3.7.1 \(\int \frac {(c+d x)^{3/2}}{\sqrt {a+\frac {b}{x^2}}} \, dx\) [601]

Optimal. Leaf size=406 \[ \frac {2 c \sqrt {c+d x} \left (b+a x^2\right )}{5 a \sqrt {a+\frac {b}{x^2}} x}+\frac {2 (c+d x)^{3/2} \left (b+a x^2\right )}{5 a \sqrt {a+\frac {b}{x^2}} x}+\frac {2 \sqrt {b} \left (a c^2-3 b d^2\right ) \sqrt {c+d x} \sqrt {1+\frac {a x^2}{b}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {-a} x}{\sqrt {b}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-a} \sqrt {b} d}{a c-\sqrt {-a} \sqrt {b} d}\right )}{5 (-a)^{3/2} d \sqrt {a+\frac {b}{x^2}} x \sqrt {\frac {a (c+d x)}{a c-\sqrt {-a} \sqrt {b} d}}}-\frac {2 \sqrt {b} c \left (a c^2+b d^2\right ) \sqrt {\frac {a (c+d x)}{a c-\sqrt {-a} \sqrt {b} d}} \sqrt {1+\frac {a x^2}{b}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {-a} x}{\sqrt {b}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-a} \sqrt {b} d}{a c-\sqrt {-a} \sqrt {b} d}\right )}{5 (-a)^{3/2} d \sqrt {a+\frac {b}{x^2}} x \sqrt {c+d x}} \]

[Out]

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

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Rubi [A]
time = 0.32, antiderivative size = 406, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1464, 847, 858, 733, 435, 430} \begin {gather*} -\frac {2 \sqrt {b} c \sqrt {\frac {a x^2}{b}+1} \left (a c^2+b d^2\right ) \sqrt {\frac {a (c+d x)}{a c-\sqrt {-a} \sqrt {b} d}} F\left (\text {ArcSin}\left (\frac {\sqrt {1-\frac {\sqrt {-a} x}{\sqrt {b}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-a} \sqrt {b} d}{a c-\sqrt {-a} \sqrt {b} d}\right )}{5 (-a)^{3/2} d x \sqrt {a+\frac {b}{x^2}} \sqrt {c+d x}}+\frac {2 \sqrt {b} \sqrt {\frac {a x^2}{b}+1} \sqrt {c+d x} \left (a c^2-3 b d^2\right ) E\left (\text {ArcSin}\left (\frac {\sqrt {1-\frac {\sqrt {-a} x}{\sqrt {b}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-a} \sqrt {b} d}{a c-\sqrt {-a} \sqrt {b} d}\right )}{5 (-a)^{3/2} d x \sqrt {a+\frac {b}{x^2}} \sqrt {\frac {a (c+d x)}{a c-\sqrt {-a} \sqrt {b} d}}}+\frac {2 \left (a x^2+b\right ) (c+d x)^{3/2}}{5 a x \sqrt {a+\frac {b}{x^2}}}+\frac {2 c \left (a x^2+b\right ) \sqrt {c+d x}}{5 a x \sqrt {a+\frac {b}{x^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^(3/2)/Sqrt[a + b/x^2],x]

[Out]

(2*c*Sqrt[c + d*x]*(b + a*x^2))/(5*a*Sqrt[a + b/x^2]*x) + (2*(c + d*x)^(3/2)*(b + a*x^2))/(5*a*Sqrt[a + b/x^2]
*x) + (2*Sqrt[b]*(a*c^2 - 3*b*d^2)*Sqrt[c + d*x]*Sqrt[1 + (a*x^2)/b]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[-a]*x)/Sq
rt[b]]/Sqrt[2]], (-2*Sqrt[-a]*Sqrt[b]*d)/(a*c - Sqrt[-a]*Sqrt[b]*d)])/(5*(-a)^(3/2)*d*Sqrt[a + b/x^2]*x*Sqrt[(
a*(c + d*x))/(a*c - Sqrt[-a]*Sqrt[b]*d)]) - (2*Sqrt[b]*c*(a*c^2 + b*d^2)*Sqrt[(a*(c + d*x))/(a*c - Sqrt[-a]*Sq
rt[b]*d)]*Sqrt[1 + (a*x^2)/b]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[-a]*x)/Sqrt[b]]/Sqrt[2]], (-2*Sqrt[-a]*Sqrt[b]*d
)/(a*c - Sqrt[-a]*Sqrt[b]*d)])/(5*(-a)^(3/2)*d*Sqrt[a + b/x^2]*x*Sqrt[c + d*x])

Rule 430

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]
))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && Gt
Q[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0
]

Rule 733

Int[((d_) + (e_.)*(x_))^(m_)/Sqrt[(a_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2*a*Rt[-c/a, 2]*(d + e*x)^m*(Sqrt[1
+ c*(x^2/a)]/(c*Sqrt[a + c*x^2]*(c*((d + e*x)/(c*d - a*e*Rt[-c/a, 2])))^m)), Subst[Int[(1 + 2*a*e*Rt[-c/a, 2]*
(x^2/(c*d - a*e*Rt[-c/a, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(1 - Rt[-c/a, 2]*x)/2]], x] /; FreeQ[{a, c, d, e},
x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m^2, 1/4]

Rule 847

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g*(d + e*x)^
m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*
Simp[c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p
}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p]) &&  !(IGtQ[m, 0] && EqQ[f, 0])

Rule 858

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rule 1464

Int[((a_.) + (c_.)*(x_)^(mn2_.))^(p_)*((d_) + (e_.)*(x_)^(n_.))^(q_.), x_Symbol] :> Dist[x^(2*n*FracPart[p])*(
(a + c/x^(2*n))^FracPart[p]/(c + a*x^(2*n))^FracPart[p]), Int[((d + e*x^n)^q*(c + a*x^(2*n))^p)/x^(2*n*p), x],
x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[mn2, -2*n] &&  !IntegerQ[p] &&  !IntegerQ[q] && PosQ[n]

Rubi steps

\begin {align*} \int \frac {(c+d x)^{3/2}}{\sqrt {a+\frac {b}{x^2}}} \, dx &=\frac {\sqrt {b+a x^2} \int \frac {x (c+d x)^{3/2}}{\sqrt {b+a x^2}} \, dx}{\sqrt {a+\frac {b}{x^2}} x}\\ &=\frac {2 (c+d x)^{3/2} \left (b+a x^2\right )}{5 a \sqrt {a+\frac {b}{x^2}} x}+\frac {\left (2 \sqrt {b+a x^2}\right ) \int \frac {\left (-\frac {3 b d}{2}+\frac {3 a c x}{2}\right ) \sqrt {c+d x}}{\sqrt {b+a x^2}} \, dx}{5 a \sqrt {a+\frac {b}{x^2}} x}\\ &=\frac {2 c \sqrt {c+d x} \left (b+a x^2\right )}{5 a \sqrt {a+\frac {b}{x^2}} x}+\frac {2 (c+d x)^{3/2} \left (b+a x^2\right )}{5 a \sqrt {a+\frac {b}{x^2}} x}+\frac {\left (4 \sqrt {b+a x^2}\right ) \int \frac {-3 a b c d+\frac {3}{4} a \left (a c^2-3 b d^2\right ) x}{\sqrt {c+d x} \sqrt {b+a x^2}} \, dx}{15 a^2 \sqrt {a+\frac {b}{x^2}} x}\\ &=\frac {2 c \sqrt {c+d x} \left (b+a x^2\right )}{5 a \sqrt {a+\frac {b}{x^2}} x}+\frac {2 (c+d x)^{3/2} \left (b+a x^2\right )}{5 a \sqrt {a+\frac {b}{x^2}} x}+\frac {\left (\left (a c^2-3 b d^2\right ) \sqrt {b+a x^2}\right ) \int \frac {\sqrt {c+d x}}{\sqrt {b+a x^2}} \, dx}{5 a d \sqrt {a+\frac {b}{x^2}} x}-\frac {\left (c \left (a c^2+b d^2\right ) \sqrt {b+a x^2}\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {b+a x^2}} \, dx}{5 a d \sqrt {a+\frac {b}{x^2}} x}\\ &=\frac {2 c \sqrt {c+d x} \left (b+a x^2\right )}{5 a \sqrt {a+\frac {b}{x^2}} x}+\frac {2 (c+d x)^{3/2} \left (b+a x^2\right )}{5 a \sqrt {a+\frac {b}{x^2}} x}+\frac {\left (2 \sqrt {-a} \sqrt {b} \left (a c^2-3 b d^2\right ) \sqrt {c+d x} \sqrt {1+\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 \sqrt {-a} \sqrt {b} d x^2}{a c-\sqrt {-a} \sqrt {b} d}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {-a} x}{\sqrt {b}}}}{\sqrt {2}}\right )}{5 a^2 d \sqrt {a+\frac {b}{x^2}} x \sqrt {\frac {a (c+d x)}{a c-\sqrt {-a} \sqrt {b} d}}}-\frac {\left (2 \sqrt {-a} \sqrt {b} c \left (a c^2+b d^2\right ) \sqrt {\frac {a (c+d x)}{a c-\sqrt {-a} \sqrt {b} d}} \sqrt {1+\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 \sqrt {-a} \sqrt {b} d x^2}{a c-\sqrt {-a} \sqrt {b} d}}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {-a} x}{\sqrt {b}}}}{\sqrt {2}}\right )}{5 a^2 d \sqrt {a+\frac {b}{x^2}} x \sqrt {c+d x}}\\ &=\frac {2 c \sqrt {c+d x} \left (b+a x^2\right )}{5 a \sqrt {a+\frac {b}{x^2}} x}+\frac {2 (c+d x)^{3/2} \left (b+a x^2\right )}{5 a \sqrt {a+\frac {b}{x^2}} x}+\frac {2 \sqrt {b} \left (a c^2-3 b d^2\right ) \sqrt {c+d x} \sqrt {1+\frac {a x^2}{b}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {-a} x}{\sqrt {b}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-a} \sqrt {b} d}{a c-\sqrt {-a} \sqrt {b} d}\right )}{5 (-a)^{3/2} d \sqrt {a+\frac {b}{x^2}} x \sqrt {\frac {a (c+d x)}{a c-\sqrt {-a} \sqrt {b} d}}}-\frac {2 \sqrt {b} c \left (a c^2+b d^2\right ) \sqrt {\frac {a (c+d x)}{a c-\sqrt {-a} \sqrt {b} d}} \sqrt {1+\frac {a x^2}{b}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {-a} x}{\sqrt {b}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-a} \sqrt {b} d}{a c-\sqrt {-a} \sqrt {b} d}\right )}{5 (-a)^{3/2} d \sqrt {a+\frac {b}{x^2}} x \sqrt {c+d x}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 21.94, size = 540, normalized size = 1.33 \begin {gather*} \frac {\sqrt {c+d x} \left (\frac {2 (2 c+d x) \left (b+a x^2\right )}{a}+\frac {2 \left (d^2 \sqrt {-c-\frac {i \sqrt {b} d}{\sqrt {a}}} \left (-3 b^2 d^2+a^2 c^2 x^2+a b \left (c^2-3 d^2 x^2\right )\right )+\sqrt {a} \left (-i a^{3/2} c^3+a \sqrt {b} c^2 d+3 i \sqrt {a} b c d^2-3 b^{3/2} d^3\right ) \sqrt {\frac {d \left (\frac {i \sqrt {b}}{\sqrt {a}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {i \sqrt {b} d}{\sqrt {a}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \sinh ^{-1}\left (\frac {\sqrt {-c-\frac {i \sqrt {b} d}{\sqrt {a}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {a} c-i \sqrt {b} d}{\sqrt {a} c+i \sqrt {b} d}\right )-\sqrt {a} \sqrt {b} d \left (a c^2+4 i \sqrt {a} \sqrt {b} c d-3 b d^2\right ) \sqrt {\frac {d \left (\frac {i \sqrt {b}}{\sqrt {a}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {i \sqrt {b} d}{\sqrt {a}}-d x}{c+d x}} (c+d x)^{3/2} F\left (i \sinh ^{-1}\left (\frac {\sqrt {-c-\frac {i \sqrt {b} d}{\sqrt {a}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {a} c-i \sqrt {b} d}{\sqrt {a} c+i \sqrt {b} d}\right )\right )}{a^2 d^2 \sqrt {-c-\frac {i \sqrt {b} d}{\sqrt {a}}} (c+d x)}\right )}{5 \sqrt {a+\frac {b}{x^2}} x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c + d*x]*((2*(2*c + d*x)*(b + a*x^2))/a + (2*(d^2*Sqrt[-c - (I*Sqrt[b]*d)/Sqrt[a]]*(-3*b^2*d^2 + a^2*c^2
*x^2 + a*b*(c^2 - 3*d^2*x^2)) + Sqrt[a]*((-I)*a^(3/2)*c^3 + a*Sqrt[b]*c^2*d + (3*I)*Sqrt[a]*b*c*d^2 - 3*b^(3/2
)*d^3)*Sqrt[(d*((I*Sqrt[b])/Sqrt[a] + x))/(c + d*x)]*Sqrt[-(((I*Sqrt[b]*d)/Sqrt[a] - d*x)/(c + d*x))]*(c + d*x
)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-c - (I*Sqrt[b]*d)/Sqrt[a]]/Sqrt[c + d*x]], (Sqrt[a]*c - I*Sqrt[b]*d)/(Sqrt[a
]*c + I*Sqrt[b]*d)] - Sqrt[a]*Sqrt[b]*d*(a*c^2 + (4*I)*Sqrt[a]*Sqrt[b]*c*d - 3*b*d^2)*Sqrt[(d*((I*Sqrt[b])/Sqr
t[a] + x))/(c + d*x)]*Sqrt[-(((I*Sqrt[b]*d)/Sqrt[a] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticF[I*ArcSinh[Sqr
t[-c - (I*Sqrt[b]*d)/Sqrt[a]]/Sqrt[c + d*x]], (Sqrt[a]*c - I*Sqrt[b]*d)/(Sqrt[a]*c + I*Sqrt[b]*d)]))/(a^2*d^2*
Sqrt[-c - (I*Sqrt[b]*d)/Sqrt[a]]*(c + d*x))))/(5*Sqrt[a + b/x^2]*x)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1144\) vs. \(2(330)=660\).
time = 0.36, size = 1145, normalized size = 2.82

method	result	size
risch	\(\frac {2 \left (d x +2 c \right ) \left (a \,x^{2}+b \right ) \sqrt {d x +c}}{5 a \sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, x}+\frac {\left (\frac {2 \left (c^{2} a -3 d^{2} b \right ) \left (\frac {c}{d}-\frac {\sqrt {-a b}}{a}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {-a b}}{a}}}\, \sqrt {\frac {x -\frac {\sqrt {-a b}}{a}}{-\frac {c}{d}-\frac {\sqrt {-a b}}{a}}}\, \sqrt {\frac {x +\frac {\sqrt {-a b}}{a}}{-\frac {c}{d}+\frac {\sqrt {-a b}}{a}}}\, \left (\left (-\frac {c}{d}-\frac {\sqrt {-a b}}{a}\right ) \EllipticE \left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {-a b}}{a}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {-a b}}{a}}{-\frac {c}{d}-\frac {\sqrt {-a b}}{a}}}\right )+\frac {\sqrt {-a b}\, \EllipticF \left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {-a b}}{a}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {-a b}}{a}}{-\frac {c}{d}-\frac {\sqrt {-a b}}{a}}}\right )}{a}\right )}{\sqrt {a d \,x^{3}+a c \,x^{2}+b d x +b c}}-\frac {8 b c d \left (\frac {c}{d}-\frac {\sqrt {-a b}}{a}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {-a b}}{a}}}\, \sqrt {\frac {x -\frac {\sqrt {-a b}}{a}}{-\frac {c}{d}-\frac {\sqrt {-a b}}{a}}}\, \sqrt {\frac {x +\frac {\sqrt {-a b}}{a}}{-\frac {c}{d}+\frac {\sqrt {-a b}}{a}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {-a b}}{a}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {-a b}}{a}}{-\frac {c}{d}-\frac {\sqrt {-a b}}{a}}}\right )}{\sqrt {a d \,x^{3}+a c \,x^{2}+b d x +b c}}\right ) \sqrt {\left (a \,x^{2}+b \right ) \left (d x +c \right )}}{5 a \sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, x \sqrt {d x +c}}\)	\(623\)
default	\(\frac {\frac {2 \sqrt {-\frac {\left (d x +c \right ) a}{\sqrt {-a b}\, d -a c}}\, \sqrt {\frac {\left (-a x +\sqrt {-a b}\right ) d}{\sqrt {-a b}\, d +a c}}\, \sqrt {\frac {\left (a x +\sqrt {-a b}\right ) d}{\sqrt {-a b}\, d -a c}}\, \EllipticF \left (\sqrt {-\frac {\left (d x +c \right ) a}{\sqrt {-a b}\, d -a c}}, \sqrt {-\frac {\sqrt {-a b}\, d -a c}{\sqrt {-a b}\, d +a c}}\right ) \sqrt {-a b}\, a \,c^{3} d}{5}+\frac {2 \sqrt {-\frac {\left (d x +c \right ) a}{\sqrt {-a b}\, d -a c}}\, \sqrt {\frac {\left (-a x +\sqrt {-a b}\right ) d}{\sqrt {-a b}\, d +a c}}\, \sqrt {\frac {\left (a x +\sqrt {-a b}\right ) d}{\sqrt {-a b}\, d -a c}}\, \EllipticF \left (\sqrt {-\frac {\left (d x +c \right ) a}{\sqrt {-a b}\, d -a c}}, \sqrt {-\frac {\sqrt {-a b}\, d -a c}{\sqrt {-a b}\, d +a c}}\right ) \sqrt {-a b}\, b c \,d^{3}}{5}-\frac {6 \sqrt {-\frac {\left (d x +c \right ) a}{\sqrt {-a b}\, d -a c}}\, \sqrt {\frac {\left (-a x +\sqrt {-a b}\right ) d}{\sqrt {-a b}\, d +a c}}\, \sqrt {\frac {\left (a x +\sqrt {-a b}\right ) d}{\sqrt {-a b}\, d -a c}}\, \EllipticF \left (\sqrt {-\frac {\left (d x +c \right ) a}{\sqrt {-a b}\, d -a c}}, \sqrt {-\frac {\sqrt {-a b}\, d -a c}{\sqrt {-a b}\, d +a c}}\right ) a b \,c^{2} d^{2}}{5}-\frac {6 \sqrt {-\frac {\left (d x +c \right ) a}{\sqrt {-a b}\, d -a c}}\, \sqrt {\frac {\left (-a x +\sqrt {-a b}\right ) d}{\sqrt {-a b}\, d +a c}}\, \sqrt {\frac {\left (a x +\sqrt {-a b}\right ) d}{\sqrt {-a b}\, d -a c}}\, \EllipticF \left (\sqrt {-\frac {\left (d x +c \right ) a}{\sqrt {-a b}\, d -a c}}, \sqrt {-\frac {\sqrt {-a b}\, d -a c}{\sqrt {-a b}\, d +a c}}\right ) b^{2} d^{4}}{5}-\frac {2 \sqrt {-\frac {\left (d x +c \right ) a}{\sqrt {-a b}\, d -a c}}\, \sqrt {\frac {\left (-a x +\sqrt {-a b}\right ) d}{\sqrt {-a b}\, d +a c}}\, \sqrt {\frac {\left (a x +\sqrt {-a b}\right ) d}{\sqrt {-a b}\, d -a c}}\, \EllipticE \left (\sqrt {-\frac {\left (d x +c \right ) a}{\sqrt {-a b}\, d -a c}}, \sqrt {-\frac {\sqrt {-a b}\, d -a c}{\sqrt {-a b}\, d +a c}}\right ) a^{2} c^{4}}{5}+\frac {4 \sqrt {-\frac {\left (d x +c \right ) a}{\sqrt {-a b}\, d -a c}}\, \sqrt {\frac {\left (-a x +\sqrt {-a b}\right ) d}{\sqrt {-a b}\, d +a c}}\, \sqrt {\frac {\left (a x +\sqrt {-a b}\right ) d}{\sqrt {-a b}\, d -a c}}\, \EllipticE \left (\sqrt {-\frac {\left (d x +c \right ) a}{\sqrt {-a b}\, d -a c}}, \sqrt {-\frac {\sqrt {-a b}\, d -a c}{\sqrt {-a b}\, d +a c}}\right ) a b \,c^{2} d^{2}}{5}+\frac {6 \sqrt {-\frac {\left (d x +c \right ) a}{\sqrt {-a b}\, d -a c}}\, \sqrt {\frac {\left (-a x +\sqrt {-a b}\right ) d}{\sqrt {-a b}\, d +a c}}\, \sqrt {\frac {\left (a x +\sqrt {-a b}\right ) d}{\sqrt {-a b}\, d -a c}}\, \EllipticE \left (\sqrt {-\frac {\left (d x +c \right ) a}{\sqrt {-a b}\, d -a c}}, \sqrt {-\frac {\sqrt {-a b}\, d -a c}{\sqrt {-a b}\, d +a c}}\right ) b^{2} d^{4}}{5}+\frac {2 a^{2} d^{4} x^{4}}{5}+\frac {6 a^{2} c \,d^{3} x^{3}}{5}+\frac {4 a^{2} c^{2} d^{2} x^{2}}{5}+\frac {2 a b \,d^{4} x^{2}}{5}+\frac {6 a b c \,d^{3} x}{5}+\frac {4 a b \,c^{2} d^{2}}{5}}{\sqrt {d x +c}\, d^{2} a^{2} x \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}\)	\(1145\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(3/2)/(a+b/x^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/5*((-(d*x+c)*a/((-a*b)^(1/2)*d-a*c))^(1/2)*((-a*x+(-a*b)^(1/2))*d/((-a*b)^(1/2)*d+a*c))^(1/2)*((a*x+(-a*b)^(
1/2))*d/((-a*b)^(1/2)*d-a*c))^(1/2)*EllipticF((-(d*x+c)*a/((-a*b)^(1/2)*d-a*c))^(1/2),(-((-a*b)^(1/2)*d-a*c)/(
(-a*b)^(1/2)*d+a*c))^(1/2))*(-a*b)^(1/2)*a*c^3*d+(-(d*x+c)*a/((-a*b)^(1/2)*d-a*c))^(1/2)*((-a*x+(-a*b)^(1/2))*
d/((-a*b)^(1/2)*d+a*c))^(1/2)*((a*x+(-a*b)^(1/2))*d/((-a*b)^(1/2)*d-a*c))^(1/2)*EllipticF((-(d*x+c)*a/((-a*b)^
(1/2)*d-a*c))^(1/2),(-((-a*b)^(1/2)*d-a*c)/((-a*b)^(1/2)*d+a*c))^(1/2))*(-a*b)^(1/2)*b*c*d^3-3*(-(d*x+c)*a/((-
a*b)^(1/2)*d-a*c))^(1/2)*((-a*x+(-a*b)^(1/2))*d/((-a*b)^(1/2)*d+a*c))^(1/2)*((a*x+(-a*b)^(1/2))*d/((-a*b)^(1/2
)*d-a*c))^(1/2)*EllipticF((-(d*x+c)*a/((-a*b)^(1/2)*d-a*c))^(1/2),(-((-a*b)^(1/2)*d-a*c)/((-a*b)^(1/2)*d+a*c))
^(1/2))*a*b*c^2*d^2-3*(-(d*x+c)*a/((-a*b)^(1/2)*d-a*c))^(1/2)*((-a*x+(-a*b)^(1/2))*d/((-a*b)^(1/2)*d+a*c))^(1/
2)*((a*x+(-a*b)^(1/2))*d/((-a*b)^(1/2)*d-a*c))^(1/2)*EllipticF((-(d*x+c)*a/((-a*b)^(1/2)*d-a*c))^(1/2),(-((-a*
b)^(1/2)*d-a*c)/((-a*b)^(1/2)*d+a*c))^(1/2))*b^2*d^4-(-(d*x+c)*a/((-a*b)^(1/2)*d-a*c))^(1/2)*((-a*x+(-a*b)^(1/
2))*d/((-a*b)^(1/2)*d+a*c))^(1/2)*((a*x+(-a*b)^(1/2))*d/((-a*b)^(1/2)*d-a*c))^(1/2)*EllipticE((-(d*x+c)*a/((-a
*b)^(1/2)*d-a*c))^(1/2),(-((-a*b)^(1/2)*d-a*c)/((-a*b)^(1/2)*d+a*c))^(1/2))*a^2*c^4+2*(-(d*x+c)*a/((-a*b)^(1/2
)*d-a*c))^(1/2)*((-a*x+(-a*b)^(1/2))*d/((-a*b)^(1/2)*d+a*c))^(1/2)*((a*x+(-a*b)^(1/2))*d/((-a*b)^(1/2)*d-a*c))
^(1/2)*EllipticE((-(d*x+c)*a/((-a*b)^(1/2)*d-a*c))^(1/2),(-((-a*b)^(1/2)*d-a*c)/((-a*b)^(1/2)*d+a*c))^(1/2))*a
*b*c^2*d^2+3*(-(d*x+c)*a/((-a*b)^(1/2)*d-a*c))^(1/2)*((-a*x+(-a*b)^(1/2))*d/((-a*b)^(1/2)*d+a*c))^(1/2)*((a*x+
(-a*b)^(1/2))*d/((-a*b)^(1/2)*d-a*c))^(1/2)*EllipticE((-(d*x+c)*a/((-a*b)^(1/2)*d-a*c))^(1/2),(-((-a*b)^(1/2)*
d-a*c)/((-a*b)^(1/2)*d+a*c))^(1/2))*b^2*d^4+a^2*d^4*x^4+3*a^2*c*d^3*x^3+2*a^2*c^2*d^2*x^2+a*b*d^4*x^2+3*a*b*c*
d^3*x+2*a*b*c^2*d^2)/(d*x+c)^(1/2)/d^2/a^2/x/((a*x^2+b)/x^2)^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(3/2)/(a+b/x^2)^(1/2),x, algorithm="maxima")

[Out]

integrate((d*x + c)^(3/2)/sqrt(a + b/x^2), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.10, size = 235, normalized size = 0.58 \begin {gather*} -\frac {2 \, {\left ({\left (a c^{3} + 9 \, b c d^{2}\right )} \sqrt {a d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (a c^{2} - 3 \, b d^{2}\right )}}{3 \, a d^{2}}, -\frac {8 \, {\left (a c^{3} + 9 \, b c d^{2}\right )}}{27 \, a d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) + 3 \, {\left (a c^{2} d - 3 \, b d^{3}\right )} \sqrt {a d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (a c^{2} - 3 \, b d^{2}\right )}}{3 \, a d^{2}}, -\frac {8 \, {\left (a c^{3} + 9 \, b c d^{2}\right )}}{27 \, a d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (a c^{2} - 3 \, b d^{2}\right )}}{3 \, a d^{2}}, -\frac {8 \, {\left (a c^{3} + 9 \, b c d^{2}\right )}}{27 \, a d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) - 3 \, {\left (a d^{3} x^{2} + 2 \, a c d^{2} x\right )} \sqrt {d x + c} \sqrt {\frac {a x^{2} + b}{x^{2}}}\right )}}{15 \, a^{2} d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(3/2)/(a+b/x^2)^(1/2),x, algorithm="fricas")

[Out]

-2/15*((a*c^3 + 9*b*c*d^2)*sqrt(a*d)*weierstrassPInverse(4/3*(a*c^2 - 3*b*d^2)/(a*d^2), -8/27*(a*c^3 + 9*b*c*d
^2)/(a*d^3), 1/3*(3*d*x + c)/d) + 3*(a*c^2*d - 3*b*d^3)*sqrt(a*d)*weierstrassZeta(4/3*(a*c^2 - 3*b*d^2)/(a*d^2
), -8/27*(a*c^3 + 9*b*c*d^2)/(a*d^3), weierstrassPInverse(4/3*(a*c^2 - 3*b*d^2)/(a*d^2), -8/27*(a*c^3 + 9*b*c*
d^2)/(a*d^3), 1/3*(3*d*x + c)/d)) - 3*(a*d^3*x^2 + 2*a*c*d^2*x)*sqrt(d*x + c)*sqrt((a*x^2 + b)/x^2))/(a^2*d^2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c + d x\right )^{\frac {3}{2}}}{\sqrt {a + \frac {b}{x^{2}}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**(3/2)/(a+b/x**2)**(1/2),x)

[Out]

Integral((c + d*x)**(3/2)/sqrt(a + b/x**2), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(3/2)/(a+b/x^2)^(1/2),x, algorithm="giac")

[Out]

integrate((d*x + c)^(3/2)/sqrt(a + b/x^2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^{3/2}}{\sqrt {a+\frac {b}{x^2}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^(3/2)/(a + b/x^2)^(1/2),x)

[Out]

int((c + d*x)^(3/2)/(a + b/x^2)^(1/2), x)

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