3.1655 \(\int \frac{\sqrt{a+b x}}{(c+d x)^{3/4}} \, dx\)

Optimal. Leaf size=111 \[ \frac{4 \sqrt{a+b x} \sqrt [4]{c+d x}}{3 d}-\frac{8 (b c-a d)^{5/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{3 \sqrt [4]{b} d^2 \sqrt{a+b x}} \]

[Out]

(4*Sqrt[a + b*x]*(c + d*x)^(1/4))/(3*d) - (8*(b*c - a*d)^(5/4)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*EllipticF[Ar
cSin[(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(3*b^(1/4)*d^2*Sqrt[a + b*x])

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Rubi [A]  time = 0.0672476, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {50, 63, 224, 221} \[ \frac{4 \sqrt{a+b x} \sqrt [4]{c+d x}}{3 d}-\frac{8 (b c-a d)^{5/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{3 \sqrt [4]{b} d^2 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*x]/(c + d*x)^(3/4),x]

[Out]

(4*Sqrt[a + b*x]*(c + d*x)^(1/4))/(3*d) - (8*(b*c - a*d)^(5/4)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*EllipticF[Ar
cSin[(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(3*b^(1/4)*d^2*Sqrt[a + b*x])

Rule 50

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] + Dist[(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] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

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 224

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 221

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]

Rubi steps

\begin{align*} \int \frac{\sqrt{a+b x}}{(c+d x)^{3/4}} \, dx &=\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}\\ &=\frac{4 \sqrt{a+b x} \sqrt [4]{c+d x}}{3 d}-\frac{(8 (b c-a d)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-\frac{b c}{d}+\frac{b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{3 d^2}\\ &=\frac{4 \sqrt{a+b x} \sqrt [4]{c+d x}}{3 d}-\frac{\left (8 (b c-a d) \sqrt{\frac{d (a+b x)}{-b c+a d}}\right ) \operatorname{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 )}{3 d^2 \sqrt{a+b x}}\\ &=\frac{4 \sqrt{a+b x} \sqrt [4]{c+d x}}{3 d}-\frac{8 (b c-a d)^{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 )}{3 \sqrt [4]{b} d^2 \sqrt{a+b x}}\\ \end{align*}

Mathematica [C]  time = 0.0262108, size = 73, normalized size = 0.66 \[ \frac{2 (a+b x)^{3/2} \left (\frac{b (c+d x)}{b c-a d}\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{2};\frac{5}{2};\frac{d (a+b x)}{a d-b c}\right )}{3 b (c+d x)^{3/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b*x]/(c + d*x)^(3/4),x]

[Out]

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

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Maple [F]  time = 0.016, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{bx+a} \left ( dx+c \right ) ^{-{\frac{3}{4}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(1/2)/(d*x+c)^(3/4),x)

[Out]

int((b*x+a)^(1/2)/(d*x+c)^(3/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{b x + a}}{{\left (d x + c\right )}^{\frac{3}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(1/2)/(d*x+c)^(3/4),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b x + a}}{{\left (d x + c\right )}^{\frac{3}{4}}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(1/2)/(d*x+c)^(3/4),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a + b x}}{\left (c + d x\right )^{\frac{3}{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(1/2)/(d*x+c)**(3/4),x)

[Out]

Integral(sqrt(a + b*x)/(c + d*x)**(3/4), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{b x + a}}{{\left (d x + c\right )}^{\frac{3}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(1/2)/(d*x+c)^(3/4),x, algorithm="giac")

[Out]

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