Integrand size = 19, antiderivative size = 457 \[ \int (a+b x)^{3/2} \sqrt [3]{c+d x} \, dx=-\frac {108 (b c-a d)^2 \sqrt {a+b x} \sqrt [3]{c+d x}}{935 b d^2}+\frac {12 (b c-a d) (a+b x)^{3/2} \sqrt [3]{c+d x}}{187 b d}+\frac {6 (a+b x)^{5/2} \sqrt [3]{c+d x}}{17 b}-\frac {108\ 3^{3/4} \sqrt {2-\sqrt {3}} (b c-a d)^3 \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{b c-a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right ),-7+4 \sqrt {3}\right )}{935 b^{4/3} d^3 \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}} \] Output:
-108/935*(-a*d+b*c)^2*(b*x+a)^(1/2)*(d*x+c)^(1/3)/b/d^2+12/187*(-a*d+b*c)* (b*x+a)^(3/2)*(d*x+c)^(1/3)/b/d+6/17*(b*x+a)^(5/2)*(d*x+c)^(1/3)/b-108/935 *3^(3/4)*(1/2*6^(1/2)-1/2*2^(1/2))*(-a*d+b*c)^3*((-a*d+b*c)^(1/3)-b^(1/3)* (d*x+c)^(1/3))*(((-a*d+b*c)^(2/3)+b^(1/3)*(-a*d+b*c)^(1/3)*(d*x+c)^(1/3)+b ^(2/3)*(d*x+c)^(2/3))/((1-3^(1/2))*(-a*d+b*c)^(1/3)-b^(1/3)*(d*x+c)^(1/3)) ^2)^(1/2)*EllipticF(((1+3^(1/2))*(-a*d+b*c)^(1/3)-b^(1/3)*(d*x+c)^(1/3))/( (1-3^(1/2))*(-a*d+b*c)^(1/3)-b^(1/3)*(d*x+c)^(1/3)),2*I-I*3^(1/2))/b^(4/3) /d^3/(b*x+a)^(1/2)/(-(-a*d+b*c)^(1/3)*((-a*d+b*c)^(1/3)-b^(1/3)*(d*x+c)^(1 /3))/((1-3^(1/2))*(-a*d+b*c)^(1/3)-b^(1/3)*(d*x+c)^(1/3))^2)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.16 \[ \int (a+b x)^{3/2} \sqrt [3]{c+d x} \, dx=\frac {2 (a+b x)^{5/2} \sqrt [3]{c+d x} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {5}{2},\frac {7}{2},\frac {d (a+b x)}{-b c+a d}\right )}{5 b \sqrt [3]{\frac {b (c+d x)}{b c-a d}}} \] Input:
Integrate[(a + b*x)^(3/2)*(c + d*x)^(1/3),x]
Output:
(2*(a + b*x)^(5/2)*(c + d*x)^(1/3)*Hypergeometric2F1[-1/3, 5/2, 7/2, (d*(a + b*x))/(-(b*c) + a*d)])/(5*b*((b*(c + d*x))/(b*c - a*d))^(1/3))
Time = 0.43 (sec) , antiderivative size = 477, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {60, 60, 60, 73, 760}
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.
| \(\displaystyle \int (a+b x)^{3/2} \sqrt [3]{c+d x} \, dx\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {2 (b c-a d) \int \frac {(a+b x)^{3/2}}{(c+d x)^{2/3}}dx}{17 b}+\frac {6 (a+b x)^{5/2} \sqrt [3]{c+d x}}{17 b}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {2 (b c-a d) \left (\frac {6 (a+b x)^{3/2} \sqrt [3]{c+d x}}{11 d}-\frac {9 (b c-a d) \int \frac {\sqrt {a+b x}}{(c+d x)^{2/3}}dx}{11 d}\right )}{17 b}+\frac {6 (a+b x)^{5/2} \sqrt [3]{c+d x}}{17 b}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {2 (b c-a d) \left (\frac {6 (a+b x)^{3/2} \sqrt [3]{c+d x}}{11 d}-\frac {9 (b c-a d) \left (\frac {6 \sqrt {a+b x} \sqrt [3]{c+d x}}{5 d}-\frac {3 (b c-a d) \int \frac {1}{\sqrt {a+b x} (c+d x)^{2/3}}dx}{5 d}\right )}{11 d}\right )}{17 b}+\frac {6 (a+b x)^{5/2} \sqrt [3]{c+d x}}{17 b}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {2 (b c-a d) \left (\frac {6 (a+b x)^{3/2} \sqrt [3]{c+d x}}{11 d}-\frac {9 (b c-a d) \left (\frac {6 \sqrt {a+b x} \sqrt [3]{c+d x}}{5 d}-\frac {9 (b c-a d) \int \frac {1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [3]{c+d x}}{5 d^2}\right )}{11 d}\right )}{17 b}+\frac {6 (a+b x)^{5/2} \sqrt [3]{c+d x}}{17 b}\) |
| \(\Big \downarrow \) 760 |
| \(\displaystyle \frac {2 (b c-a d) \left (\frac {6 (a+b x)^{3/2} \sqrt [3]{c+d x}}{11 d}-\frac {9 (b c-a d) \left (\frac {6\ 3^{3/4} \sqrt {2-\sqrt {3}} (b c-a d) \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right ),-7+4 \sqrt {3}\right )}{5 \sqrt [3]{b} d^2 \sqrt {-\frac {\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}+\frac {6 \sqrt {a+b x} \sqrt [3]{c+d x}}{5 d}\right )}{11 d}\right )}{17 b}+\frac {6 (a+b x)^{5/2} \sqrt [3]{c+d x}}{17 b}\) |
Input:
Int[(a + b*x)^(3/2)*(c + d*x)^(1/3),x]
Output:
(6*(a + b*x)^(5/2)*(c + d*x)^(1/3))/(17*b) + (2*(b*c - a*d)*((6*(a + b*x)^ (3/2)*(c + d*x)^(1/3))/(11*d) - (9*(b*c - a*d)*((6*Sqrt[a + b*x]*(c + d*x) ^(1/3))/(5*d) + (6*3^(3/4)*Sqrt[2 - Sqrt[3]]*(b*c - a*d)*((b*c - a*d)^(1/3 ) - b^(1/3)*(c + d*x)^(1/3))*Sqrt[((b*c - a*d)^(2/3) + b^(1/3)*(b*c - a*d) ^(1/3)*(c + d*x)^(1/3) + b^(2/3)*(c + d*x)^(2/3))/((1 - Sqrt[3])*(b*c - a* d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*(b* c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))/((1 - Sqrt[3])*(b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))], -7 + 4*Sqrt[3]])/(5*b^(1/3)*d^2*Sqrt[-(((b*c - a*d)^(1/3)*((b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3)))/((1 - Sqrt[3] )*(b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))^2)]*Sqrt[a - (b*c)/d + (b*( c + d*x))/d])))/(11*d)))/(17*b)
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
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]
\[\int \left (b x +a \right )^{\frac {3}{2}} \left (x d +c \right )^{\frac {1}{3}}d x\]
Input:
int((b*x+a)^(3/2)*(d*x+c)^(1/3),x)
Output:
int((b*x+a)^(3/2)*(d*x+c)^(1/3),x)
\[ \int (a+b x)^{3/2} \sqrt [3]{c+d x} \, dx=\int { {\left (b x + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {1}{3}} \,d x } \] Input:
integrate((b*x+a)^(3/2)*(d*x+c)^(1/3),x, algorithm="fricas")
Output:
integral((b*x + a)^(3/2)*(d*x + c)^(1/3), x)
\[ \int (a+b x)^{3/2} \sqrt [3]{c+d x} \, dx=\int \left (a + b x\right )^{\frac {3}{2}} \sqrt [3]{c + d x}\, dx \] Input:
integrate((b*x+a)**(3/2)*(d*x+c)**(1/3),x)
Output:
Integral((a + b*x)**(3/2)*(c + d*x)**(1/3), x)
\[ \int (a+b x)^{3/2} \sqrt [3]{c+d x} \, dx=\int { {\left (b x + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {1}{3}} \,d x } \] Input:
integrate((b*x+a)^(3/2)*(d*x+c)^(1/3),x, algorithm="maxima")
Output:
integrate((b*x + a)^(3/2)*(d*x + c)^(1/3), x)
\[ \int (a+b x)^{3/2} \sqrt [3]{c+d x} \, dx=\int { {\left (b x + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {1}{3}} \,d x } \] Input:
integrate((b*x+a)^(3/2)*(d*x+c)^(1/3),x, algorithm="giac")
Output:
integrate((b*x + a)^(3/2)*(d*x + c)^(1/3), x)
Timed out. \[ \int (a+b x)^{3/2} \sqrt [3]{c+d x} \, dx=\int {\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{1/3} \,d x \] Input:
int((a + b*x)^(3/2)*(c + d*x)^(1/3),x)
Output:
int((a + b*x)^(3/2)*(c + d*x)^(1/3), x)
\[ \int (a+b x)^{3/2} \sqrt [3]{c+d x} \, dx=\frac {\frac {402 \left (d x +c \right )^{\frac {1}{3}} \sqrt {b x +a}\, a^{2} c d}{187}+\frac {240 \left (d x +c \right )^{\frac {1}{3}} \sqrt {b x +a}\, a^{2} d^{2} x}{187}-\frac {72 \left (d x +c \right )^{\frac {1}{3}} \sqrt {b x +a}\, a b \,c^{2}}{187}+\frac {384 \left (d x +c \right )^{\frac {1}{3}} \sqrt {b x +a}\, a b c d x}{187}+\frac {12 \left (d x +c \right )^{\frac {1}{3}} \sqrt {b x +a}\, a b \,d^{2} x^{2}}{17}+\frac {36 \left (d x +c \right )^{\frac {1}{3}} \sqrt {b x +a}\, b^{2} c^{2} x}{187}+\frac {18 \left (d x +c \right )^{\frac {1}{3}} \sqrt {b x +a}\, b^{2} c d \,x^{2}}{17}+\frac {108 \left (\int \frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b x +a}\, x}{2 a b \,d^{2} x^{2}+3 b^{2} c d \,x^{2}+2 a^{2} d^{2} x +5 a b c d x +3 b^{2} c^{2} x +2 a^{2} c d +3 a b \,c^{2}}d x \right ) a^{4} d^{4}}{187}-\frac {162 \left (\int \frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b x +a}\, x}{2 a b \,d^{2} x^{2}+3 b^{2} c d \,x^{2}+2 a^{2} d^{2} x +5 a b c d x +3 b^{2} c^{2} x +2 a^{2} c d +3 a b \,c^{2}}d x \right ) a^{3} b c \,d^{3}}{187}-\frac {162 \left (\int \frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b x +a}\, x}{2 a b \,d^{2} x^{2}+3 b^{2} c d \,x^{2}+2 a^{2} d^{2} x +5 a b c d x +3 b^{2} c^{2} x +2 a^{2} c d +3 a b \,c^{2}}d x \right ) a^{2} b^{2} c^{2} d^{2}}{187}+\frac {378 \left (\int \frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b x +a}\, x}{2 a b \,d^{2} x^{2}+3 b^{2} c d \,x^{2}+2 a^{2} d^{2} x +5 a b c d x +3 b^{2} c^{2} x +2 a^{2} c d +3 a b \,c^{2}}d x \right ) a \,b^{3} c^{3} d}{187}-\frac {162 \left (\int \frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b x +a}\, x}{2 a b \,d^{2} x^{2}+3 b^{2} c d \,x^{2}+2 a^{2} d^{2} x +5 a b c d x +3 b^{2} c^{2} x +2 a^{2} c d +3 a b \,c^{2}}d x \right ) b^{4} c^{4}}{187}}{d \left (2 a d +3 b c \right )} \] Input:
int((b*x+a)^(3/2)*(d*x+c)^(1/3),x)
Output:
(6*(67*(c + d*x)**(1/3)*sqrt(a + b*x)*a**2*c*d + 40*(c + d*x)**(1/3)*sqrt( a + b*x)*a**2*d**2*x - 12*(c + d*x)**(1/3)*sqrt(a + b*x)*a*b*c**2 + 64*(c + d*x)**(1/3)*sqrt(a + b*x)*a*b*c*d*x + 22*(c + d*x)**(1/3)*sqrt(a + b*x)* a*b*d**2*x**2 + 6*(c + d*x)**(1/3)*sqrt(a + b*x)*b**2*c**2*x + 33*(c + d*x )**(1/3)*sqrt(a + b*x)*b**2*c*d*x**2 + 18*int(((c + d*x)**(1/3)*sqrt(a + b *x)*x)/(2*a**2*c*d + 2*a**2*d**2*x + 3*a*b*c**2 + 5*a*b*c*d*x + 2*a*b*d**2 *x**2 + 3*b**2*c**2*x + 3*b**2*c*d*x**2),x)*a**4*d**4 - 27*int(((c + d*x)* *(1/3)*sqrt(a + b*x)*x)/(2*a**2*c*d + 2*a**2*d**2*x + 3*a*b*c**2 + 5*a*b*c *d*x + 2*a*b*d**2*x**2 + 3*b**2*c**2*x + 3*b**2*c*d*x**2),x)*a**3*b*c*d**3 - 27*int(((c + d*x)**(1/3)*sqrt(a + b*x)*x)/(2*a**2*c*d + 2*a**2*d**2*x + 3*a*b*c**2 + 5*a*b*c*d*x + 2*a*b*d**2*x**2 + 3*b**2*c**2*x + 3*b**2*c*d*x **2),x)*a**2*b**2*c**2*d**2 + 63*int(((c + d*x)**(1/3)*sqrt(a + b*x)*x)/(2 *a**2*c*d + 2*a**2*d**2*x + 3*a*b*c**2 + 5*a*b*c*d*x + 2*a*b*d**2*x**2 + 3 *b**2*c**2*x + 3*b**2*c*d*x**2),x)*a*b**3*c**3*d - 27*int(((c + d*x)**(1/3 )*sqrt(a + b*x)*x)/(2*a**2*c*d + 2*a**2*d**2*x + 3*a*b*c**2 + 5*a*b*c*d*x + 2*a*b*d**2*x**2 + 3*b**2*c**2*x + 3*b**2*c*d*x**2),x)*b**4*c**4))/(187*d *(2*a*d + 3*b*c))