 d^2\right ) x \sqrt{a+b x^2}}{3 c^2 \sqrt{c+d x^2}}-\frac{2 a (3 b c-a d) \sqrt{a+b x^2} \sqrt{c+d x^2}}{3 c^2 x}-\frac{a \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{3 c x^3}+\frac{b (9 b c-a d) \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 \sqrt{c} \sqrt{d} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{c+d x^2}}-\frac{\left (3 b^2 c^2+7 a b c d-2 a^2 d^2\right ) \int \frac{\sqrt{a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{3 c}\\ &=\frac{\left (3 b^2 c^2+7 a b c d-2 a^2 d^2\right ) x \sqrt{a+b x^2}}{3 c^2 \sqrt{c+d x^2}}-\frac{2 a (3 b c-a d) \sqrt{a+b x^2} \sqrt{c+d x^2}}{3 c^2 x}-\frac{a \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{3 c x^3}-\frac{\left (3 b^2 c^2+7 a b c d-2 a^2 d^2\right ) \sqrt{a+b x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 c^{3/2} \sqrt{d} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{c+d x^2}}+\frac{b (9 b c-a d) \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 \sqrt{c} \sqrt{d} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{c+d x^2}}\\ \end{align*}

Mathematica [C]  time = 0.454699, size = 261, normalized size = 0.78 \[ \frac{-i b c x^3 \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (a^2 d^2+2 a b c d-3 b^2 c^2\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{\frac{b}{a}}\right ),\frac{a d}{b c}\right )+i b c x^3 \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (2 a^2 d^2-7 a b c d-3 b^2 c^2\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )+a d \sqrt{\frac{b}{a}} \left (a+b x^2\right ) \left (c+d x^2\right ) \left (-a c+2 a d x^2-7 b c x^2\right )}{3 c^2 d x^3 \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(5/2)/(x^4*Sqrt[c + d*x^2]),x]

[Out]

(a*Sqrt[b/a]*d*(a + b*x^2)*(c + d*x^2)*(-(a*c) - 7*b*c*x^2 + 2*a*d*x^2) + I*b*c*(-3*b^2*c^2 - 7*a*b*c*d + 2*a^
2*d^2)*x^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*b*c*(-3*
b^2*c^2 + 2*a*b*c*d + a^2*d^2)*x^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (
a*d)/(b*c)])/(3*Sqrt[b/a]*c^2*d*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])

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Maple [A]  time = 0.019, size = 583, normalized size = 1.7 \begin{align*}{\frac{1}{ \left ( 3\,bd{x}^{4}+3\,ad{x}^{2}+3\,bc{x}^{2}+3\,ac \right ){c}^{2}d{x}^{3}}\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c} \left ( 2\,\sqrt{-{\frac{b}{a}}}{x}^{6}{a}^{2}b{d}^{3}-7\,\sqrt{-{\frac{b}{a}}}{x}^{6}a{b}^{2}c{d}^{2}+\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){x}^{3}{a}^{2}bc{d}^{2}+2\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){x}^{3}a{b}^{2}{c}^{2}d-3\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){x}^{3}{b}^{3}{c}^{3}-2\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){x}^{3}{a}^{2}bc{d}^{2}+7\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){x}^{3}a{b}^{2}{c}^{2}d+3\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){x}^{3}{b}^{3}{c}^{3}+2\,\sqrt{-{\frac{b}{a}}}{x}^{4}{a}^{3}{d}^{3}-6\,\sqrt{-{\frac{b}{a}}}{x}^{4}{a}^{2}bc{d}^{2}-7\,\sqrt{-{\frac{b}{a}}}{x}^{4}a{b}^{2}{c}^{2}d+\sqrt{-{\frac{b}{a}}}{x}^{2}{a}^{3}c{d}^{2}-8\,\sqrt{-{\frac{b}{a}}}{x}^{2}{a}^{2}b{c}^{2}d-\sqrt{-{\frac{b}{a}}}{a}^{3}{c}^{2}d \right ){\frac{1}{\sqrt{-{\frac{b}{a}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(5/2)/x^4/(d*x^2+c)^(1/2),x)

[Out]

1/3*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(2*(-b/a)^(1/2)*x^6*a^2*b*d^3-7*(-b/a)^(1/2)*x^6*a*b^2*c*d^2+((b*x^2+a)/a)
^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*x^3*a^2*b*c*d^2+2*((b*x^2+a)/a)^(1/2)*((d
*x^2+c)/c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*x^3*a*b^2*c^2*d-3*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)
^(1/2)*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*x^3*b^3*c^3-2*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*Ellipti
cE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*x^3*a^2*b*c*d^2+7*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticE(x*(-b/a
)^(1/2),(a*d/b/c)^(1/2))*x^3*a*b^2*c^2*d+3*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticE(x*(-b/a)^(1/2),(a
*d/b/c)^(1/2))*x^3*b^3*c^3+2*(-b/a)^(1/2)*x^4*a^3*d^3-6*(-b/a)^(1/2)*x^4*a^2*b*c*d^2-7*(-b/a)^(1/2)*x^4*a*b^2*
c^2*d+(-b/a)^(1/2)*x^2*a^3*c*d^2-8*(-b/a)^(1/2)*x^2*a^2*b*c^2*d-(-b/a)^(1/2)*a^3*c^2*d)/(b*d*x^4+a*d*x^2+b*c*x
^2+a*c)/c^2/(-b/a)^(1/2)/d/x^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{2} + a\right )}^{\frac{5}{2}}}{\sqrt{d x^{2} + c} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(5/2)/x^4/(d*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^(5/2)/(sqrt(d*x^2 + c)*x^4), 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{b x^{2} + a} \sqrt{d x^{2} + c}}{d x^{6} + c x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(5/2)/x^4/(d*x^2+c)^(1/2),x, algorithm="fricas")

[Out]

integral((b^2*x^4 + 2*a*b*x^2 + a^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)/(d*x^6 + c*x^4), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x^{2}\right )^{\frac{5}{2}}}{x^{4} \sqrt{c + d x^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(5/2)/x**4/(d*x**2+c)**(1/2),x)

[Out]

Integral((a + b*x**2)**(5/2)/(x**4*sqrt(c + d*x**2)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{2} + a\right )}^{\frac{5}{2}}}{\sqrt{d x^{2} + c} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(5/2)/x^4/(d*x^2+c)^(1/2),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^(5/2)/(sqrt(d*x^2 + c)*x^4), x)