Integrand size = 19, antiderivative size = 185 \[ \int \sqrt {a+b x} (c+d x)^{3/4} \, dx=-\frac {4 (b c-a d)^2 \sqrt {a+b x}}{15 b^2 d \sqrt [4]{c+d x}}+\frac {4 (b c-a d) (a+b x)^{3/2}}{15 b^2 \sqrt [4]{c+d x}}+\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 b}+\frac {8 (b c-a d)^{5/2} \sqrt [4]{\frac {b (c+d x)}{b c-a d}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )\right |2\right )}{15 b^2 d^{3/2} \sqrt [4]{c+d x}} \] Output:
-4/15*(-a*d+b*c)^2*(b*x+a)^(1/2)/b^2/d/(d*x+c)^(1/4)+4/15*(-a*d+b*c)*(b*x+ a)^(3/2)/b^2/(d*x+c)^(1/4)+4/9*(b*x+a)^(3/2)*(d*x+c)^(3/4)/b+8/15*(-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))/b^2/d^(3/2)/(d*x+c)^(1/4)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.39 \[ \int \sqrt {a+b x} (c+d x)^{3/4} \, dx=\frac {2 (a+b x)^{3/2} (c+d x)^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {3}{2},\frac {5}{2},\frac {d (a+b x)}{-b c+a d}\right )}{3 b \left (\frac {b (c+d x)}{b c-a d}\right )^{3/4}} \] Input:
Integrate[Sqrt[a + b*x]*(c + d*x)^(3/4),x]
Output:
(2*(a + b*x)^(3/2)*(c + d*x)^(3/4)*Hypergeometric2F1[-3/4, 3/2, 5/2, (d*(a + b*x))/(-(b*c) + a*d)])/(3*b*((b*(c + d*x))/(b*c - a*d))^(3/4))
Time = 0.44 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.48, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {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.
\(\displaystyle \int \sqrt {a+b x} (c+d x)^{3/4} \, dx\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {(b c-a d) \int \frac {\sqrt {a+b x}}{\sqrt [4]{c+d x}}dx}{3 b}+\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 b}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {(b c-a d) \left (\frac {4 \sqrt {a+b x} (c+d x)^{3/4}}{5 d}-\frac {2 (b c-a d) \int \frac {1}{\sqrt {a+b x} \sqrt [4]{c+d x}}dx}{5 d}\right )}{3 b}+\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 b}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {(b c-a d) \left (\frac {4 \sqrt {a+b x} (c+d x)^{3/4}}{5 d}-\frac {8 (b c-a d) \int \frac {\sqrt {c+d x}}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{5 d^2}\right )}{3 b}+\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 b}\) |
\(\Big \downarrow \) 836 |
\(\displaystyle \frac {(b c-a d) \left (\frac {4 \sqrt {a+b x} (c+d x)^{3/4}}{5 d}-\frac {8 (b c-a d) \left (\frac {\sqrt {b c-a d} \int \frac {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{\sqrt {b}}-\frac {\sqrt {b c-a d} \int \frac {1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{\sqrt {b}}\right )}{5 d^2}\right )}{3 b}+\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 b}\) |
\(\Big \downarrow \) 765 |
\(\displaystyle \frac {(b c-a d) \left (\frac {4 \sqrt {a+b x} (c+d x)^{3/4}}{5 d}-\frac {8 (b c-a d) \left (\frac {\sqrt {b c-a d} \int \frac {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{\sqrt {b}}-\frac {\sqrt {b c-a d} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \int \frac {1}{\sqrt {1-\frac {b (c+d x)}{b c-a d}}}d\sqrt [4]{c+d x}}{\sqrt {b} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{5 d^2}\right )}{3 b}+\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 b}\) |
\(\Big \downarrow \) 762 |
\(\displaystyle \frac {(b c-a d) \left (\frac {4 \sqrt {a+b x} (c+d x)^{3/4}}{5 d}-\frac {8 (b c-a d) \left (\frac {\sqrt {b c-a d} \int \frac {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{\sqrt {b}}-\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{5 d^2}\right )}{3 b}+\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 b}\) |
\(\Big \downarrow \) 1390 |
\(\displaystyle \frac {(b c-a d) \left (\frac {4 \sqrt {a+b x} (c+d x)^{3/4}}{5 d}-\frac {8 (b c-a d) \left (\frac {\sqrt {b c-a d} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \int \frac {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}{\sqrt {1-\frac {b (c+d x)}{b c-a d}}}d\sqrt [4]{c+d x}}{\sqrt {b} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}-\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{5 d^2}\right )}{3 b}+\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 b}\) |
\(\Big \downarrow \) 1388 |
\(\displaystyle \frac {(b c-a d) \left (\frac {4 \sqrt {a+b x} (c+d x)^{3/4}}{5 d}-\frac {8 (b c-a d) \left (\frac {\sqrt {b c-a d} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \int \frac {\sqrt {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}}{\sqrt {1-\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}}}d\sqrt [4]{c+d x}}{\sqrt {b} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}-\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{5 d^2}\right )}{3 b}+\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 b}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle \frac {(b c-a d) \left (\frac {4 \sqrt {a+b x} (c+d x)^{3/4}}{5 d}-\frac {8 (b c-a d) \left (\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}-\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{5 d^2}\right )}{3 b}+\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 b}\) |
Input:
Int[Sqrt[a + b*x]*(c + d*x)^(3/4),x]
Output:
(4*(a + b*x)^(3/2)*(c + d*x)^(3/4))/(9*b) + ((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*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[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])
\[\int \sqrt {b x +a}\, \left (x d +c \right )^{\frac {3}{4}}d x\]
Input:
int((b*x+a)^(1/2)*(d*x+c)^(3/4),x)
Output:
int((b*x+a)^(1/2)*(d*x+c)^(3/4),x)
\[ \int \sqrt {a+b x} (c+d x)^{3/4} \, dx=\int { \sqrt {b x + a} {\left (d x + c\right )}^{\frac {3}{4}} \,d x } \] Input:
integrate((b*x+a)^(1/2)*(d*x+c)^(3/4),x, algorithm="fricas")
Output:
integral(sqrt(b*x + a)*(d*x + c)^(3/4), x)
\[ \int \sqrt {a+b x} (c+d x)^{3/4} \, dx=\int \sqrt {a + b x} \left (c + d x\right )^{\frac {3}{4}}\, dx \] Input:
integrate((b*x+a)**(1/2)*(d*x+c)**(3/4),x)
Output:
Integral(sqrt(a + b*x)*(c + d*x)**(3/4), x)
\[ \int \sqrt {a+b x} (c+d x)^{3/4} \, dx=\int { \sqrt {b x + a} {\left (d x + c\right )}^{\frac {3}{4}} \,d x } \] Input:
integrate((b*x+a)^(1/2)*(d*x+c)^(3/4),x, algorithm="maxima")
Output:
integrate(sqrt(b*x + a)*(d*x + c)^(3/4), x)
\[ \int \sqrt {a+b x} (c+d x)^{3/4} \, dx=\int { \sqrt {b x + a} {\left (d x + c\right )}^{\frac {3}{4}} \,d x } \] Input:
integrate((b*x+a)^(1/2)*(d*x+c)^(3/4),x, algorithm="giac")
Output:
integrate(sqrt(b*x + a)*(d*x + c)^(3/4), x)
Timed out. \[ \int \sqrt {a+b x} (c+d x)^{3/4} \, dx=\int \sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{3/4} \,d x \] Input:
int((a + b*x)^(1/2)*(c + d*x)^(3/4),x)
Output:
int((a + b*x)^(1/2)*(c + d*x)^(3/4), x)
\[ \int \sqrt {a+b x} (c+d x)^{3/4} \, dx=\frac {20 \left (d x +c \right )^{\frac {3}{4}} \sqrt {b x +a}\, a c +12 \left (d x +c \right )^{\frac {3}{4}} \sqrt {b x +a}\, a d x +8 \left (d x +c \right )^{\frac {3}{4}} \sqrt {b x +a}\, b c x +18 \left (\int \frac {\left (d x +c \right )^{\frac {3}{4}} \sqrt {b x +a}\, x}{3 a b \,d^{2} x^{2}+2 b^{2} c d \,x^{2}+3 a^{2} d^{2} x +5 a b c d x +2 b^{2} c^{2} x +3 a^{2} c d +2 a b \,c^{2}}d x \right ) a^{3} d^{3}-24 \left (\int \frac {\left (d x +c \right )^{\frac {3}{4}} \sqrt {b x +a}\, x}{3 a b \,d^{2} x^{2}+2 b^{2} c d \,x^{2}+3 a^{2} d^{2} x +5 a b c d x +2 b^{2} c^{2} x +3 a^{2} c d +2 a b \,c^{2}}d x \right ) a^{2} b c \,d^{2}-6 \left (\int \frac {\left (d x +c \right )^{\frac {3}{4}} \sqrt {b x +a}\, x}{3 a b \,d^{2} x^{2}+2 b^{2} c d \,x^{2}+3 a^{2} d^{2} x +5 a b c d x +2 b^{2} c^{2} x +3 a^{2} c d +2 a b \,c^{2}}d x \right ) a \,b^{2} c^{2} d +12 \left (\int \frac {\left (d x +c \right )^{\frac {3}{4}} \sqrt {b x +a}\, x}{3 a b \,d^{2} x^{2}+2 b^{2} c d \,x^{2}+3 a^{2} d^{2} x +5 a b c d x +2 b^{2} c^{2} x +3 a^{2} c d +2 a b \,c^{2}}d x \right ) b^{3} c^{3}}{27 a d +18 b c} \] Input:
int((b*x+a)^(1/2)*(d*x+c)^(3/4),x)
Output:
(2*(10*(c + d*x)**(3/4)*sqrt(a + b*x)*a*c + 6*(c + d*x)**(3/4)*sqrt(a + b* x)*a*d*x + 4*(c + d*x)**(3/4)*sqrt(a + b*x)*b*c*x + 9*int(((c + d*x)**(3/4 )*sqrt(a + b*x)*x)/(3*a**2*c*d + 3*a**2*d**2*x + 2*a*b*c**2 + 5*a*b*c*d*x + 3*a*b*d**2*x**2 + 2*b**2*c**2*x + 2*b**2*c*d*x**2),x)*a**3*d**3 - 12*int (((c + d*x)**(3/4)*sqrt(a + b*x)*x)/(3*a**2*c*d + 3*a**2*d**2*x + 2*a*b*c* *2 + 5*a*b*c*d*x + 3*a*b*d**2*x**2 + 2*b**2*c**2*x + 2*b**2*c*d*x**2),x)*a **2*b*c*d**2 - 3*int(((c + d*x)**(3/4)*sqrt(a + b*x)*x)/(3*a**2*c*d + 3*a* *2*d**2*x + 2*a*b*c**2 + 5*a*b*c*d*x + 3*a*b*d**2*x**2 + 2*b**2*c**2*x + 2 *b**2*c*d*x**2),x)*a*b**2*c**2*d + 6*int(((c + d*x)**(3/4)*sqrt(a + b*x)*x )/(3*a**2*c*d + 3*a**2*d**2*x + 2*a*b*c**2 + 5*a*b*c*d*x + 3*a*b*d**2*x**2 + 2*b**2*c**2*x + 2*b**2*c*d*x**2),x)*b**3*c**3))/(9*(3*a*d + 2*b*c))