3.6.100 \(\int \sqrt {c x} (a+b x^2)^{3/2} \, dx\) [600]

Optimal. Leaf size=297 \[ \frac {4 a (c x)^{3/2} \sqrt {a+b x^2}}{15 c}+\frac {8 a^2 \sqrt {c x} \sqrt {a+b x^2}}{15 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {2 (c x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 c}-\frac {8 a^{9/4} \sqrt {c} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a+b x^2}}+\frac {4 a^{9/4} \sqrt {c} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a+b x^2}} \]

[Out]

2/9*(c*x)^(3/2)*(b*x^2+a)^(3/2)/c+4/15*a*(c*x)^(3/2)*(b*x^2+a)^(1/2)/c+8/15*a^2*(c*x)^(1/2)*(b*x^2+a)^(1/2)/b^
(1/2)/(a^(1/2)+x*b^(1/2))-8/15*a^(9/4)*(cos(2*arctan(b^(1/4)*(c*x)^(1/2)/a^(1/4)/c^(1/2)))^2)^(1/2)/cos(2*arct
an(b^(1/4)*(c*x)^(1/2)/a^(1/4)/c^(1/2)))*EllipticE(sin(2*arctan(b^(1/4)*(c*x)^(1/2)/a^(1/4)/c^(1/2))),1/2*2^(1
/2))*(a^(1/2)+x*b^(1/2))*c^(1/2)*((b*x^2+a)/(a^(1/2)+x*b^(1/2))^2)^(1/2)/b^(3/4)/(b*x^2+a)^(1/2)+4/15*a^(9/4)*
(cos(2*arctan(b^(1/4)*(c*x)^(1/2)/a^(1/4)/c^(1/2)))^2)^(1/2)/cos(2*arctan(b^(1/4)*(c*x)^(1/2)/a^(1/4)/c^(1/2))
)*EllipticF(sin(2*arctan(b^(1/4)*(c*x)^(1/2)/a^(1/4)/c^(1/2))),1/2*2^(1/2))*(a^(1/2)+x*b^(1/2))*c^(1/2)*((b*x^
2+a)/(a^(1/2)+x*b^(1/2))^2)^(1/2)/b^(3/4)/(b*x^2+a)^(1/2)

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Rubi [A]
time = 0.16, antiderivative size = 297, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {285, 335, 311, 226, 1210} \begin {gather*} \frac {4 a^{9/4} \sqrt {c} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a+b x^2}}-\frac {8 a^{9/4} \sqrt {c} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a+b x^2}}+\frac {8 a^2 \sqrt {c x} \sqrt {a+b x^2}}{15 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {4 a (c x)^{3/2} \sqrt {a+b x^2}}{15 c}+\frac {2 (c x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[c*x]*(a + b*x^2)^(3/2),x]

[Out]

(4*a*(c*x)^(3/2)*Sqrt[a + b*x^2])/(15*c) + (8*a^2*Sqrt[c*x]*Sqrt[a + b*x^2])/(15*Sqrt[b]*(Sqrt[a] + Sqrt[b]*x)
) + (2*(c*x)^(3/2)*(a + b*x^2)^(3/2))/(9*c) - (8*a^(9/4)*Sqrt[c]*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[
a] + Sqrt[b]*x)^2]*EllipticE[2*ArcTan[(b^(1/4)*Sqrt[c*x])/(a^(1/4)*Sqrt[c])], 1/2])/(15*b^(3/4)*Sqrt[a + b*x^2
]) + (4*a^(9/4)*Sqrt[c]*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticF[2*ArcTan[(b^
(1/4)*Sqrt[c*x])/(a^(1/4)*Sqrt[c])], 1/2])/(15*b^(3/4)*Sqrt[a + b*x^2])

Rule 226

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)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 285

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 311

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

Rule 335

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 1210

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a +
c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4
]))*EllipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]

Rubi steps

\begin {align*} \int \sqrt {c x} \left (a+b x^2\right )^{3/2} \, dx &=\frac {2 (c x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 c}+\frac {1}{3} (2 a) \int \sqrt {c x} \sqrt {a+b x^2} \, dx\\ &=\frac {4 a (c x)^{3/2} \sqrt {a+b x^2}}{15 c}+\frac {2 (c x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 c}+\frac {1}{15} \left (4 a^2\right ) \int \frac {\sqrt {c x}}{\sqrt {a+b x^2}} \, dx\\ &=\frac {4 a (c x)^{3/2} \sqrt {a+b x^2}}{15 c}+\frac {2 (c x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 c}+\frac {\left (8 a^2\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {a+\frac {b x^4}{c^2}}} \, dx,x,\sqrt {c x}\right )}{15 c}\\ &=\frac {4 a (c x)^{3/2} \sqrt {a+b x^2}}{15 c}+\frac {2 (c x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 c}+\frac {\left (8 a^{5/2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{c^2}}} \, dx,x,\sqrt {c x}\right )}{15 \sqrt {b}}-\frac {\left (8 a^{5/2}\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a} c}}{\sqrt {a+\frac {b x^4}{c^2}}} \, dx,x,\sqrt {c x}\right )}{15 \sqrt {b}}\\ &=\frac {4 a (c x)^{3/2} \sqrt {a+b x^2}}{15 c}+\frac {8 a^2 \sqrt {c x} \sqrt {a+b x^2}}{15 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {2 (c x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 c}-\frac {8 a^{9/4} \sqrt {c} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a+b x^2}}+\frac {4 a^{9/4} \sqrt {c} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a+b x^2}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.01, size = 57, normalized size = 0.19 \begin {gather*} \frac {2 a x \sqrt {c x} \sqrt {a+b x^2} \, _2F_1\left (-\frac {3}{2},\frac {3}{4};\frac {7}{4};-\frac {b x^2}{a}\right )}{3 \sqrt {1+\frac {b x^2}{a}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*x*Sqrt[c*x]*Sqrt[a + b*x^2]*Hypergeometric2F1[-3/2, 3/4, 7/4, -((b*x^2)/a)])/(3*Sqrt[1 + (b*x^2)/a])

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Maple [A]
time = 0.04, size = 218, normalized size = 0.73

method	result	size
default	\(\frac {2 \sqrt {c x}\, \left (5 b^{3} x^{6}+12 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticE \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{3}-6 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{3}+16 a \,b^{2} x^{4}+11 a^{2} b \,x^{2}\right )}{45 \sqrt {b \,x^{2}+a}\, b x}\)	\(218\)
risch	\(\frac {2 x^{2} \left (5 b \,x^{2}+11 a \right ) \sqrt {b \,x^{2}+a}\, c}{45 \sqrt {c x}}+\frac {4 a^{2} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right ) c \sqrt {c x \left (b \,x^{2}+a \right )}}{15 b \sqrt {b c \,x^{3}+a c x}\, \sqrt {c x}\, \sqrt {b \,x^{2}+a}}\)	\(222\)
elliptic	\(\frac {\sqrt {c x}\, \sqrt {c x \left (b \,x^{2}+a \right )}\, \left (\frac {2 b \,x^{3} \sqrt {b c \,x^{3}+a c x}}{9}+\frac {22 a x \sqrt {b c \,x^{3}+a c x}}{45}+\frac {4 a^{2} c \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{15 b \sqrt {b c \,x^{3}+a c x}}\right )}{c x \sqrt {b \,x^{2}+a}}\)	\(235\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45*(c*x)^(1/2)/(b*x^2+a)^(1/2)/b*(5*b^3*x^6+12*((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*2^(1/2)*((-b*x+(-a*b)
^(1/2))/(-a*b)^(1/2))^(1/2)*(-x*b/(-a*b)^(1/2))^(1/2)*EllipticE(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^
(1/2))*a^3-6*((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*2^(1/2)*((-b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-x*b/(-
a*b)^(1/2))^(1/2)*EllipticF(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*a^3+16*a*b^2*x^4+11*a^2*b*x^2
)/x

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.33, size = 63, normalized size = 0.21 \begin {gather*} -\frac {2 \, {\left (12 \, \sqrt {b c} a^{2} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) - {\left (5 \, b^{2} x^{3} + 11 \, a b x\right )} \sqrt {b x^{2} + a} \sqrt {c x}\right )}}{45 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/45*(12*sqrt(b*c)*a^2*weierstrassZeta(-4*a/b, 0, weierstrassPInverse(-4*a/b, 0, x)) - (5*b^2*x^3 + 11*a*b*x)
*sqrt(b*x^2 + a)*sqrt(c*x))/b

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Sympy [C] Result contains complex when optimal does not.
time = 1.39, size = 46, normalized size = 0.15 \begin {gather*} \frac {a^{\frac {3}{2}} \sqrt {c} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {7}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x)^(1/2)*(a + b*x^2)^(3/2),x)

[Out]

int((c*x)^(1/2)*(a + b*x^2)^(3/2), x)

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