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

Optimal. Leaf size=237 \[ -\frac{c^{3/2} \sqrt{d} \sqrt{a+b x^2} \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right ),1-\frac{b c}{a d}\right )}{3 a^2 \sqrt{c+d x^2} (b c-a d) \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{\sqrt{c+d x^2} (2 b c-a d) E\left (\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|1-\frac{a d}{b c}\right )}{3 a^{3/2} \sqrt{b} \sqrt{a+b x^2} (b c-a d) \sqrt{\frac{a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac{x \sqrt{c+d x^2}}{3 a \left (a+b x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.120085, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {412, 525, 418, 411} \[ -\frac{c^{3/2} \sqrt{d} \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 a^2 \sqrt{c+d x^2} (b c-a d) \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{\sqrt{c+d x^2} (2 b c-a d) E\left (\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|1-\frac{a d}{b c}\right )}{3 a^{3/2} \sqrt{b} \sqrt{a+b x^2} (b c-a d) \sqrt{\frac{a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac{x \sqrt{c+d x^2}}{3 a \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 412

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

Rule 525

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)^(3/2)), x_Symbol] :> Dist[(b*e - a*
f)/(b*c - a*d), Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[Sqrt[a + b
*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[b/a] && PosQ[d/c]

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 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{\sqrt{c+d x^2}}{\left (a+b x^2\right )^{5/2}} \, dx &=\frac{x \sqrt{c+d x^2}}{3 a \left (a+b x^2\right )^{3/2}}-\frac{\int \frac{-2 c-d x^2}{\left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}} \, dx}{3 a}\\ &=\frac{x \sqrt{c+d x^2}}{3 a \left (a+b x^2\right )^{3/2}}-\frac{(c d) \int \frac{1}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{3 a (b c-a d)}+\frac{(2 b c-a d) \int \frac{\sqrt{c+d x^2}}{\left (a+b x^2\right )^{3/2}} \, dx}{3 a (b c-a d)}\\ &=\frac{x \sqrt{c+d x^2}}{3 a \left (a+b x^2\right )^{3/2}}+\frac{(2 b c-a d) \sqrt{c+d x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|1-\frac{a d}{b c}\right )}{3 a^{3/2} \sqrt{b} (b c-a d) \sqrt{a+b x^2} \sqrt{\frac{a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac{c^{3/2} \sqrt{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 a^2 (b c-a 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 = 0.443624, size = 243, normalized size = 1.03 \[ \frac{-2 i c \left (a+b x^2\right ) \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d-b c) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{\frac{b}{a}}\right ),\frac{a d}{b c}\right )+x \sqrt{\frac{b}{a}} \left (c+d x^2\right ) \left (2 a^2 d+a b \left (d x^2-3 c\right )-2 b^2 c x^2\right )+i c \left (a+b x^2\right ) \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d-2 b c) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )}{3 a^2 \sqrt{\frac{b}{a}} \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2} (a d-b c)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.036, size = 617, normalized size = 2.6 \begin{align*}{\frac{1}{ \left ( 3\,ad-3\,bc \right ){a}^{2}} \left ({x}^{5}ab{d}^{2}\sqrt{-{\frac{b}{a}}}-2\,{x}^{5}{b}^{2}cd\sqrt{-{\frac{b}{a}}}+2\,{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){x}^{2}abcd\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}-2\,{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){x}^{2}{b}^{2}{c}^{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}^{2}abcd\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}+2\,{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){x}^{2}{b}^{2}{c}^{2}\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}+2\,{x}^{3}{a}^{2}{d}^{2}\sqrt{-{\frac{b}{a}}}-2\,{x}^{3}abcd\sqrt{-{\frac{b}{a}}}-2\,{x}^{3}{b}^{2}{c}^{2}\sqrt{-{\frac{b}{a}}}+2\,{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){a}^{2}cd\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}-2\,{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) ab{c}^{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 ){a}^{2}cd\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}+2\,{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) ab{c}^{2}\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}+2\,x{a}^{2}cd\sqrt{-{\frac{b}{a}}}-3\,xab{c}^{2}\sqrt{-{\frac{b}{a}}} \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}{\frac{1}{\sqrt{-{\frac{b}{a}}}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(x^5*a*b*d^2*(-b/a)^(1/2)-2*x^5*b^2*c*d*(-b/a)^(1/2)+2*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*x^2*a*b*c
*d*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)-2*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*x^2*b^2*c^2*((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^2*a*b*c*d*((b*x^2+a)/a)^(1/2)*((d*x^2
+c)/c)^(1/2)+2*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*x^2*b^2*c^2*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)+2
*x^3*a^2*d^2*(-b/a)^(1/2)-2*x^3*a*b*c*d*(-b/a)^(1/2)-2*x^3*b^2*c^2*(-b/a)^(1/2)+2*EllipticF(x*(-b/a)^(1/2),(a*
d/b/c)^(1/2))*a^2*c*d*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)-2*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a*b*
c^2*((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))*a^2*c*d*((b*x^2+a)/a)^(1
/2)*((d*x^2+c)/c)^(1/2)+2*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a*b*c^2*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^
(1/2)+2*x*a^2*c*d*(-b/a)^(1/2)-3*x*a*b*c^2*(-b/a)^(1/2))/(d*x^2+c)^(1/2)/(-b/a)^(1/2)/(a*d-b*c)/a^2/(b*x^2+a)^
(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}{b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)/(b^3*x^6 + 3*a*b^2*x^4 + 3*a^2*b*x^2 + a^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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