Integrand size = 19, antiderivative size = 182 \[ \int \sqrt {a+b x} (c+d x)^{5/4} \, dx=\frac {20 (b c-a d)^2 \sqrt {a+b x} \sqrt [4]{c+d x}}{231 b^2 d}+\frac {20 (b c-a d) (a+b x)^{3/2} \sqrt [4]{c+d x}}{77 b^2}+\frac {4 (a+b x)^{3/2} (c+d x)^{5/4}}{11 b}-\frac {40 (b c-a d)^{13/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{231 b^{9/4} d^2 \sqrt {a+b x}} \]
20/77*(-a*d+b*c)*(b*x+a)^(3/2)*(d*x+c)^(1/4)/b^2+4/11*(b*x+a)^(3/2)*(d*x+c )^(5/4)/b+20/231*(-a*d+b*c)^2*(d*x+c)^(1/4)*(b*x+a)^(1/2)/b^2/d-40/231*(-a *d+b*c)^(13/4)*EllipticF(b^(1/4)*(d*x+c)^(1/4)/(-a*d+b*c)^(1/4),I)*(-d*(b* x+a)/(-a*d+b*c))^(1/2)/b^(9/4)/d^2/(b*x+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.40 \[ \int \sqrt {a+b x} (c+d x)^{5/4} \, dx=\frac {2 (a+b x)^{3/2} (c+d x)^{5/4} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {3}{2},\frac {5}{2},\frac {d (a+b x)}{-b c+a d}\right )}{3 b \left (\frac {b (c+d x)}{b c-a d}\right )^{5/4}} \]
(2*(a + b*x)^(3/2)*(c + d*x)^(5/4)*Hypergeometric2F1[-5/4, 3/2, 5/2, (d*(a + b*x))/(-(b*c) + a*d)])/(3*b*((b*(c + d*x))/(b*c - a*d))^(5/4))
Time = 0.27 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.15, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {60, 60, 60, 73, 765, 762}
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.
| \(\displaystyle \int \sqrt {a+b x} (c+d x)^{5/4} \, dx\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {5 (b c-a d) \int \sqrt {a+b x} \sqrt [4]{c+d x}dx}{11 b}+\frac {4 (a+b x)^{3/2} (c+d x)^{5/4}}{11 b}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {5 (b c-a d) \left (\frac {(b c-a d) \int \frac {\sqrt {a+b x}}{(c+d x)^{3/4}}dx}{7 b}+\frac {4 (a+b x)^{3/2} \sqrt [4]{c+d x}}{7 b}\right )}{11 b}+\frac {4 (a+b x)^{3/2} (c+d x)^{5/4}}{11 b}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {5 (b c-a d) \left (\frac {(b c-a d) \left (\frac {4 \sqrt {a+b x} \sqrt [4]{c+d x}}{3 d}-\frac {2 (b c-a d) \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/4}}dx}{3 d}\right )}{7 b}+\frac {4 (a+b x)^{3/2} \sqrt [4]{c+d x}}{7 b}\right )}{11 b}+\frac {4 (a+b x)^{3/2} (c+d x)^{5/4}}{11 b}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {5 (b c-a d) \left (\frac {(b c-a d) \left (\frac {4 \sqrt {a+b x} \sqrt [4]{c+d x}}{3 d}-\frac {8 (b c-a d) \int \frac {1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{3 d^2}\right )}{7 b}+\frac {4 (a+b x)^{3/2} \sqrt [4]{c+d x}}{7 b}\right )}{11 b}+\frac {4 (a+b x)^{3/2} (c+d x)^{5/4}}{11 b}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {5 (b c-a d) \left (\frac {(b c-a d) \left (\frac {4 \sqrt {a+b x} \sqrt [4]{c+d x}}{3 d}-\frac {8 (b c-a d) \sqrt {1-\frac {b (c+d x)}{b c-a d}} \int \frac {1}{\sqrt {1-\frac {b (c+d x)}{b c-a d}}}d\sqrt [4]{c+d x}}{3 d^2 \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{7 b}+\frac {4 (a+b x)^{3/2} \sqrt [4]{c+d x}}{7 b}\right )}{11 b}+\frac {4 (a+b x)^{3/2} (c+d x)^{5/4}}{11 b}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {5 (b c-a d) \left (\frac {(b c-a d) \left (\frac {4 \sqrt {a+b x} \sqrt [4]{c+d x}}{3 d}-\frac {8 (b c-a d)^{5/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{3 \sqrt [4]{b} d^2 \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{7 b}+\frac {4 (a+b x)^{3/2} \sqrt [4]{c+d x}}{7 b}\right )}{11 b}+\frac {4 (a+b x)^{3/2} (c+d x)^{5/4}}{11 b}\) |
(4*(a + b*x)^(3/2)*(c + d*x)^(5/4))/(11*b) + (5*(b*c - a*d)*((4*(a + b*x)^ (3/2)*(c + d*x)^(1/4))/(7*b) + ((b*c - a*d)*((4*Sqrt[a + b*x]*(c + d*x)^(1 /4))/(3*d) - (8*(b*c - a*d)^(5/4)*Sqrt[1 - (b*(c + d*x))/(b*c - a*d)]*Elli pticF[ArcSin[(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(3*b^(1/4) *d^2*Sqrt[a - (b*c)/d + (b*(c + d*x))/d])))/(7*b)))/(11*b)
3.17.42.3.1 Defintions of rubi rules used
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_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) )*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt [a + b*x^4] Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ [b/a] && !GtQ[a, 0]
\[\int \sqrt {b x +a}\, \left (d x +c \right )^{\frac {5}{4}}d x\]
\[ \int \sqrt {a+b x} (c+d x)^{5/4} \, dx=\int { \sqrt {b x + a} {\left (d x + c\right )}^{\frac {5}{4}} \,d x } \]
\[ \int \sqrt {a+b x} (c+d x)^{5/4} \, dx=\int \sqrt {a + b x} \left (c + d x\right )^{\frac {5}{4}}\, dx \]
\[ \int \sqrt {a+b x} (c+d x)^{5/4} \, dx=\int { \sqrt {b x + a} {\left (d x + c\right )}^{\frac {5}{4}} \,d x } \]
\[ \int \sqrt {a+b x} (c+d x)^{5/4} \, dx=\int { \sqrt {b x + a} {\left (d x + c\right )}^{\frac {5}{4}} \,d x } \]
Timed out. \[ \int \sqrt {a+b x} (c+d x)^{5/4} \, dx=\int \sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{5/4} \,d x \]
\[ \int \sqrt {a+b x} (c+d x)^{5/4} \, dx=\frac {-\frac {32 \left (d x +c \right )^{\frac {1}{4}} \sqrt {b x +a}\, a^{2} c d}{77}+\frac {8 \left (d x +c \right )^{\frac {1}{4}} \sqrt {b x +a}\, a^{2} d^{2} x}{77}+\frac {116 \left (d x +c \right )^{\frac {1}{4}} \sqrt {b x +a}\, a b \,c^{2}}{77}+\frac {64 \left (d x +c \right )^{\frac {1}{4}} \sqrt {b x +a}\, a b c d x}{77}+\frac {4 \left (d x +c \right )^{\frac {1}{4}} \sqrt {b x +a}\, a b \,d^{2} x^{2}}{11}+\frac {96 \left (d x +c \right )^{\frac {1}{4}} \sqrt {b x +a}\, b^{2} c^{2} x}{77}+\frac {8 \left (d x +c \right )^{\frac {1}{4}} \sqrt {b x +a}\, b^{2} c d \,x^{2}}{11}-\frac {10 \left (\int \frac {\left (d x +c \right )^{\frac {1}{4}} \sqrt {b x +a}\, x}{a b \,d^{2} x^{2}+2 b^{2} c d \,x^{2}+a^{2} d^{2} x +3 a b c d x +2 b^{2} c^{2} x +a^{2} c d +2 a b \,c^{2}}d x \right ) a^{4} d^{4}}{77}+\frac {10 \left (\int \frac {\left (d x +c \right )^{\frac {1}{4}} \sqrt {b x +a}\, x}{a b \,d^{2} x^{2}+2 b^{2} c d \,x^{2}+a^{2} d^{2} x +3 a b c d x +2 b^{2} c^{2} x +a^{2} c d +2 a b \,c^{2}}d x \right ) a^{3} b c \,d^{3}}{77}+\frac {30 \left (\int \frac {\left (d x +c \right )^{\frac {1}{4}} \sqrt {b x +a}\, x}{a b \,d^{2} x^{2}+2 b^{2} c d \,x^{2}+a^{2} d^{2} x +3 a b c d x +2 b^{2} c^{2} x +a^{2} c d +2 a b \,c^{2}}d x \right ) a^{2} b^{2} c^{2} d^{2}}{77}-\frac {50 \left (\int \frac {\left (d x +c \right )^{\frac {1}{4}} \sqrt {b x +a}\, x}{a b \,d^{2} x^{2}+2 b^{2} c d \,x^{2}+a^{2} d^{2} x +3 a b c d x +2 b^{2} c^{2} x +a^{2} c d +2 a b \,c^{2}}d x \right ) a \,b^{3} c^{3} d}{77}+\frac {20 \left (\int \frac {\left (d x +c \right )^{\frac {1}{4}} \sqrt {b x +a}\, x}{a b \,d^{2} x^{2}+2 b^{2} c d \,x^{2}+a^{2} d^{2} x +3 a b c d x +2 b^{2} c^{2} x +a^{2} c d +2 a b \,c^{2}}d x \right ) b^{4} c^{4}}{77}}{b \left (a d +2 b c \right )} \]
(2*( - 16*(c + d*x)**(1/4)*sqrt(a + b*x)*a**2*c*d + 4*(c + d*x)**(1/4)*sqr t(a + b*x)*a**2*d**2*x + 58*(c + d*x)**(1/4)*sqrt(a + b*x)*a*b*c**2 + 32*( c + d*x)**(1/4)*sqrt(a + b*x)*a*b*c*d*x + 14*(c + d*x)**(1/4)*sqrt(a + b*x )*a*b*d**2*x**2 + 48*(c + d*x)**(1/4)*sqrt(a + b*x)*b**2*c**2*x + 28*(c + d*x)**(1/4)*sqrt(a + b*x)*b**2*c*d*x**2 - 5*int(((c + d*x)**(1/4)*sqrt(a + b*x)*x)/(a**2*c*d + a**2*d**2*x + 2*a*b*c**2 + 3*a*b*c*d*x + a*b*d**2*x** 2 + 2*b**2*c**2*x + 2*b**2*c*d*x**2),x)*a**4*d**4 + 5*int(((c + d*x)**(1/4 )*sqrt(a + b*x)*x)/(a**2*c*d + a**2*d**2*x + 2*a*b*c**2 + 3*a*b*c*d*x + a* b*d**2*x**2 + 2*b**2*c**2*x + 2*b**2*c*d*x**2),x)*a**3*b*c*d**3 + 15*int(( (c + d*x)**(1/4)*sqrt(a + b*x)*x)/(a**2*c*d + a**2*d**2*x + 2*a*b*c**2 + 3 *a*b*c*d*x + a*b*d**2*x**2 + 2*b**2*c**2*x + 2*b**2*c*d*x**2),x)*a**2*b**2 *c**2*d**2 - 25*int(((c + d*x)**(1/4)*sqrt(a + b*x)*x)/(a**2*c*d + a**2*d* *2*x + 2*a*b*c**2 + 3*a*b*c*d*x + a*b*d**2*x**2 + 2*b**2*c**2*x + 2*b**2*c *d*x**2),x)*a*b**3*c**3*d + 10*int(((c + d*x)**(1/4)*sqrt(a + b*x)*x)/(a** 2*c*d + a**2*d**2*x + 2*a*b*c**2 + 3*a*b*c*d*x + a*b*d**2*x**2 + 2*b**2*c* *2*x + 2*b**2*c*d*x**2),x)*b**4*c**4))/(77*b*(a*d + 2*b*c))