Integrand size = 19, antiderivative size = 550 \[ \int \frac {(c x)^{15/2}}{\left (a+b x^3\right )^{3/2}} \, dx=-\frac {2 c^2 (c x)^{11/2}}{3 b \sqrt {a+b x^3}}+\frac {11 c^5 (c x)^{5/2} \sqrt {a+b x^3}}{12 b^2}-\frac {55 \left (1+\sqrt {3}\right ) a c^7 \sqrt {c x} \sqrt {a+b x^3}}{24 b^{8/3} \left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )}+\frac {55 a^{4/3} c^7 \sqrt {c x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} E\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{8\ 3^{3/4} b^{8/3} \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {55 \left (1-\sqrt {3}\right ) a^{4/3} c^7 \sqrt {c x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{48 \sqrt [4]{3} b^{8/3} \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:
-2/3*c^2*(c*x)^(11/2)/b/(b*x^3+a)^(1/2)+11/12*c^5*(c*x)^(5/2)*(b*x^3+a)^(1 /2)/b^2-55/24*(1+3^(1/2))*a*c^7*(c*x)^(1/2)*(b*x^3+a)^(1/2)/b^(8/3)/(a^(1/ 3)+(1+3^(1/2))*b^(1/3)*x)+55/24*a^(4/3)*c^7*(c*x)^(1/2)*(a^(1/3)+b^(1/3)*x )*((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(1/3)+(1+3^(1/2))*b^(1/3)*x) ^2)^(1/2)*EllipticE((1-(a^(1/3)+(1-3^(1/2))*b^(1/3)*x)^2/(a^(1/3)+(1+3^(1/ 2))*b^(1/3)*x)^2)^(1/2),1/4*6^(1/2)+1/4*2^(1/2))*3^(1/4)/b^(8/3)/(b^(1/3)* x*(a^(1/3)+b^(1/3)*x)/(a^(1/3)+(1+3^(1/2))*b^(1/3)*x)^2)^(1/2)/(b*x^3+a)^( 1/2)+55/144*(1-3^(1/2))*a^(4/3)*c^7*(c*x)^(1/2)*(a^(1/3)+b^(1/3)*x)*((a^(2 /3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(1/3)+(1+3^(1/2))*b^(1/3)*x)^2)^(1/2 )*InverseJacobiAM(arccos((a^(1/3)+(1-3^(1/2))*b^(1/3)*x)/(a^(1/3)+(1+3^(1/ 2))*b^(1/3)*x)),1/4*6^(1/2)+1/4*2^(1/2))*3^(3/4)/b^(8/3)/(b^(1/3)*x*(a^(1/ 3)+b^(1/3)*x)/(a^(1/3)+(1+3^(1/2))*b^(1/3)*x)^2)^(1/2)/(b*x^3+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.04 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.13 \[ \int \frac {(c x)^{15/2}}{\left (a+b x^3\right )^{3/2}} \, dx=\frac {c^5 (c x)^{5/2} \left (-11 a+2 b x^3+11 a \sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (\frac {5}{6},\frac {3}{2},\frac {11}{6},-\frac {b x^3}{a}\right )\right )}{8 b^2 \sqrt {a+b x^3}} \] Input:
Integrate[(c*x)^(15/2)/(a + b*x^3)^(3/2),x]
Output:
(c^5*(c*x)^(5/2)*(-11*a + 2*b*x^3 + 11*a*Sqrt[1 + (b*x^3)/a]*Hypergeometri c2F1[5/6, 3/2, 11/6, -((b*x^3)/a)]))/(8*b^2*Sqrt[a + b*x^3])
Time = 1.09 (sec) , antiderivative size = 624, normalized size of antiderivative = 1.13, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {817, 843, 851, 837, 25, 766, 2420}
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.
| \(\displaystyle \int \frac {(c x)^{15/2}}{\left (a+b x^3\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle \frac {11 c^3 \int \frac {(c x)^{9/2}}{\sqrt {b x^3+a}}dx}{3 b}-\frac {2 c^2 (c x)^{11/2}}{3 b \sqrt {a+b x^3}}\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {11 c^3 \left (\frac {c^2 (c x)^{5/2} \sqrt {a+b x^3}}{4 b}-\frac {5 a c^3 \int \frac {(c x)^{3/2}}{\sqrt {b x^3+a}}dx}{8 b}\right )}{3 b}-\frac {2 c^2 (c x)^{11/2}}{3 b \sqrt {a+b x^3}}\) |
| \(\Big \downarrow \) 851 |
| \(\displaystyle \frac {11 c^3 \left (\frac {c^2 (c x)^{5/2} \sqrt {a+b x^3}}{4 b}-\frac {5 a c^2 \int \frac {c^2 x^2}{\sqrt {b x^3+a}}d\sqrt {c x}}{4 b}\right )}{3 b}-\frac {2 c^2 (c x)^{11/2}}{3 b \sqrt {a+b x^3}}\) |
| \(\Big \downarrow \) 837 |
| \(\displaystyle \frac {11 c^3 \left (\frac {c^2 (c x)^{5/2} \sqrt {a+b x^3}}{4 b}-\frac {5 a c^2 \left (-\frac {\left (1-\sqrt {3}\right ) a^{2/3} c^2 \int \frac {1}{\sqrt {b x^3+a}}d\sqrt {c x}}{2 b^{2/3}}-\frac {\int -\frac {2 b^{2/3} x^2 c^2+\left (1-\sqrt {3}\right ) a^{2/3} c^2}{\sqrt {b x^3+a}}d\sqrt {c x}}{2 b^{2/3}}\right )}{4 b}\right )}{3 b}-\frac {2 c^2 (c x)^{11/2}}{3 b \sqrt {a+b x^3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {11 c^3 \left (\frac {c^2 (c x)^{5/2} \sqrt {a+b x^3}}{4 b}-\frac {5 a c^2 \left (\frac {\int \frac {2 b^{2/3} x^2 c^2+\left (1-\sqrt {3}\right ) a^{2/3} c^2}{\sqrt {b x^3+a}}d\sqrt {c x}}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) a^{2/3} c^2 \int \frac {1}{\sqrt {b x^3+a}}d\sqrt {c x}}{2 b^{2/3}}\right )}{4 b}\right )}{3 b}-\frac {2 c^2 (c x)^{11/2}}{3 b \sqrt {a+b x^3}}\) |
| \(\Big \downarrow \) 766 |
| \(\displaystyle \frac {11 c^3 \left (\frac {c^2 (c x)^{5/2} \sqrt {a+b x^3}}{4 b}-\frac {5 a c^2 \left (\frac {\int \frac {2 b^{2/3} x^2 c^2+\left (1-\sqrt {3}\right ) a^{2/3} c^2}{\sqrt {b x^3+a}}d\sqrt {c x}}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} c \sqrt {c x} \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right ) \sqrt {\frac {a^{2/3} c^2-\sqrt [3]{a} \sqrt [3]{b} c^2 x+b^{2/3} c^2 x^2}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt [4]{3} b^{2/3} \sqrt {a+b x^3} \sqrt {\frac {\sqrt [3]{b} c x \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right )}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}}}\right )}{4 b}\right )}{3 b}-\frac {2 c^2 (c x)^{11/2}}{3 b \sqrt {a+b x^3}}\) |
| \(\Big \downarrow \) 2420 |
| \(\displaystyle \frac {11 c^3 \left (\frac {c^2 (c x)^{5/2} \sqrt {a+b x^3}}{4 b}-\frac {5 a c^2 \left (\frac {\frac {\left (1+\sqrt {3}\right ) c^3 \sqrt {c x} \sqrt {a+b x^3}}{\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x}-\frac {\sqrt [4]{3} \sqrt [3]{a} c \sqrt {c x} \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right ) \sqrt {\frac {a^{2/3} c^2-\sqrt [3]{a} \sqrt [3]{b} c^2 x+b^{2/3} c^2 x^2}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}} E\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt {a+b x^3} \sqrt {\frac {\sqrt [3]{b} c x \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right )}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}}}}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} c \sqrt {c x} \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right ) \sqrt {\frac {a^{2/3} c^2-\sqrt [3]{a} \sqrt [3]{b} c^2 x+b^{2/3} c^2 x^2}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt [4]{3} b^{2/3} \sqrt {a+b x^3} \sqrt {\frac {\sqrt [3]{b} c x \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right )}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}}}\right )}{4 b}\right )}{3 b}-\frac {2 c^2 (c x)^{11/2}}{3 b \sqrt {a+b x^3}}\) |
Input:
Int[(c*x)^(15/2)/(a + b*x^3)^(3/2),x]
Output:
(-2*c^2*(c*x)^(11/2))/(3*b*Sqrt[a + b*x^3]) + (11*c^3*((c^2*(c*x)^(5/2)*Sq rt[a + b*x^3])/(4*b) - (5*a*c^2*((((1 + Sqrt[3])*c^3*Sqrt[c*x]*Sqrt[a + b* x^3])/(a^(1/3)*c + (1 + Sqrt[3])*b^(1/3)*c*x) - (3^(1/4)*a^(1/3)*c*Sqrt[c* x]*(a^(1/3)*c + b^(1/3)*c*x)*Sqrt[(a^(2/3)*c^2 - a^(1/3)*b^(1/3)*c^2*x + b ^(2/3)*c^2*x^2)/(a^(1/3)*c + (1 + Sqrt[3])*b^(1/3)*c*x)^2]*EllipticE[ArcCo s[(a^(1/3)*c + (1 - Sqrt[3])*b^(1/3)*c*x)/(a^(1/3)*c + (1 + Sqrt[3])*b^(1/ 3)*c*x)], (2 + Sqrt[3])/4])/(Sqrt[(b^(1/3)*c*x*(a^(1/3)*c + b^(1/3)*c*x))/ (a^(1/3)*c + (1 + Sqrt[3])*b^(1/3)*c*x)^2]*Sqrt[a + b*x^3]))/(2*b^(2/3)) - ((1 - Sqrt[3])*a^(1/3)*c*Sqrt[c*x]*(a^(1/3)*c + b^(1/3)*c*x)*Sqrt[(a^(2/3 )*c^2 - a^(1/3)*b^(1/3)*c^2*x + b^(2/3)*c^2*x^2)/(a^(1/3)*c + (1 + Sqrt[3] )*b^(1/3)*c*x)^2]*EllipticF[ArcCos[(a^(1/3)*c + (1 - Sqrt[3])*b^(1/3)*c*x) /(a^(1/3)*c + (1 + Sqrt[3])*b^(1/3)*c*x)], (2 + Sqrt[3])/4])/(4*3^(1/4)*b^ (2/3)*Sqrt[(b^(1/3)*c*x*(a^(1/3)*c + b^(1/3)*c*x))/(a^(1/3)*c + (1 + Sqrt[ 3])*b^(1/3)*c*x)^2]*Sqrt[a + b*x^3])))/(4*b)))/(3*b)
Int[1/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[x*(s + r*x^2)*(Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/ (s + (1 + Sqrt[3])*r*x^2)^2]/(2*3^(1/4)*s*Sqrt[a + b*x^6]*Sqrt[r*x^2*((s + r*x^2)/(s + (1 + Sqrt[3])*r*x^2)^2)]))*EllipticF[ArcCos[(s + (1 - Sqrt[3])* r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4], x]] /; FreeQ[{a, b}, x ]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^( n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*n*(p + 1))), x] - Simp[c^n *((m - n + 1)/(b*n*(p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x], x ] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] && ! ILtQ[(m + n*(p + 1) + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Int[(x_)^4/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(Sqrt[3] - 1)*(s^2/(2*r^2)) Int[1/Sqrt[ a + b*x^6], x], x] - Simp[1/(2*r^2) Int[((Sqrt[3] - 1)*s^2 - 2*r^2*x^4)/S qrt[a + b*x^6], x], x]] /; FreeQ[{a, b}, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[ a*c^n*((m - n + 1)/(b*(m + n*p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^p, x] , x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/c Subst[Int[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]
Int[((c_) + (d_.)*(x_)^4)/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(1 + Sqrt[3])*d*s^3*x*(Sqr t[a + b*x^6]/(2*a*r^2*(s + (1 + Sqrt[3])*r*x^2))), x] - Simp[3^(1/4)*d*s*x* (s + r*x^2)*(Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/(s + (1 + Sqrt[3])*r*x^2)^2]/(2 *r^2*Sqrt[(r*x^2*(s + r*x^2))/(s + (1 + Sqrt[3])*r*x^2)^2]*Sqrt[a + b*x^6]) )*EllipticE[ArcCos[(s + (1 - Sqrt[3])*r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[2*Rt[b/a, 3]^2*c - (1 - Sqrt[3])*d, 0]
Result contains complex when optimal does not.
Time = 3.52 (sec) , antiderivative size = 1148, normalized size of antiderivative = 2.09
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1148\) |
| risch | \(\text {Expression too large to display}\) | \(2203\) |
| default | \(\text {Expression too large to display}\) | \(2909\) |
Input:
int((c*x)^(15/2)/(b*x^3+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/c/x*(c*x)^(1/2)/(b*x^3+a)^(1/2)*(c*x*(b*x^3+a))^(1/2)*(2/3/b^2*c^8*x^3*a /((x^3+a/b)*b*c*x)^(1/2)+1/4/b^2*c^7*x^2*(b*c*x^4+a*c*x)^(1/2)-55/24*a*c^8 /b^2*(x*(x+1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*(x+1/2/b*( -a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))+(1/2/b*(-a*b^2)^(1/3)-1/2*I* 3^(1/2)/b*(-a*b^2)^(1/3))*((-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2) ^(1/3))*x/(-1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(x-1/b*(- a*b^2)^(1/3)))^(1/2)*(x-1/b*(-a*b^2)^(1/3))^2*(1/b*(-a*b^2)^(1/3)*(x+1/2/b *(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(-1/2/b*(-a*b^2)^(1/3)-1/2 *I*3^(1/2)/b*(-a*b^2)^(1/3))/(x-1/b*(-a*b^2)^(1/3)))^(1/2)*(1/b*(-a*b^2)^( 1/3)*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(-1/2/b*(-a*b ^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(x-1/b*(-a*b^2)^(1/3)))^(1/2)*(( (-1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/b*(-a*b^2)^(1/3)+1/ b^2*(-a*b^2)^(2/3))/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)) *b/(-a*b^2)^(1/3)*EllipticF(((-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^ 2)^(1/3))*x/(-1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(x-1/b* (-a*b^2)^(1/3)))^(1/2),((3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/ 3))*(1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(1/2/b*(-a*b^2)^ (1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(3/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/ b*(-a*b^2)^(1/3)))^(1/2))+(1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^( 1/3))*EllipticE(((-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))...
\[ \int \frac {(c x)^{15/2}}{\left (a+b x^3\right )^{3/2}} \, dx=\int { \frac {\left (c x\right )^{\frac {15}{2}}}{{\left (b x^{3} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((c*x)^(15/2)/(b*x^3+a)^(3/2),x, algorithm="fricas")
Output:
integral(sqrt(b*x^3 + a)*sqrt(c*x)*c^7*x^7/(b^2*x^6 + 2*a*b*x^3 + a^2), x)
Timed out. \[ \int \frac {(c x)^{15/2}}{\left (a+b x^3\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((c*x)**(15/2)/(b*x**3+a)**(3/2),x)
Output:
Timed out
\[ \int \frac {(c x)^{15/2}}{\left (a+b x^3\right )^{3/2}} \, dx=\int { \frac {\left (c x\right )^{\frac {15}{2}}}{{\left (b x^{3} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((c*x)^(15/2)/(b*x^3+a)^(3/2),x, algorithm="maxima")
Output:
integrate((c*x)^(15/2)/(b*x^3 + a)^(3/2), x)
\[ \int \frac {(c x)^{15/2}}{\left (a+b x^3\right )^{3/2}} \, dx=\int { \frac {\left (c x\right )^{\frac {15}{2}}}{{\left (b x^{3} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((c*x)^(15/2)/(b*x^3+a)^(3/2),x, algorithm="giac")
Output:
integrate((c*x)^(15/2)/(b*x^3 + a)^(3/2), x)
Timed out. \[ \int \frac {(c x)^{15/2}}{\left (a+b x^3\right )^{3/2}} \, dx=\int \frac {{\left (c\,x\right )}^{15/2}}{{\left (b\,x^3+a\right )}^{3/2}} \,d x \] Input:
int((c*x)^(15/2)/(a + b*x^3)^(3/2),x)
Output:
int((c*x)^(15/2)/(a + b*x^3)^(3/2), x)
\[ \int \frac {(c x)^{15/2}}{\left (a+b x^3\right )^{3/2}} \, dx=\frac {\sqrt {c}\, c^{7} \left (-22 \sqrt {x}\, \sqrt {b \,x^{3}+a}\, a \,x^{2}+4 \sqrt {x}\, \sqrt {b \,x^{3}+a}\, b \,x^{5}+55 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{3}+a}\, x}{b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) a^{3}+55 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{3}+a}\, x}{b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) a^{2} b \,x^{3}\right )}{16 b^{2} \left (b \,x^{3}+a \right )} \] Input:
int((c*x)^(15/2)/(b*x^3+a)^(3/2),x)
Output:
(sqrt(c)*c**7*( - 22*sqrt(x)*sqrt(a + b*x**3)*a*x**2 + 4*sqrt(x)*sqrt(a + b*x**3)*b*x**5 + 55*int((sqrt(x)*sqrt(a + b*x**3)*x)/(a**2 + 2*a*b*x**3 + b**2*x**6),x)*a**3 + 55*int((sqrt(x)*sqrt(a + b*x**3)*x)/(a**2 + 2*a*b*x** 3 + b**2*x**6),x)*a**2*b*x**3))/(16*b**2*(a + b*x**3))