3.1668 \(\int \frac{1}{\sqrt{a+b x} (c+d x)^{7/4}} \, dx\)

Optimal. Leaf size=118 \[ \frac{4 b^{3/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{3 d \sqrt{a+b x} (b c-a d)^{3/4}}+\frac{4 \sqrt{a+b x}}{3 (c+d x)^{3/4} (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.0700133, antiderivative size = 118, 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 = {51, 63, 224, 221} \[ \frac{4 b^{3/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{3 d \sqrt{a+b x} (b c-a d)^{3/4}}+\frac{4 \sqrt{a+b x}}{3 (c+d x)^{3/4} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b*x]*(c + d*x)^(7/4)),x]

[Out]

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

Rule 51

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 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{1}{\sqrt{a+b x} (c+d x)^{7/4}} \, dx &=\frac{4 \sqrt{a+b x}}{3 (b c-a d) (c+d x)^{3/4}}+\frac{b \int \frac{1}{\sqrt{a+b x} (c+d x)^{3/4}} \, dx}{3 (b c-a d)}\\ &=\frac{4 \sqrt{a+b x}}{3 (b c-a d) (c+d x)^{3/4}}+\frac{(4 b) \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 (b c-a d)}\\ &=\frac{4 \sqrt{a+b x}}{3 (b c-a d) (c+d x)^{3/4}}+\frac{\left (4 b \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 (b c-a d) \sqrt{a+b x}}\\ &=\frac{4 \sqrt{a+b x}}{3 (b c-a d) (c+d x)^{3/4}}+\frac{4 b^{3/4} \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 d (b c-a d)^{3/4} \sqrt{a+b x}}\\ \end{align*}

Mathematica [C]  time = 0.0369033, size = 71, normalized size = 0.6 \[ \frac{2 \sqrt{a+b x} \left (\frac{b (c+d x)}{b c-a d}\right )^{7/4} \, _2F_1\left (\frac{1}{2},\frac{7}{4};\frac{3}{2};\frac{d (a+b x)}{a d-b c}\right )}{b (c+d x)^{7/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[a + b*x]*(c + d*x)^(7/4)),x]

[Out]

(2*Sqrt[a + b*x]*((b*(c + d*x))/(b*c - a*d))^(7/4)*Hypergeometric2F1[1/2, 7/4, 3/2, (d*(a + b*x))/(-(b*c) + a*
d)])/(b*(c + d*x)^(7/4))

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Maple [F]  time = 0.042, size = 0, normalized size = 0. \begin{align*} \int{{\frac{1}{\sqrt{bx+a}}} \left ( dx+c \right ) ^{-{\frac{7}{4}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)^(1/2)/(d*x+c)^(7/4),x)

[Out]

int(1/(b*x+a)^(1/2)/(d*x+c)^(7/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b x + a}{\left (d x + c\right )}^{\frac{7}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(1/2)/(d*x+c)^(7/4),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(b*x + a)*(d*x + c)^(7/4)), 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{1}{4}}}{b d^{2} x^{3} + a c^{2} +{\left (2 \, b c d + a d^{2}\right )} x^{2} +{\left (b c^{2} + 2 \, a c d\right )} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(1/2)/(d*x+c)^(7/4),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a + b x} \left (c + d x\right )^{\frac{7}{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)**(1/2)/(d*x+c)**(7/4),x)

[Out]

Integral(1/(sqrt(a + b*x)*(c + d*x)**(7/4)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b x + a}{\left (d x + c\right )}^{\frac{7}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(1/2)/(d*x+c)^(7/4),x, algorithm="giac")

[Out]

integrate(1/(sqrt(b*x + a)*(d*x + c)^(7/4)), x)