### 3.719 $$\int \frac{1}{(d+e x)^{3/2} \sqrt [4]{a+c x^2}} \, dx$$

Optimal. Leaf size=200 $-\frac{2 \left (\sqrt{-a}-\sqrt{c} x\right ) \sqrt [4]{-\frac{\left (\sqrt{-a}+\sqrt{c} x\right ) \left (\sqrt{-a} e+\sqrt{c} d\right )}{\left (\sqrt{-a}-\sqrt{c} x\right ) \left (\sqrt{c} d-\sqrt{-a} e\right )}} \, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{1}{2};\frac{2 \sqrt{-a} \sqrt{c} (d+e x)}{\left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}\right )}{\sqrt [4]{a+c x^2} \sqrt{d+e x} \left (\sqrt{-a} e+\sqrt{c} d\right )}$

[Out]

(-2*(Sqrt[-a] - Sqrt[c]*x)*(-(((Sqrt[c]*d + Sqrt[-a]*e)*(Sqrt[-a] + Sqrt[c]*x))/((Sqrt[c]*d - Sqrt[-a]*e)*(Sqr
t[-a] - Sqrt[c]*x))))^(1/4)*Hypergeometric2F1[-1/2, 1/4, 1/2, (2*Sqrt[-a]*Sqrt[c]*(d + e*x))/((Sqrt[c]*d - Sqr
t[-a]*e)*(Sqrt[-a] - Sqrt[c]*x))])/((Sqrt[c]*d + Sqrt[-a]*e)*Sqrt[d + e*x]*(a + c*x^2)^(1/4))

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Rubi [A]  time = 0.0944947, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.048, Rules used = {727} $-\frac{2 \left (\sqrt{-a}-\sqrt{c} x\right ) \sqrt [4]{-\frac{\left (\sqrt{-a}+\sqrt{c} x\right ) \left (\sqrt{-a} e+\sqrt{c} d\right )}{\left (\sqrt{-a}-\sqrt{c} x\right ) \left (\sqrt{c} d-\sqrt{-a} e\right )}} \, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{1}{2};\frac{2 \sqrt{-a} \sqrt{c} (d+e x)}{\left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}\right )}{\sqrt [4]{a+c x^2} \sqrt{d+e x} \left (\sqrt{-a} e+\sqrt{c} d\right )}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/((d + e*x)^(3/2)*(a + c*x^2)^(1/4)),x]

[Out]

(-2*(Sqrt[-a] - Sqrt[c]*x)*(-(((Sqrt[c]*d + Sqrt[-a]*e)*(Sqrt[-a] + Sqrt[c]*x))/((Sqrt[c]*d - Sqrt[-a]*e)*(Sqr
t[-a] - Sqrt[c]*x))))^(1/4)*Hypergeometric2F1[-1/2, 1/4, 1/2, (2*Sqrt[-a]*Sqrt[c]*(d + e*x))/((Sqrt[c]*d - Sqr
t[-a]*e)*(Sqrt[-a] - Sqrt[c]*x))])/((Sqrt[c]*d + Sqrt[-a]*e)*Sqrt[d + e*x]*(a + c*x^2)^(1/4))

Rule 727

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((Rt[-(a*c), 2] - c*x)*(d + e*x)^(m
+ 1)*(a + c*x^2)^p*Hypergeometric2F1[m + 1, -p, m + 2, (2*c*Rt[-(a*c), 2]*(d + e*x))/((c*d - e*Rt[-(a*c), 2])
*(Rt[-(a*c), 2] - c*x))])/((m + 1)*(c*d + e*Rt[-(a*c), 2])*(((c*d + e*Rt[-(a*c), 2])*(Rt[-(a*c), 2] + c*x))/((
c*d - e*Rt[-(a*c), 2])*(-Rt[-(a*c), 2] + c*x)))^p), x] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0
] &&  !IntegerQ[p] && EqQ[m + 2*p + 2, 0]

Rubi steps

\begin{align*} \int \frac{1}{(d+e x)^{3/2} \sqrt [4]{a+c x^2}} \, dx &=-\frac{2 \left (\sqrt{-a}-\sqrt{c} x\right ) \sqrt [4]{-\frac{\left (\sqrt{c} d+\sqrt{-a} e\right ) \left (\sqrt{-a}+\sqrt{c} x\right )}{\left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}} \, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{1}{2};\frac{2 \sqrt{-a} \sqrt{c} (d+e x)}{\left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}\right )}{\left (\sqrt{c} d+\sqrt{-a} e\right ) \sqrt{d+e x} \sqrt [4]{a+c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.517099, size = 108, normalized size = 0.54 $\frac{\left (a+c x^2\right )^{3/4} (c d x-a e) \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{(a e-c d x)^2}{a c (d+e x)^2}\right )}{a c (d+e x)^{5/2} \left (\frac{\left (a+c x^2\right ) \left (a e^2+c d^2\right )}{a c (d+e x)^2}\right )^{3/4}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/((d + e*x)^(3/2)*(a + c*x^2)^(1/4)),x]

[Out]

((-(a*e) + c*d*x)*(a + c*x^2)^(3/4)*Hypergeometric2F1[1/4, 1/2, 3/2, -((a*e - c*d*x)^2/(a*c*(d + e*x)^2))])/(a
*c*(d + e*x)^(5/2)*(((c*d^2 + a*e^2)*(a + c*x^2))/(a*c*(d + e*x)^2))^(3/4))

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Maple [F]  time = 0.668, size = 0, normalized size = 0. \begin{align*} \int{ \left ( ex+d \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt [4]{c{x}^{2}+a}}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c x^{2} + a\right )}^{\frac{1}{4}}{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (c x^{2} + a\right )}^{\frac{3}{4}} \sqrt{e x + d}}{c e^{2} x^{4} + 2 \, c d e x^{3} + 2 \, a d e x + a d^{2} +{\left (c d^{2} + a e^{2}\right )} x^{2}}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt [4]{a + c x^{2}} \left (d + e x\right )^{\frac{3}{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/((a + c*x**2)**(1/4)*(d + e*x)**(3/2)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c x^{2} + a\right )}^{\frac{1}{4}}{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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