Integrand size = 19, antiderivative size = 179 \[ \int \frac {(a+b x)^{5/2}}{(c+d x)^{5/4}} \, dx=\frac {16 (b c-a d)^2 \sqrt {a+b x}}{3 d^3 \sqrt [4]{c+d x}}-\frac {8 (b c-a d) (a+b x)^{3/2}}{9 d^2 \sqrt [4]{c+d x}}+\frac {4 (a+b x)^{5/2}}{9 d \sqrt [4]{c+d x}}-\frac {32 (b c-a d)^{5/2} \sqrt [4]{\frac {b (c+d x)}{b c-a d}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )\right |2\right )}{3 d^{7/2} \sqrt [4]{c+d x}} \] Output:
16/3*(-a*d+b*c)^2*(b*x+a)^(1/2)/d^3/(d*x+c)^(1/4)-8/9*(-a*d+b*c)*(b*x+a)^( 3/2)/d^2/(d*x+c)^(1/4)+4/9*(b*x+a)^(5/2)/d/(d*x+c)^(1/4)-32/3*(-a*d+b*c)^( 5/2)*(b*(d*x+c)/(-a*d+b*c))^(1/4)*EllipticE(sin(1/2*arctan(d^(1/2)*(b*x+a) ^(1/2)/(-a*d+b*c)^(1/2))),2^(1/2))/d^(7/2)/(d*x+c)^(1/4)
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 = 0.41 \[ \int \frac {(a+b x)^{5/2}}{(c+d x)^{5/4}} \, dx=\frac {2 (a+b x)^{7/2} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/4} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {7}{2},\frac {9}{2},\frac {d (a+b x)}{-b c+a d}\right )}{7 b (c+d x)^{5/4}} \] Input:
Integrate[(a + b*x)^(5/2)/(c + d*x)^(5/4),x]
Output:
(2*(a + b*x)^(7/2)*((b*(c + d*x))/(b*c - a*d))^(5/4)*Hypergeometric2F1[5/4 , 7/2, 9/2, (d*(a + b*x))/(-(b*c) + a*d)])/(7*b*(c + d*x)^(5/4))
Time = 0.47 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.70, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {57, 60, 60, 73, 836, 765, 762, 1390, 1388, 327}
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.
| \(\displaystyle \int \frac {(a+b x)^{5/2}}{(c+d x)^{5/4}} \, dx\) |
| \(\Big \downarrow \) 57 |
| \(\displaystyle \frac {10 b \int \frac {(a+b x)^{3/2}}{\sqrt [4]{c+d x}}dx}{d}-\frac {4 (a+b x)^{5/2}}{d \sqrt [4]{c+d x}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {10 b \left (\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 d}-\frac {2 (b c-a d) \int \frac {\sqrt {a+b x}}{\sqrt [4]{c+d x}}dx}{3 d}\right )}{d}-\frac {4 (a+b x)^{5/2}}{d \sqrt [4]{c+d x}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {10 b \left (\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 d}-\frac {2 (b c-a d) \left (\frac {4 \sqrt {a+b x} (c+d x)^{3/4}}{5 d}-\frac {2 (b c-a d) \int \frac {1}{\sqrt {a+b x} \sqrt [4]{c+d x}}dx}{5 d}\right )}{3 d}\right )}{d}-\frac {4 (a+b x)^{5/2}}{d \sqrt [4]{c+d x}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {10 b \left (\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 d}-\frac {2 (b c-a d) \left (\frac {4 \sqrt {a+b x} (c+d x)^{3/4}}{5 d}-\frac {8 (b c-a d) \int \frac {\sqrt {c+d x}}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{5 d^2}\right )}{3 d}\right )}{d}-\frac {4 (a+b x)^{5/2}}{d \sqrt [4]{c+d x}}\) |
| \(\Big \downarrow \) 836 |
| \(\displaystyle \frac {10 b \left (\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 d}-\frac {2 (b c-a d) \left (\frac {4 \sqrt {a+b x} (c+d x)^{3/4}}{5 d}-\frac {8 (b c-a d) \left (\frac {\sqrt {b c-a d} \int \frac {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{\sqrt {b}}-\frac {\sqrt {b c-a d} \int \frac {1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{\sqrt {b}}\right )}{5 d^2}\right )}{3 d}\right )}{d}-\frac {4 (a+b x)^{5/2}}{d \sqrt [4]{c+d x}}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {10 b \left (\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 d}-\frac {2 (b c-a d) \left (\frac {4 \sqrt {a+b x} (c+d x)^{3/4}}{5 d}-\frac {8 (b c-a d) \left (\frac {\sqrt {b c-a d} \int \frac {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{\sqrt {b}}-\frac {\sqrt {b c-a d} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \int \frac {1}{\sqrt {1-\frac {b (c+d x)}{b c-a d}}}d\sqrt [4]{c+d x}}{\sqrt {b} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{5 d^2}\right )}{3 d}\right )}{d}-\frac {4 (a+b x)^{5/2}}{d \sqrt [4]{c+d x}}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {10 b \left (\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 d}-\frac {2 (b c-a d) \left (\frac {4 \sqrt {a+b x} (c+d x)^{3/4}}{5 d}-\frac {8 (b c-a d) \left (\frac {\sqrt {b c-a d} \int \frac {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{\sqrt {b}}-\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{5 d^2}\right )}{3 d}\right )}{d}-\frac {4 (a+b x)^{5/2}}{d \sqrt [4]{c+d x}}\) |
| \(\Big \downarrow \) 1390 |
| \(\displaystyle \frac {10 b \left (\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 d}-\frac {2 (b c-a d) \left (\frac {4 \sqrt {a+b x} (c+d x)^{3/4}}{5 d}-\frac {8 (b c-a d) \left (\frac {\sqrt {b c-a d} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \int \frac {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}{\sqrt {1-\frac {b (c+d x)}{b c-a d}}}d\sqrt [4]{c+d x}}{\sqrt {b} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}-\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{5 d^2}\right )}{3 d}\right )}{d}-\frac {4 (a+b x)^{5/2}}{d \sqrt [4]{c+d x}}\) |
| \(\Big \downarrow \) 1388 |
| \(\displaystyle \frac {10 b \left (\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 d}-\frac {2 (b c-a d) \left (\frac {4 \sqrt {a+b x} (c+d x)^{3/4}}{5 d}-\frac {8 (b c-a d) \left (\frac {\sqrt {b c-a d} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \int \frac {\sqrt {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}}{\sqrt {1-\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}}}d\sqrt [4]{c+d x}}{\sqrt {b} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}-\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{5 d^2}\right )}{3 d}\right )}{d}-\frac {4 (a+b x)^{5/2}}{d \sqrt [4]{c+d x}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {10 b \left (\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 d}-\frac {2 (b c-a d) \left (\frac {4 \sqrt {a+b x} (c+d x)^{3/4}}{5 d}-\frac {8 (b c-a d) \left (\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}-\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{5 d^2}\right )}{3 d}\right )}{d}-\frac {4 (a+b x)^{5/2}}{d \sqrt [4]{c+d x}}\) |
Input:
Int[(a + b*x)^(5/2)/(c + d*x)^(5/4),x]
Output:
(-4*(a + b*x)^(5/2))/(d*(c + d*x)^(1/4)) + (10*b*((4*(a + b*x)^(3/2)*(c + d*x)^(3/4))/(9*d) - (2*(b*c - a*d)*((4*Sqrt[a + b*x]*(c + d*x)^(3/4))/(5*d ) - (8*(b*c - a*d)*(((b*c - a*d)^(3/4)*Sqrt[1 - (b*(c + d*x))/(b*c - a*d)] *EllipticE[ArcSin[(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(b^(3 /4)*Sqrt[a - (b*c)/d + (b*(c + d*x))/d]) - ((b*c - a*d)^(3/4)*Sqrt[1 - (b* (c + d*x))/(b*c - a*d)]*EllipticF[ArcSin[(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(b^(3/4)*Sqrt[a - (b*c)/d + (b*(c + d*x))/d])))/(5*d^2)) )/(3*d)))/d
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] & & GtQ[n, 0] && LtQ[m, -1] && !(IntegerQ[n] && !IntegerQ[m]) && !(ILeQ[m + n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c , d, m, n, x]
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[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
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[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[-q^(-1) Int[1/Sqrt[a + b*x^4], x], x] + Simp[1/q Int[(1 + q*x^2)/S qrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer Q[p] || (GtQ[a, 0] && GtQ[d, 0]))
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[Sqrt [1 + c*(x^4/a)]/Sqrt[a + c*x^4] Int[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && !GtQ [a, 0] && !(LtQ[a, 0] && GtQ[c, 0])
\[\int \frac {\left (b x +a \right )^{\frac {5}{2}}}{\left (x d +c \right )^{\frac {5}{4}}}d x\]
Input:
int((b*x+a)^(5/2)/(d*x+c)^(5/4),x)
Output:
int((b*x+a)^(5/2)/(d*x+c)^(5/4),x)
\[ \int \frac {(a+b x)^{5/2}}{(c+d x)^{5/4}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {5}{2}}}{{\left (d x + c\right )}^{\frac {5}{4}}} \,d x } \] Input:
integrate((b*x+a)^(5/2)/(d*x+c)^(5/4),x, algorithm="fricas")
Output:
integral((b^2*x^2 + 2*a*b*x + a^2)*sqrt(b*x + a)*(d*x + c)^(3/4)/(d^2*x^2 + 2*c*d*x + c^2), x)
\[ \int \frac {(a+b x)^{5/2}}{(c+d x)^{5/4}} \, dx=\int \frac {\left (a + b x\right )^{\frac {5}{2}}}{\left (c + d x\right )^{\frac {5}{4}}}\, dx \] Input:
integrate((b*x+a)**(5/2)/(d*x+c)**(5/4),x)
Output:
Integral((a + b*x)**(5/2)/(c + d*x)**(5/4), x)
\[ \int \frac {(a+b x)^{5/2}}{(c+d x)^{5/4}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {5}{2}}}{{\left (d x + c\right )}^{\frac {5}{4}}} \,d x } \] Input:
integrate((b*x+a)^(5/2)/(d*x+c)^(5/4),x, algorithm="maxima")
Output:
integrate((b*x + a)^(5/2)/(d*x + c)^(5/4), x)
\[ \int \frac {(a+b x)^{5/2}}{(c+d x)^{5/4}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {5}{2}}}{{\left (d x + c\right )}^{\frac {5}{4}}} \,d x } \] Input:
integrate((b*x+a)^(5/2)/(d*x+c)^(5/4),x, algorithm="giac")
Output:
integrate((b*x + a)^(5/2)/(d*x + c)^(5/4), x)
Timed out. \[ \int \frac {(a+b x)^{5/2}}{(c+d x)^{5/4}} \, dx=\int \frac {{\left (a+b\,x\right )}^{5/2}}{{\left (c+d\,x\right )}^{5/4}} \,d x \] Input:
int((a + b*x)^(5/2)/(c + d*x)^(5/4),x)
Output:
int((a + b*x)^(5/2)/(c + d*x)^(5/4), x)
\[ \int \frac {(a+b x)^{5/2}}{(c+d x)^{5/4}} \, dx=\left (\int \frac {\sqrt {b x +a}}{\left (d x +c \right )^{\frac {1}{4}} c +\left (d x +c \right )^{\frac {1}{4}} d x}d x \right ) a^{2}+\left (\int \frac {\sqrt {b x +a}\, x^{2}}{\left (d x +c \right )^{\frac {1}{4}} c +\left (d x +c \right )^{\frac {1}{4}} d x}d x \right ) b^{2}+2 \left (\int \frac {\sqrt {b x +a}\, x}{\left (d x +c \right )^{\frac {1}{4}} c +\left (d x +c \right )^{\frac {1}{4}} d x}d x \right ) a b \] Input:
int((b*x+a)^(5/2)/(d*x+c)^(5/4),x)
Output:
int(sqrt(a + b*x)/((c + d*x)**(1/4)*c + (c + d*x)**(1/4)*d*x),x)*a**2 + in t((sqrt(a + b*x)*x**2)/((c + d*x)**(1/4)*c + (c + d*x)**(1/4)*d*x),x)*b**2 + 2*int((sqrt(a + b*x)*x)/((c + d*x)**(1/4)*c + (c + d*x)**(1/4)*d*x),x)* a*b