\(\int \frac {(A+B x^2) (b x^2+c x^4)^{3/2}}{\sqrt {x}} \, dx\) [239]

Optimal result

Integrand size = 28, antiderivative size = 408 \[ \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{\sqrt {x}} \, dx=\frac {8 b^3 (7 b B-17 A c) x^{3/2} \left (b+c x^2\right )}{1105 c^{5/2} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}-\frac {8 b^2 (7 b B-17 A c) \sqrt {x} \sqrt {b x^2+c x^4}}{3315 c^2}-\frac {4 b (7 b B-17 A c) x^{5/2} \sqrt {b x^2+c x^4}}{663 c}-\frac {2 (7 b B-17 A c) \sqrt {x} \left (b x^2+c x^4\right )^{3/2}}{221 c}+\frac {2 B \left (b x^2+c x^4\right )^{5/2}}{17 c x^{3/2}}-\frac {8 b^{13/4} (7 b B-17 A c) x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{1105 c^{11/4} \sqrt {b x^2+c x^4}}+\frac {4 b^{13/4} (7 b B-17 A c) x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{1105 c^{11/4} \sqrt {b x^2+c x^4}} \] Output:

8/1105*b^3*(-17*A*c+7*B*b)*x^(3/2)*(c*x^2+b)/c^(5/2)/(b^(1/2)+c^(1/2)*x)/( 
c*x^4+b*x^2)^(1/2)-8/3315*b^2*(-17*A*c+7*B*b)*x^(1/2)*(c*x^4+b*x^2)^(1/2)/ 
c^2-4/663*b*(-17*A*c+7*B*b)*x^(5/2)*(c*x^4+b*x^2)^(1/2)/c-2/221*(-17*A*c+7 
*B*b)*x^(1/2)*(c*x^4+b*x^2)^(3/2)/c+2/17*B*(c*x^4+b*x^2)^(5/2)/c/x^(3/2)-8 
/1105*b^(13/4)*(-17*A*c+7*B*b)*x*(b^(1/2)+c^(1/2)*x)*((c*x^2+b)/(b^(1/2)+c 
^(1/2)*x)^2)^(1/2)*EllipticE(sin(2*arctan(c^(1/4)*x^(1/2)/b^(1/4))),1/2*2^ 
(1/2))/c^(11/4)/(c*x^4+b*x^2)^(1/2)+4/1105*b^(13/4)*(-17*A*c+7*B*b)*x*(b^( 
1/2)+c^(1/2)*x)*((c*x^2+b)/(b^(1/2)+c^(1/2)*x)^2)^(1/2)*InverseJacobiAM(2* 
arctan(c^(1/4)*x^(1/2)/b^(1/4)),1/2*2^(1/2))/c^(11/4)/(c*x^4+b*x^2)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.12 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.28 \[ \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{\sqrt {x}} \, dx=\frac {2 \sqrt {x} \sqrt {x^2 \left (b+c x^2\right )} \left (-\left (b+c x^2\right )^2 \sqrt {1+\frac {c x^2}{b}} \left (7 b B-17 A c-13 B c x^2\right )+b^2 (7 b B-17 A c) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {3}{4},\frac {7}{4},-\frac {c x^2}{b}\right )\right )}{221 c^2 \sqrt {1+\frac {c x^2}{b}}} \] Input:

Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/Sqrt[x],x]
 

Output:

(2*Sqrt[x]*Sqrt[x^2*(b + c*x^2)]*(-((b + c*x^2)^2*Sqrt[1 + (c*x^2)/b]*(7*b 
*B - 17*A*c - 13*B*c*x^2)) + b^2*(7*b*B - 17*A*c)*Hypergeometric2F1[-3/2, 
3/4, 7/4, -((c*x^2)/b)]))/(221*c^2*Sqrt[1 + (c*x^2)/b])
 

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 383, normalized size of antiderivative = 0.94, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {1945, 1426, 1426, 1429, 1431, 266, 834, 27, 761, 1510}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{\sqrt {x}} \, dx\)

\(\Big \downarrow \) 1945

\(\displaystyle \frac {2 B \left (b x^2+c x^4\right )^{5/2}}{17 c x^{3/2}}-\frac {(7 b B-17 A c) \int \frac {\left (c x^4+b x^2\right )^{3/2}}{\sqrt {x}}dx}{17 c}\)

\(\Big \downarrow \) 1426

\(\displaystyle \frac {2 B \left (b x^2+c x^4\right )^{5/2}}{17 c x^{3/2}}-\frac {(7 b B-17 A c) \left (\frac {6}{13} b \int x^{3/2} \sqrt {c x^4+b x^2}dx+\frac {2}{13} \sqrt {x} \left (b x^2+c x^4\right )^{3/2}\right )}{17 c}\)

\(\Big \downarrow \) 1426

\(\displaystyle \frac {2 B \left (b x^2+c x^4\right )^{5/2}}{17 c x^{3/2}}-\frac {(7 b B-17 A c) \left (\frac {6}{13} b \left (\frac {2}{9} b \int \frac {x^{7/2}}{\sqrt {c x^4+b x^2}}dx+\frac {2}{9} x^{5/2} \sqrt {b x^2+c x^4}\right )+\frac {2}{13} \sqrt {x} \left (b x^2+c x^4\right )^{3/2}\right )}{17 c}\)

\(\Big \downarrow \) 1429

\(\displaystyle \frac {2 B \left (b x^2+c x^4\right )^{5/2}}{17 c x^{3/2}}-\frac {(7 b B-17 A c) \left (\frac {6}{13} b \left (\frac {2}{9} b \left (\frac {2 \sqrt {x} \sqrt {b x^2+c x^4}}{5 c}-\frac {3 b \int \frac {x^{3/2}}{\sqrt {c x^4+b x^2}}dx}{5 c}\right )+\frac {2}{9} x^{5/2} \sqrt {b x^2+c x^4}\right )+\frac {2}{13} \sqrt {x} \left (b x^2+c x^4\right )^{3/2}\right )}{17 c}\)

\(\Big \downarrow \) 1431

\(\displaystyle \frac {2 B \left (b x^2+c x^4\right )^{5/2}}{17 c x^{3/2}}-\frac {(7 b B-17 A c) \left (\frac {6}{13} b \left (\frac {2}{9} b \left (\frac {2 \sqrt {x} \sqrt {b x^2+c x^4}}{5 c}-\frac {3 b x \sqrt {b+c x^2} \int \frac {\sqrt {x}}{\sqrt {c x^2+b}}dx}{5 c \sqrt {b x^2+c x^4}}\right )+\frac {2}{9} x^{5/2} \sqrt {b x^2+c x^4}\right )+\frac {2}{13} \sqrt {x} \left (b x^2+c x^4\right )^{3/2}\right )}{17 c}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {2 B \left (b x^2+c x^4\right )^{5/2}}{17 c x^{3/2}}-\frac {(7 b B-17 A c) \left (\frac {6}{13} b \left (\frac {2}{9} b \left (\frac {2 \sqrt {x} \sqrt {b x^2+c x^4}}{5 c}-\frac {6 b x \sqrt {b+c x^2} \int \frac {x}{\sqrt {c x^2+b}}d\sqrt {x}}{5 c \sqrt {b x^2+c x^4}}\right )+\frac {2}{9} x^{5/2} \sqrt {b x^2+c x^4}\right )+\frac {2}{13} \sqrt {x} \left (b x^2+c x^4\right )^{3/2}\right )}{17 c}\)

\(\Big \downarrow \) 834

\(\displaystyle \frac {2 B \left (b x^2+c x^4\right )^{5/2}}{17 c x^{3/2}}-\frac {(7 b B-17 A c) \left (\frac {6}{13} b \left (\frac {2}{9} b \left (\frac {2 \sqrt {x} \sqrt {b x^2+c x^4}}{5 c}-\frac {6 b x \sqrt {b+c x^2} \left (\frac {\sqrt {b} \int \frac {1}{\sqrt {c x^2+b}}d\sqrt {x}}{\sqrt {c}}-\frac {\sqrt {b} \int \frac {\sqrt {b}-\sqrt {c} x}{\sqrt {b} \sqrt {c x^2+b}}d\sqrt {x}}{\sqrt {c}}\right )}{5 c \sqrt {b x^2+c x^4}}\right )+\frac {2}{9} x^{5/2} \sqrt {b x^2+c x^4}\right )+\frac {2}{13} \sqrt {x} \left (b x^2+c x^4\right )^{3/2}\right )}{17 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 B \left (b x^2+c x^4\right )^{5/2}}{17 c x^{3/2}}-\frac {(7 b B-17 A c) \left (\frac {6}{13} b \left (\frac {2}{9} b \left (\frac {2 \sqrt {x} \sqrt {b x^2+c x^4}}{5 c}-\frac {6 b x \sqrt {b+c x^2} \left (\frac {\sqrt {b} \int \frac {1}{\sqrt {c x^2+b}}d\sqrt {x}}{\sqrt {c}}-\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{\sqrt {c x^2+b}}d\sqrt {x}}{\sqrt {c}}\right )}{5 c \sqrt {b x^2+c x^4}}\right )+\frac {2}{9} x^{5/2} \sqrt {b x^2+c x^4}\right )+\frac {2}{13} \sqrt {x} \left (b x^2+c x^4\right )^{3/2}\right )}{17 c}\)

\(\Big \downarrow \) 761

\(\displaystyle \frac {2 B \left (b x^2+c x^4\right )^{5/2}}{17 c x^{3/2}}-\frac {(7 b B-17 A c) \left (\frac {6}{13} b \left (\frac {2}{9} b \left (\frac {2 \sqrt {x} \sqrt {b x^2+c x^4}}{5 c}-\frac {6 b x \sqrt {b+c x^2} \left (\frac {\sqrt [4]{b} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{2 c^{3/4} \sqrt {b+c x^2}}-\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{\sqrt {c x^2+b}}d\sqrt {x}}{\sqrt {c}}\right )}{5 c \sqrt {b x^2+c x^4}}\right )+\frac {2}{9} x^{5/2} \sqrt {b x^2+c x^4}\right )+\frac {2}{13} \sqrt {x} \left (b x^2+c x^4\right )^{3/2}\right )}{17 c}\)

\(\Big \downarrow \) 1510

\(\displaystyle \frac {2 B \left (b x^2+c x^4\right )^{5/2}}{17 c x^{3/2}}-\frac {(7 b B-17 A c) \left (\frac {6}{13} b \left (\frac {2}{9} b \left (\frac {2 \sqrt {x} \sqrt {b x^2+c x^4}}{5 c}-\frac {6 b x \sqrt {b+c x^2} \left (\frac {\sqrt [4]{b} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{2 c^{3/4} \sqrt {b+c x^2}}-\frac {\frac {\sqrt [4]{b} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {b+c x^2}}-\frac {\sqrt {x} \sqrt {b+c x^2}}{\sqrt {b}+\sqrt {c} x}}{\sqrt {c}}\right )}{5 c \sqrt {b x^2+c x^4}}\right )+\frac {2}{9} x^{5/2} \sqrt {b x^2+c x^4}\right )+\frac {2}{13} \sqrt {x} \left (b x^2+c x^4\right )^{3/2}\right )}{17 c}\)

Input:

Int[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/Sqrt[x],x]
 

Output:

(2*B*(b*x^2 + c*x^4)^(5/2))/(17*c*x^(3/2)) - ((7*b*B - 17*A*c)*((2*Sqrt[x] 
*(b*x^2 + c*x^4)^(3/2))/13 + (6*b*((2*x^(5/2)*Sqrt[b*x^2 + c*x^4])/9 + (2* 
b*((2*Sqrt[x]*Sqrt[b*x^2 + c*x^4])/(5*c) - (6*b*x*Sqrt[b + c*x^2]*(-((-((S 
qrt[x]*Sqrt[b + c*x^2])/(Sqrt[b] + Sqrt[c]*x)) + (b^(1/4)*(Sqrt[b] + Sqrt[ 
c]*x)*Sqrt[(b + c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4 
)*Sqrt[x])/b^(1/4)], 1/2])/(c^(1/4)*Sqrt[b + c*x^2]))/Sqrt[c]) + (b^(1/4)* 
(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2]*EllipticF[ 
2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(2*c^(3/4)*Sqrt[b + c*x^2])))/( 
5*c*Sqrt[b*x^2 + c*x^4])))/9))/13))/(17*c)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 266

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De 
nominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) 
^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I 
ntBinomialQ[a, b, c, 2, m, p, x]
 

rule 761

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 834

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

rule 1426

Int[((d_.)*(x_))^(m_)*((b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp 
[(d*x)^(m + 1)*((b*x^2 + c*x^4)^p/(d*(m + 4*p + 1))), x] + Simp[2*b*(p/(d^2 
*(m + 4*p + 1)))   Int[(d*x)^(m + 2)*(b*x^2 + c*x^4)^(p - 1), x], x] /; Fre 
eQ[{b, c, d, m, p}, x] &&  !IntegerQ[p] && GtQ[p, 0] && NeQ[m + 4*p + 1, 0]
 

rule 1429

Int[((d_.)*(x_))^(m_)*((b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp 
[d^3*(d*x)^(m - 3)*((b*x^2 + c*x^4)^(p + 1)/(c*(m + 4*p + 1))), x] - Simp[b 
*d^2*((m + 2*p - 1)/(c*(m + 4*p + 1)))   Int[(d*x)^(m - 2)*(b*x^2 + c*x^4)^ 
p, x], x] /; FreeQ[{b, c, d, m, p}, x] &&  !IntegerQ[p] && GtQ[m + 2*p - 1, 
 0] && NeQ[m + 4*p + 1, 0]
 

rule 1431

Int[((d_.)*(x_))^(m_)*((b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp 
[(b*x^2 + c*x^4)^p/((d*x)^(2*p)*(b + c*x^2)^p)   Int[(d*x)^(m + 2*p)*(b + c 
*x^2)^p, x], x] /; FreeQ[{b, c, d, m, p}, x] &&  !IntegerQ[p]
 

rule 1510

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]))*E 
llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e 
}, x] && PosQ[c/a]
 

rule 1945

Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + 
(d_.)*(x_)^(n_.)), x_Symbol] :> Simp[d*e^(j - 1)*(e*x)^(m - j + 1)*((a*x^j 
+ b*x^(j + n))^(p + 1)/(b*(m + n + p*(j + n) + 1))), x] - Simp[(a*d*(m + j* 
p + 1) - b*c*(m + n + p*(j + n) + 1))/(b*(m + n + p*(j + n) + 1))   Int[(e* 
x)^m*(a*x^j + b*x^(j + n))^p, x], x] /; FreeQ[{a, b, c, d, e, j, m, n, p}, 
x] && EqQ[jn, j + n] &&  !IntegerQ[p] && NeQ[b*c - a*d, 0] && NeQ[m + n + p 
*(j + n) + 1, 0] && (GtQ[e, 0] || IntegerQ[j])
 

Maple [A] (verified)

Time = 1.11 (sec) , antiderivative size = 291, normalized size of antiderivative = 0.71

method	result	size
risch	\(\frac {2 \sqrt {x}\, \left (195 B \,c^{3} x^{6}+255 A \,c^{3} x^{4}+285 B b \,c^{2} x^{4}+425 A b \,c^{2} x^{2}+20 x^{2} B \,b^{2} c +68 A \,b^{2} c -28 B \,b^{3}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}}{3315 c^{2}}-\frac {4 b^{3} \left (17 A c -7 B b \right ) \sqrt {-b c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, \left (-\frac {2 \sqrt {-b c}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-b c}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}\, \sqrt {x \left (c \,x^{2}+b \right )}}{1105 c^{3} \sqrt {c \,x^{3}+b x}\, x^{\frac {3}{2}} \left (c \,x^{2}+b \right )}\)	\(291\)
default	\(-\frac {2 \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} \left (-195 B \,c^{5} x^{10}-255 A \,c^{5} x^{8}-480 B b \,c^{4} x^{8}-680 A b \,c^{4} x^{6}-305 B \,b^{2} c^{3} x^{6}+204 A \,b^{4} c \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )-102 A \,b^{4} c \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )-84 B \,b^{5} \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )+42 B \,b^{5} \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )-493 A \,b^{2} c^{3} x^{4}+8 B \,b^{3} c^{2} x^{4}-68 A \,b^{3} c^{2} x^{2}+28 B \,b^{4} c \,x^{2}\right )}{3315 x^{\frac {7}{2}} \left (c \,x^{2}+b \right )^{2} c^{3}}\)	\(470\)

Input:

int((B*x^2+A)*(c*x^4+b*x^2)^(3/2)/x^(1/2),x,method=_RETURNVERBOSE)
 

Output:

2/3315/c^2*x^(1/2)*(195*B*c^3*x^6+255*A*c^3*x^4+285*B*b*c^2*x^4+425*A*b*c^ 
2*x^2+20*B*b^2*c*x^2+68*A*b^2*c-28*B*b^3)*(x^2*(c*x^2+b))^(1/2)-4/1105*b^3 
/c^3*(17*A*c-7*B*b)*(-b*c)^(1/2)*((x+1/c*(-b*c)^(1/2))*c/(-b*c)^(1/2))^(1/ 
2)*(-2*(x-1/c*(-b*c)^(1/2))*c/(-b*c)^(1/2))^(1/2)*(-c/(-b*c)^(1/2)*x)^(1/2 
)/(c*x^3+b*x)^(1/2)*(-2/c*(-b*c)^(1/2)*EllipticE(((x+1/c*(-b*c)^(1/2))*c/( 
-b*c)^(1/2))^(1/2),1/2*2^(1/2))+1/c*(-b*c)^(1/2)*EllipticF(((x+1/c*(-b*c)^ 
(1/2))*c/(-b*c)^(1/2))^(1/2),1/2*2^(1/2)))*(x^2*(c*x^2+b))^(1/2)/x^(3/2)/( 
c*x^2+b)*(x*(c*x^2+b))^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.31 \[ \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{\sqrt {x}} \, dx=-\frac {2 \, {\left (12 \, {\left (7 \, B b^{4} - 17 \, A b^{3} c\right )} \sqrt {c} {\rm weierstrassZeta}\left (-\frac {4 \, b}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right )\right ) - {\left (195 \, B c^{4} x^{6} - 28 \, B b^{3} c + 68 \, A b^{2} c^{2} + 15 \, {\left (19 \, B b c^{3} + 17 \, A c^{4}\right )} x^{4} + 5 \, {\left (4 \, B b^{2} c^{2} + 85 \, A b c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}\right )}}{3315 \, c^{3}} \] Input:

integrate((B*x^2+A)*(c*x^4+b*x^2)^(3/2)/x^(1/2),x, algorithm="fricas")
 

Output:

-2/3315*(12*(7*B*b^4 - 17*A*b^3*c)*sqrt(c)*weierstrassZeta(-4*b/c, 0, weie 
rstrassPInverse(-4*b/c, 0, x)) - (195*B*c^4*x^6 - 28*B*b^3*c + 68*A*b^2*c^ 
2 + 15*(19*B*b*c^3 + 17*A*c^4)*x^4 + 5*(4*B*b^2*c^2 + 85*A*b*c^3)*x^2)*sqr 
t(c*x^4 + b*x^2)*sqrt(x))/c^3
 

Sympy [F]

\[ \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{\sqrt {x}} \, dx=\int \frac {\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}} \left (A + B x^{2}\right )}{\sqrt {x}}\, dx \] Input:

integrate((B*x**2+A)*(c*x**4+b*x**2)**(3/2)/x**(1/2),x)
 

Output:

Integral((x**2*(b + c*x**2))**(3/2)*(A + B*x**2)/sqrt(x), x)
 

Maxima [F]

\[ \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{\sqrt {x}} \, dx=\int { \frac {{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )}}{\sqrt {x}} \,d x } \] Input:

integrate((B*x^2+A)*(c*x^4+b*x^2)^(3/2)/x^(1/2),x, algorithm="maxima")
 

Output:

integrate((c*x^4 + b*x^2)^(3/2)*(B*x^2 + A)/sqrt(x), x)
 

Giac [F]

\[ \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{\sqrt {x}} \, dx=\int { \frac {{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )}}{\sqrt {x}} \,d x } \] Input:

integrate((B*x^2+A)*(c*x^4+b*x^2)^(3/2)/x^(1/2),x, algorithm="giac")
 

Output:

integrate((c*x^4 + b*x^2)^(3/2)*(B*x^2 + A)/sqrt(x), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{\sqrt {x}} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (c\,x^4+b\,x^2\right )}^{3/2}}{\sqrt {x}} \,d x \] Input:

int(((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^(1/2),x)
 

Output:

int(((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^(1/2), x)
 

Reduce [F]

\[ \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{\sqrt {x}} \, dx=\frac {\frac {8 \sqrt {x}\, \sqrt {c \,x^{2}+b}\, a \,b^{2} c x}{195}+\frac {10 \sqrt {x}\, \sqrt {c \,x^{2}+b}\, a b \,c^{2} x^{3}}{39}+\frac {2 \sqrt {x}\, \sqrt {c \,x^{2}+b}\, a \,c^{3} x^{5}}{13}-\frac {56 \sqrt {x}\, \sqrt {c \,x^{2}+b}\, b^{4} x}{3315}+\frac {8 \sqrt {x}\, \sqrt {c \,x^{2}+b}\, b^{3} c \,x^{3}}{663}+\frac {38 \sqrt {x}\, \sqrt {c \,x^{2}+b}\, b^{2} c^{2} x^{5}}{221}+\frac {2 \sqrt {x}\, \sqrt {c \,x^{2}+b}\, b \,c^{3} x^{7}}{17}-\frac {4 \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b}}{c \,x^{2}+b}d x \right ) a \,b^{3} c}{65}+\frac {28 \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b}}{c \,x^{2}+b}d x \right ) b^{5}}{1105}}{c^{2}} \] Input:

int((B*x^2+A)*(c*x^4+b*x^2)^(3/2)/x^(1/2),x)
 

Output:

(2*(68*sqrt(x)*sqrt(b + c*x**2)*a*b**2*c*x + 425*sqrt(x)*sqrt(b + c*x**2)* 
a*b*c**2*x**3 + 255*sqrt(x)*sqrt(b + c*x**2)*a*c**3*x**5 - 28*sqrt(x)*sqrt 
(b + c*x**2)*b**4*x + 20*sqrt(x)*sqrt(b + c*x**2)*b**3*c*x**3 + 285*sqrt(x 
)*sqrt(b + c*x**2)*b**2*c**2*x**5 + 195*sqrt(x)*sqrt(b + c*x**2)*b*c**3*x* 
*7 - 102*int((sqrt(x)*sqrt(b + c*x**2))/(b + c*x**2),x)*a*b**3*c + 42*int( 
(sqrt(x)*sqrt(b + c*x**2))/(b + c*x**2),x)*b**5))/(3315*c**2)