### 3.2548 $$\int \frac{1}{(d+e x)^{3/2} \sqrt [4]{a+b x+c x^2}} \, dx$$

Optimal. Leaf size=239 $\frac{2 \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \sqrt [4]{\frac{\left (\sqrt{b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )}{\left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )}} \, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{1}{2};-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right )}\right )}{\sqrt{d+e x} \sqrt [4]{a+b x+c x^2} \left (e \sqrt{b^2-4 a c}-b e+2 c d\right )}$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.153645, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.042, Rules used = {726} $\frac{2 \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \sqrt [4]{\frac{\left (\sqrt{b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )}{\left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )}} \, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{1}{2};-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right )}\right )}{\sqrt{d+e x} \sqrt [4]{a+b x+c x^2} \left (e \sqrt{b^2-4 a c}-b e+2 c d\right )}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 726

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((b - Rt[b^2 - 4*a*
c, 2] + 2*c*x)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Hypergeometric2F1[m + 1, -p, m + 2, (-4*c*Rt[b^2 - 4*a*c,
2]*(d + e*x))/((2*c*d - b*e - e*Rt[b^2 - 4*a*c, 2])*(b - Rt[b^2 - 4*a*c, 2] + 2*c*x))])/((m + 1)*(2*c*d - b*e
+ e*Rt[b^2 - 4*a*c, 2])*(((2*c*d - b*e + e*Rt[b^2 - 4*a*c, 2])*(b + Rt[b^2 - 4*a*c, 2] + 2*c*x))/((2*c*d - b*
e - e*Rt[b^2 - 4*a*c, 2])*(b - Rt[b^2 - 4*a*c, 2] + 2*c*x)))^p), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[
b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] &&  !IntegerQ[p] && EqQ[m + 2*p + 2, 0
]

Rubi steps

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

Mathematica [A]  time = 0.264665, size = 235, normalized size = 0.98 $\frac{2 \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \sqrt [4]{\frac{\left (\sqrt{b^2-4 a c}+b+2 c x\right ) \left (e \left (\sqrt{b^2-4 a c}-b\right )+2 c d\right )}{\left (\sqrt{b^2-4 a c}-b-2 c x\right ) \left (e \left (\sqrt{b^2-4 a c}+b\right )-2 c d\right )}} \, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{1}{2};-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left (\left (b+\sqrt{b^2-4 a c}\right ) e-2 c d\right ) \left (-b-2 c x+\sqrt{b^2-4 a c}\right )}\right )}{\sqrt{d+e x} \sqrt [4]{a+x (b+c x)} \left (e \left (\sqrt{b^2-4 a c}-b\right )+2 c d\right )}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 1.284, size = 0, normalized size = 0. \begin{align*} \int{ \left ( ex+d \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt [4]{c{x}^{2}+bx+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+b*x+a)^(1/4),x)

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c x^{2} + b x + 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+b*x+a)^(1/4),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (c x^{2} + b x + a\right )}^{\frac{3}{4}} \sqrt{e x + d}}{c e^{2} x^{4} +{\left (2 \, c d e + b e^{2}\right )} x^{3} + a d^{2} +{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{2} +{\left (b d^{2} + 2 \, a d e\right )} x}, 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+b*x+a)^(1/4),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out