\(\int \frac {\sqrt {a+b x}}{(c+d x)^{9/4}} \, dx\) [1674]

Optimal result

Integrand size = 19, antiderivative size = 232 \[ \int \frac {\sqrt {a+b x}}{(c+d x)^{9/4}} \, dx=-\frac {4 \sqrt {a+b x}}{5 d (c+d x)^{5/4}}+\frac {8 b \sqrt {a+b x}}{5 d (b c-a d) \sqrt [4]{c+d x}}-\frac {8 b^{5/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 d^2 \sqrt [4]{b c-a d} \sqrt {a+b x}}+\frac {8 b^{5/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{5 d^2 \sqrt [4]{b c-a d} \sqrt {a+b x}} \]

[Out]

-4/5*(b*x+a)^(1/2)/d/(d*x+c)^(5/4)+8/5*b*(b*x+a)^(1/2)/d/(-a*d+b*c)/(d*x+c)^(1/4)-8/5*b^(5/4)*EllipticE(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)/d^2/(-a*d+b*c)^(1/4)/(b*x+a)^(1/2)+8/5*b^(5/
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)/d^2/(-a*d+b*c)^(1/4)/(b*x
+a)^(1/2)

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {49, 53, 65, 313, 230, 227, 1214, 1213, 435} \[ \int \frac {\sqrt {a+b x}}{(c+d x)^{9/4}} \, dx=\frac {8 b^{5/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{5 d^2 \sqrt {a+b x} \sqrt [4]{b c-a d}}-\frac {8 b^{5/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 d^2 \sqrt {a+b x} \sqrt [4]{b c-a d}}+\frac {8 b \sqrt {a+b x}}{5 d \sqrt [4]{c+d x} (b c-a d)}-\frac {4 \sqrt {a+b x}}{5 d (c+d x)^{5/4}} \]

[In]

Int[Sqrt[a + b*x]/(c + d*x)^(9/4),x]

[Out]

(-4*Sqrt[a + b*x])/(5*d*(c + d*x)^(5/4)) + (8*b*Sqrt[a + b*x])/(5*d*(b*c - a*d)*(c + d*x)^(1/4)) - (8*b^(5/4)*
Sqrt[-((d*(a + b*x))/(b*c - a*d))]*EllipticE[ArcSin[(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(5*d^2*
(b*c - a*d)^(1/4)*Sqrt[a + b*x]) + (8*b^(5/4)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*EllipticF[ArcSin[(b^(1/4)*(c
+ d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(5*d^2*(b*c - a*d)^(1/4)*Sqrt[a + b*x])

Rule 49

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] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && 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]

Rule 53

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[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] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 227

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[Rt[-b, 4]*(x/Rt[a, 4])], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rule 230

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Dist[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]

Rule 313

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Dist[-q^(-1), Int[1/Sqrt[a + b*x^4]
, x], x] + Dist[1/q, Int[(1 + q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[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
]

Rule 1213

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Dist[d/Sqrt[a], Int[Sqrt[1 + e*(x^2/d)]/Sqrt
[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && EqQ[c*d^2 + a*e^2, 0] && GtQ[a, 0]

Rule 1214

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Dist[Sqrt[1 + c*(x^4/a)]/Sqrt[a + c*x^4], In
t[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && EqQ[c*d^2 + a*e^2, 0] &&
!GtQ[a, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {4 \sqrt {a+b x}}{5 d (c+d x)^{5/4}}+\frac {(2 b) \int \frac {1}{\sqrt {a+b x} (c+d x)^{5/4}} \, dx}{5 d} \\ & = -\frac {4 \sqrt {a+b x}}{5 d (c+d x)^{5/4}}+\frac {8 b \sqrt {a+b x}}{5 d (b c-a d) \sqrt [4]{c+d x}}-\frac {\left (2 b^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt [4]{c+d x}} \, dx}{5 d (b c-a d)} \\ & = -\frac {4 \sqrt {a+b x}}{5 d (c+d x)^{5/4}}+\frac {8 b \sqrt {a+b x}}{5 d (b c-a d) \sqrt [4]{c+d x}}-\frac {\left (8 b^2\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {a-\frac {b c}{d}+\frac {b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{5 d^2 (b c-a d)} \\ & = -\frac {4 \sqrt {a+b x}}{5 d (c+d x)^{5/4}}+\frac {8 b \sqrt {a+b x}}{5 d (b c-a d) \sqrt [4]{c+d x}}+\frac {\left (8 b^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-\frac {b c}{d}+\frac {b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{5 d^2 \sqrt {b c-a d}}-\frac {\left (8 b^{3/2}\right ) \text {Subst}\left (\int \frac {1+\frac {\sqrt {b} x^2}{\sqrt {b c-a d}}}{\sqrt {a-\frac {b c}{d}+\frac {b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{5 d^2 \sqrt {b c-a d}} \\ & = -\frac {4 \sqrt {a+b x}}{5 d (c+d x)^{5/4}}+\frac {8 b \sqrt {a+b x}}{5 d (b c-a d) \sqrt [4]{c+d x}}+\frac {\left (8 b^{3/2} \sqrt {\frac {d (a+b x)}{-b c+a d}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {b x^4}{\left (a-\frac {b c}{d}\right ) d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{5 d^2 \sqrt {b c-a d} \sqrt {a+b x}}-\frac {\left (8 b^{3/2} \sqrt {\frac {d (a+b x)}{-b c+a d}}\right ) \text {Subst}\left (\int \frac {1+\frac {\sqrt {b} x^2}{\sqrt {b c-a d}}}{\sqrt {1+\frac {b x^4}{\left (a-\frac {b c}{d}\right ) d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{5 d^2 \sqrt {b c-a d} \sqrt {a+b x}} \\ & = -\frac {4 \sqrt {a+b x}}{5 d (c+d x)^{5/4}}+\frac {8 b \sqrt {a+b x}}{5 d (b c-a d) \sqrt [4]{c+d x}}+\frac {8 b^{5/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 d^2 \sqrt [4]{b c-a d} \sqrt {a+b x}}-\frac {\left (8 b^{3/2} \sqrt {\frac {d (a+b x)}{-b c+a d}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {\sqrt {b} x^2}{\sqrt {b c-a d}}}}{\sqrt {1-\frac {\sqrt {b} x^2}{\sqrt {b c-a d}}}} \, dx,x,\sqrt [4]{c+d x}\right )}{5 d^2 \sqrt {b c-a d} \sqrt {a+b x}} \\ & = -\frac {4 \sqrt {a+b x}}{5 d (c+d x)^{5/4}}+\frac {8 b \sqrt {a+b x}}{5 d (b c-a d) \sqrt [4]{c+d x}}-\frac {8 b^{5/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 d^2 \sqrt [4]{b c-a d} \sqrt {a+b x}}+\frac {8 b^{5/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 d^2 \sqrt [4]{b c-a d} \sqrt {a+b x}} \\ \end{align*}

Mathematica [C] (verified)

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 = 0.31 \[ \int \frac {\sqrt {a+b x}}{(c+d x)^{9/4}} \, dx=\frac {2 (a+b x)^{3/2} \left (\frac {b (c+d x)}{b c-a d}\right )^{9/4} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {9}{4},\frac {5}{2},\frac {d (a+b x)}{-b c+a d}\right )}{3 b (c+d x)^{9/4}} \]

[In]

Integrate[Sqrt[a + b*x]/(c + d*x)^(9/4),x]

[Out]

(2*(a + b*x)^(3/2)*((b*(c + d*x))/(b*c - a*d))^(9/4)*Hypergeometric2F1[3/2, 9/4, 5/2, (d*(a + b*x))/(-(b*c) +
a*d)])/(3*b*(c + d*x)^(9/4))

Maple [F]

\[\int \frac {\sqrt {b x +a}}{\left (d x +c \right )^{\frac {9}{4}}}d x\]

[In]

int((b*x+a)^(1/2)/(d*x+c)^(9/4),x)

[Out]

int((b*x+a)^(1/2)/(d*x+c)^(9/4),x)

Fricas [F]

\[ \int \frac {\sqrt {a+b x}}{(c+d x)^{9/4}} \, dx=\int { \frac {\sqrt {b x + a}}{{\left (d x + c\right )}^{\frac {9}{4}}} \,d x } \]

[In]

integrate((b*x+a)^(1/2)/(d*x+c)^(9/4),x, algorithm="fricas")

[Out]

integral(sqrt(b*x + a)*(d*x + c)^(3/4)/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3), x)

Sympy [F]

\[ \int \frac {\sqrt {a+b x}}{(c+d x)^{9/4}} \, dx=\int \frac {\sqrt {a + b x}}{\left (c + d x\right )^{\frac {9}{4}}}\, dx \]

[In]

integrate((b*x+a)**(1/2)/(d*x+c)**(9/4),x)

[Out]

Integral(sqrt(a + b*x)/(c + d*x)**(9/4), x)

Maxima [F]

\[ \int \frac {\sqrt {a+b x}}{(c+d x)^{9/4}} \, dx=\int { \frac {\sqrt {b x + a}}{{\left (d x + c\right )}^{\frac {9}{4}}} \,d x } \]

[In]

integrate((b*x+a)^(1/2)/(d*x+c)^(9/4),x, algorithm="maxima")

[Out]

integrate(sqrt(b*x + a)/(d*x + c)^(9/4), x)

Giac [F]

\[ \int \frac {\sqrt {a+b x}}{(c+d x)^{9/4}} \, dx=\int { \frac {\sqrt {b x + a}}{{\left (d x + c\right )}^{\frac {9}{4}}} \,d x } \]

[In]

integrate((b*x+a)^(1/2)/(d*x+c)^(9/4),x, algorithm="giac")

[Out]

integrate(sqrt(b*x + a)/(d*x + c)^(9/4), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x}}{(c+d x)^{9/4}} \, dx=\int \frac {\sqrt {a+b\,x}}{{\left (c+d\,x\right )}^{9/4}} \,d x \]

[In]

int((a + b*x)^(1/2)/(c + d*x)^(9/4),x)

[Out]

int((a + b*x)^(1/2)/(c + d*x)^(9/4), x)