\(\int (a-b x^2)^{2/3} (3 a+b x^2)^3 \, dx\) [320]

Optimal result

Integrand size = 24, antiderivative size = 636 \[ \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3 \, dx=\frac {18144 a^3 x \left (a-b x^2\right )^{2/3}}{1235}-\frac {8991 a^2 x \left (a-b x^2\right )^{5/3}}{1235}-\frac {144}{95} a b x^3 \left (a-b x^2\right )^{5/3}-\frac {3}{25} b^2 x^5 \left (a-b x^2\right )^{5/3}-\frac {72576 a^4 x}{1235 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac {36288 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{13/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{1235 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac {24192 \sqrt {2} 3^{3/4} a^{13/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{1235 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}} \] Output:

18144/1235*a^3*x*(-b*x^2+a)^(2/3)-8991/1235*a^2*x*(-b*x^2+a)^(5/3)-144/95* 
a*b*x^3*(-b*x^2+a)^(5/3)-3/25*b^2*x^5*(-b*x^2+a)^(5/3)-72576*a^4*x/(1235*( 
1-3^(1/2))*a^(1/3)-1235*(-b*x^2+a)^(1/3))-36288/1235*3^(1/4)*(1/2*6^(1/2)+ 
1/2*2^(1/2))*a^(13/3)*(a^(1/3)-(-b*x^2+a)^(1/3))*((a^(2/3)+a^(1/3)*(-b*x^2 
+a)^(1/3)+(-b*x^2+a)^(2/3))/((1-3^(1/2))*a^(1/3)-(-b*x^2+a)^(1/3))^2)^(1/2 
)*EllipticE(((1+3^(1/2))*a^(1/3)-(-b*x^2+a)^(1/3))/((1-3^(1/2))*a^(1/3)-(- 
b*x^2+a)^(1/3)),2*I-I*3^(1/2))/b/x/(-a^(1/3)*(a^(1/3)-(-b*x^2+a)^(1/3))/(( 
1-3^(1/2))*a^(1/3)-(-b*x^2+a)^(1/3))^2)^(1/2)+24192/1235*2^(1/2)*3^(3/4)*a 
^(13/3)*(a^(1/3)-(-b*x^2+a)^(1/3))*((a^(2/3)+a^(1/3)*(-b*x^2+a)^(1/3)+(-b* 
x^2+a)^(2/3))/((1-3^(1/2))*a^(1/3)-(-b*x^2+a)^(1/3))^2)^(1/2)*EllipticF((( 
1+3^(1/2))*a^(1/3)-(-b*x^2+a)^(1/3))/((1-3^(1/2))*a^(1/3)-(-b*x^2+a)^(1/3) 
),2*I-I*3^(1/2))/b/x/(-a^(1/3)*(a^(1/3)-(-b*x^2+a)^(1/3))/((1-3^(1/2))*a^( 
1/3)-(-b*x^2+a)^(1/3))^2)^(1/2)
 

Mathematica [C] (warning: unable to verify)

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

Time = 10.25 (sec) , antiderivative size = 384, normalized size of antiderivative = 0.60 \[ \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3 \, dx=\frac {x \left (a-b x^2\right )^{2/3} \left (25515 a^4 \operatorname {Gamma}\left (-\frac {2}{3}\right ) \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{2},\frac {9}{2},\frac {b x^2}{a}\right )+8505 a^3 b x^2 \operatorname {Gamma}\left (-\frac {2}{3}\right ) \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{2},\frac {9}{2},\frac {b x^2}{a}\right )+1701 a^2 b^2 x^4 \operatorname {Gamma}\left (-\frac {2}{3}\right ) \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{2},\frac {9}{2},\frac {b x^2}{a}\right )+135 a b^3 x^6 \operatorname {Gamma}\left (-\frac {2}{3}\right ) \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{2},\frac {9}{2},\frac {b x^2}{a}\right )+3834 a^3 b x^2 \operatorname {Gamma}\left (\frac {1}{3}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {3}{2},\frac {11}{2},\frac {b x^2}{a}\right )+2538 a^2 b^2 x^4 \operatorname {Gamma}\left (\frac {1}{3}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {3}{2},\frac {11}{2},\frac {b x^2}{a}\right )+558 a b^3 x^6 \operatorname {Gamma}\left (\frac {1}{3}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {3}{2},\frac {11}{2},\frac {b x^2}{a}\right )+46 b^4 x^8 \operatorname {Gamma}\left (\frac {1}{3}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {3}{2},\frac {11}{2},\frac {b x^2}{a}\right )+36 b \left (5 a+b x^2\right ) \left (3 a x+b x^3\right )^2 \operatorname {Gamma}\left (\frac {1}{3}\right ) \, _3F_2\left (\frac {1}{3},\frac {3}{2},2;1,\frac {11}{2};\frac {b x^2}{a}\right )+8 b x^2 \left (3 a+b x^2\right )^3 \operatorname {Gamma}\left (\frac {1}{3}\right ) \, _4F_3\left (\frac {1}{3},\frac {3}{2},2,2;1,1,\frac {11}{2};\frac {b x^2}{a}\right )\right )}{945 a \left (1-\frac {b x^2}{a}\right )^{2/3} \operatorname {Gamma}\left (-\frac {2}{3}\right )} \] Input:

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

Output:

(x*(a - b*x^2)^(2/3)*(25515*a^4*Gamma[-2/3]*Hypergeometric2F1[-2/3, 1/2, 9 
/2, (b*x^2)/a] + 8505*a^3*b*x^2*Gamma[-2/3]*Hypergeometric2F1[-2/3, 1/2, 9 
/2, (b*x^2)/a] + 1701*a^2*b^2*x^4*Gamma[-2/3]*Hypergeometric2F1[-2/3, 1/2, 
 9/2, (b*x^2)/a] + 135*a*b^3*x^6*Gamma[-2/3]*Hypergeometric2F1[-2/3, 1/2, 
9/2, (b*x^2)/a] + 3834*a^3*b*x^2*Gamma[1/3]*Hypergeometric2F1[1/3, 3/2, 11 
/2, (b*x^2)/a] + 2538*a^2*b^2*x^4*Gamma[1/3]*Hypergeometric2F1[1/3, 3/2, 1 
1/2, (b*x^2)/a] + 558*a*b^3*x^6*Gamma[1/3]*Hypergeometric2F1[1/3, 3/2, 11/ 
2, (b*x^2)/a] + 46*b^4*x^8*Gamma[1/3]*Hypergeometric2F1[1/3, 3/2, 11/2, (b 
*x^2)/a] + 36*b*(5*a + b*x^2)*(3*a*x + b*x^3)^2*Gamma[1/3]*HypergeometricP 
FQ[{1/3, 3/2, 2}, {1, 11/2}, (b*x^2)/a] + 8*b*x^2*(3*a + b*x^2)^3*Gamma[1/ 
3]*HypergeometricPFQ[{1/3, 3/2, 2, 2}, {1, 1, 11/2}, (b*x^2)/a]))/(945*a*( 
1 - (b*x^2)/a)^(2/3)*Gamma[-2/3])
 

Rubi [A] (warning: unable to verify)

Time = 0.58 (sec) , antiderivative size = 703, normalized size of antiderivative = 1.11, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {318, 27, 403, 27, 299, 211, 233, 833, 760, 2418}

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.

\(\displaystyle \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3 \, dx\)

\(\Big \downarrow \) 318

\(\displaystyle -\frac {3 \int -6 a b \left (a-b x^2\right )^{2/3} \left (b x^2+3 a\right ) \left (7 b x^2+13 a\right )dx}{25 b}-\frac {3}{25} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )^2\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {18}{25} a \int \left (a-b x^2\right )^{2/3} \left (b x^2+3 a\right ) \left (7 b x^2+13 a\right )dx-\frac {3}{25} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )^2\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {18}{25} a \left (-\frac {3 \int -\frac {4}{3} a b \left (a-b x^2\right )^{2/3} \left (109 b x^2+201 a\right )dx}{19 b}-\frac {21}{19} x \left (3 a+b x^2\right ) \left (a-b x^2\right )^{5/3}\right )-\frac {3}{25} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )^2\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {18}{25} a \left (\frac {4}{19} a \int \left (a-b x^2\right )^{2/3} \left (109 b x^2+201 a\right )dx-\frac {21}{19} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )\right )-\frac {3}{25} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )^2\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {18}{25} a \left (\frac {4}{19} a \left (\frac {2940}{13} a \int \left (a-b x^2\right )^{2/3}dx-\frac {327}{13} x \left (a-b x^2\right )^{5/3}\right )-\frac {21}{19} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )\right )-\frac {3}{25} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )^2\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {18}{25} a \left (\frac {4}{19} a \left (\frac {2940}{13} a \left (\frac {4}{7} a \int \frac {1}{\sqrt [3]{a-b x^2}}dx+\frac {3}{7} x \left (a-b x^2\right )^{2/3}\right )-\frac {327}{13} x \left (a-b x^2\right )^{5/3}\right )-\frac {21}{19} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )\right )-\frac {3}{25} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )^2\)

\(\Big \downarrow \) 233

\(\displaystyle \frac {18}{25} a \left (\frac {4}{19} a \left (\frac {2940}{13} a \left (\frac {3}{7} x \left (a-b x^2\right )^{2/3}-\frac {6 a \sqrt {-b x^2} \int \frac {\sqrt [3]{a-b x^2}}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}}{7 b x}\right )-\frac {327}{13} x \left (a-b x^2\right )^{5/3}\right )-\frac {21}{19} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )\right )-\frac {3}{25} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )^2\)

\(\Big \downarrow \) 833

\(\displaystyle \frac {18}{25} a \left (\frac {4}{19} a \left (\frac {2940}{13} a \left (\frac {3}{7} x \left (a-b x^2\right )^{2/3}-\frac {6 a \sqrt {-b x^2} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}\right )}{7 b x}\right )-\frac {327}{13} x \left (a-b x^2\right )^{5/3}\right )-\frac {21}{19} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )\right )-\frac {3}{25} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )^2\)

\(\Big \downarrow \) 760

\(\displaystyle \frac {18}{25} a \left (\frac {4}{19} a \left (\frac {2940}{13} a \left (\frac {3}{7} x \left (a-b x^2\right )^{2/3}-\frac {6 a \sqrt {-b x^2} \left (-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}\right )}{7 b x}\right )-\frac {327}{13} x \left (a-b x^2\right )^{5/3}\right )-\frac {21}{19} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )\right )-\frac {3}{25} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )^2\)

\(\Big \downarrow \) 2418

\(\displaystyle \frac {18}{25} a \left (\frac {4}{19} a \left (\frac {2940}{13} a \left (\frac {3}{7} x \left (a-b x^2\right )^{2/3}-\frac {6 a \sqrt {-b x^2} \left (-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{\sqrt {-b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac {2 \sqrt {-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )}{7 b x}\right )-\frac {327}{13} x \left (a-b x^2\right )^{5/3}\right )-\frac {21}{19} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )\right )-\frac {3}{25} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )^2\)

Input:

Int[(a - b*x^2)^(2/3)*(3*a + b*x^2)^3,x]
 

Output:

(-3*x*(a - b*x^2)^(5/3)*(3*a + b*x^2)^2)/25 + (18*a*((-21*x*(a - b*x^2)^(5 
/3)*(3*a + b*x^2))/19 + (4*a*((-327*x*(a - b*x^2)^(5/3))/13 + (2940*a*((3* 
x*(a - b*x^2)^(2/3))/7 - (6*a*Sqrt[-(b*x^2)]*((-2*Sqrt[-(b*x^2)])/((1 - Sq 
rt[3])*a^(1/3) - (a - b*x^2)^(1/3)) + (3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(1/3)*( 
a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + ( 
a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticE 
[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) 
 - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(Sqrt[-(b*x^2)]*Sqrt[-((a^(1/3)*( 
a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^ 
2)]) - (2*Sqrt[2 - Sqrt[3]]*(1 + Sqrt[3])*a^(1/3)*(a^(1/3) - (a - b*x^2)^( 
1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - 
 Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])* 
a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], 
 -7 + 4*Sqrt[3]])/(3^(1/4)*Sqrt[-(b*x^2)]*Sqrt[-((a^(1/3)*(a^(1/3) - (a - 
b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)])))/(7*b*x)) 
)/13))/19))/25
 

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 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

rule 233

Int[((a_) + (b_.)*(x_)^2)^(-1/3), x_Symbol] :> Simp[3*(Sqrt[b*x^2]/(2*b*x)) 
   Subst[Int[x/Sqrt[-a + x^3], x], x, (a + b*x^2)^(1/3)], x] /; FreeQ[{a, b 
}, x]
 

rule 299

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

rule 318

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[d*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*(2*(p + q) + 1))), x] + S 
imp[1/(b*(2*(p + q) + 1))   Int[(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b 
*c*(2*(p + q) + 1) - a*d) + d*(b*c*(2*(p + 2*q - 1) + 1) - a*d*(2*(q - 1) + 
 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && G 
tQ[q, 1] && NeQ[2*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntBinomialQ[a, b, c, 
d, 2, p, q, x]
 

rule 403

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

rule 760

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 - Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s 
*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(- 
s)*((s + r*x)/((1 - Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 + Sqrt[3]) 
*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x 
] && NegQ[a]
 

rule 833

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] 
], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 + Sqrt[3]))*(s/r)   Int[1/Sqrt[a + b*x 
^3], x], x] + Simp[1/r   Int[((1 + Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x 
]] /; FreeQ[{a, b}, x] && NegQ[a]
 

rule 2418

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N 
umer[Simplify[(1 + Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 + Sqrt[3])*(d/c) 
]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 - Sqrt[3])*s + r*x))), x] + S 
imp[3^(1/4)*Sqrt[2 + Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( 
(1 - Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[(-s)*((s + r*x)/((1 - S 
qrt[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[ 
3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && 
 EqQ[b*c^3 - 2*(5 + 3*Sqrt[3])*a*d^3, 0]
 

Maple [F]

\[\int \left (-b \,x^{2}+a \right )^{\frac {2}{3}} \left (b \,x^{2}+3 a \right )^{3}d x\]

Input:

int((-b*x^2+a)^(2/3)*(b*x^2+3*a)^3,x)
 

Output:

int((-b*x^2+a)^(2/3)*(b*x^2+3*a)^3,x)
 

Fricas [F]

\[ \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3 \, dx=\int { {\left (b x^{2} + 3 \, a\right )}^{3} {\left (-b x^{2} + a\right )}^{\frac {2}{3}} \,d x } \] Input:

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

Output:

integral((b^3*x^6 + 9*a*b^2*x^4 + 27*a^2*b*x^2 + 27*a^3)*(-b*x^2 + a)^(2/3 
), x)
 

Sympy [A] (verification not implemented)

Time = 1.81 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.21 \[ \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3 \, dx=27 a^{\frac {11}{3}} x {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )} + 9 a^{\frac {8}{3}} b x^{3} {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )} + \frac {9 a^{\frac {5}{3}} b^{2} x^{5} {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{5} + \frac {a^{\frac {2}{3}} b^{3} x^{7} {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {7}{2} \\ \frac {9}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{7} \] Input:

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

Output:

27*a**(11/3)*x*hyper((-2/3, 1/2), (3/2,), b*x**2*exp_polar(2*I*pi)/a) + 9* 
a**(8/3)*b*x**3*hyper((-2/3, 3/2), (5/2,), b*x**2*exp_polar(2*I*pi)/a) + 9 
*a**(5/3)*b**2*x**5*hyper((-2/3, 5/2), (7/2,), b*x**2*exp_polar(2*I*pi)/a) 
/5 + a**(2/3)*b**3*x**7*hyper((-2/3, 7/2), (9/2,), b*x**2*exp_polar(2*I*pi 
)/a)/7
 

Maxima [F]

\[ \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3 \, dx=\int { {\left (b x^{2} + 3 \, a\right )}^{3} {\left (-b x^{2} + a\right )}^{\frac {2}{3}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3 \, dx=\int { {\left (b x^{2} + 3 \, a\right )}^{3} {\left (-b x^{2} + a\right )}^{\frac {2}{3}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3 \, dx=\int {\left (a-b\,x^2\right )}^{2/3}\,{\left (b\,x^2+3\,a\right )}^3 \,d x \] Input:

int((a - b*x^2)^(2/3)*(3*a + b*x^2)^3,x)
 

Output:

int((a - b*x^2)^(2/3)*(3*a + b*x^2)^3, x)
 

Reduce [F]

\[ \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3 \, dx=\frac {9153 \left (-b \,x^{2}+a \right )^{\frac {2}{3}} a^{3} x}{1235}+\frac {7119 \left (-b \,x^{2}+a \right )^{\frac {2}{3}} a^{2} b \,x^{3}}{1235}+\frac {663 \left (-b \,x^{2}+a \right )^{\frac {2}{3}} a \,b^{2} x^{5}}{475}+\frac {3 \left (-b \,x^{2}+a \right )^{\frac {2}{3}} b^{3} x^{7}}{25}+\frac {24192 \left (\int \frac {1}{\left (-b \,x^{2}+a \right )^{\frac {1}{3}}}d x \right ) a^{4}}{1235} \] Input:

int((-b*x^2+a)^(2/3)*(b*x^2+3*a)^3,x)
 

Output:

(3*(15255*(a - b*x**2)**(2/3)*a**3*x + 11865*(a - b*x**2)**(2/3)*a**2*b*x* 
*3 + 2873*(a - b*x**2)**(2/3)*a*b**2*x**5 + 247*(a - b*x**2)**(2/3)*b**3*x 
**7 + 40320*int((a - b*x**2)**(2/3)/(a - b*x**2),x)*a**4))/6175