### 3.718 $$\int \frac{1}{(a+b x) (c+d x^2)^{3/4}} \, dx$$

Optimal. Leaf size=268 $-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{c+d x^2}}{\sqrt [4]{a^2 d+b^2 c}}\right )}{\left (a^2 d+b^2 c\right )^{3/4}}-\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{c+d x^2}}{\sqrt [4]{a^2 d+b^2 c}}\right )}{\left (a^2 d+b^2 c\right )^{3/4}}+\frac{a \sqrt [4]{c} \sqrt{-\frac{d x^2}{c}} \Pi \left (-\frac{b \sqrt{c}}{\sqrt{d a^2+b^2 c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d x^2+c}}{\sqrt [4]{c}}\right )\right |-1\right )}{x \left (a^2 d+b^2 c\right )}+\frac{a \sqrt [4]{c} \sqrt{-\frac{d x^2}{c}} \Pi \left (\frac{b \sqrt{c}}{\sqrt{d a^2+b^2 c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d x^2+c}}{\sqrt [4]{c}}\right )\right |-1\right )}{x \left (a^2 d+b^2 c\right )}$

[Out]

-((Sqrt[b]*ArcTan[(Sqrt[b]*(c + d*x^2)^(1/4))/(b^2*c + a^2*d)^(1/4)])/(b^2*c + a^2*d)^(3/4)) - (Sqrt[b]*ArcTan
h[(Sqrt[b]*(c + d*x^2)^(1/4))/(b^2*c + a^2*d)^(1/4)])/(b^2*c + a^2*d)^(3/4) + (a*c^(1/4)*Sqrt[-((d*x^2)/c)]*El
lipticPi[-((b*Sqrt[c])/Sqrt[b^2*c + a^2*d]), ArcSin[(c + d*x^2)^(1/4)/c^(1/4)], -1])/((b^2*c + a^2*d)*x) + (a*
c^(1/4)*Sqrt[-((d*x^2)/c)]*EllipticPi[(b*Sqrt[c])/Sqrt[b^2*c + a^2*d], ArcSin[(c + d*x^2)^(1/4)/c^(1/4)], -1])
/((b^2*c + a^2*d)*x)

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Rubi [A]  time = 0.269515, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 19, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.526, Rules used = {747, 401, 108, 409, 1218, 444, 63, 212, 208, 205} $-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{c+d x^2}}{\sqrt [4]{a^2 d+b^2 c}}\right )}{\left (a^2 d+b^2 c\right )^{3/4}}-\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{c+d x^2}}{\sqrt [4]{a^2 d+b^2 c}}\right )}{\left (a^2 d+b^2 c\right )^{3/4}}+\frac{a \sqrt [4]{c} \sqrt{-\frac{d x^2}{c}} \Pi \left (-\frac{b \sqrt{c}}{\sqrt{d a^2+b^2 c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d x^2+c}}{\sqrt [4]{c}}\right )\right |-1\right )}{x \left (a^2 d+b^2 c\right )}+\frac{a \sqrt [4]{c} \sqrt{-\frac{d x^2}{c}} \Pi \left (\frac{b \sqrt{c}}{\sqrt{d a^2+b^2 c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d x^2+c}}{\sqrt [4]{c}}\right )\right |-1\right )}{x \left (a^2 d+b^2 c\right )}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-((Sqrt[b]*ArcTan[(Sqrt[b]*(c + d*x^2)^(1/4))/(b^2*c + a^2*d)^(1/4)])/(b^2*c + a^2*d)^(3/4)) - (Sqrt[b]*ArcTan
h[(Sqrt[b]*(c + d*x^2)^(1/4))/(b^2*c + a^2*d)^(1/4)])/(b^2*c + a^2*d)^(3/4) + (a*c^(1/4)*Sqrt[-((d*x^2)/c)]*El
lipticPi[-((b*Sqrt[c])/Sqrt[b^2*c + a^2*d]), ArcSin[(c + d*x^2)^(1/4)/c^(1/4)], -1])/((b^2*c + a^2*d)*x) + (a*
c^(1/4)*Sqrt[-((d*x^2)/c)]*EllipticPi[(b*Sqrt[c])/Sqrt[b^2*c + a^2*d], ArcSin[(c + d*x^2)^(1/4)/c^(1/4)], -1])
/((b^2*c + a^2*d)*x)

Rule 747

Int[1/(((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(3/4)), x_Symbol] :> Dist[d, Int[1/((d^2 - e^2*x^2)*(a + c*x^
2)^(3/4)), x], x] - Dist[e, Int[x/((d^2 - e^2*x^2)*(a + c*x^2)^(3/4)), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ
[c*d^2 + a*e^2, 0]

Rule 401

Int[1/(((a_) + (b_.)*(x_)^2)^(3/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Dist[Sqrt[-((b*x^2)/a)]/(2*x), Subst[I
nt[1/(Sqrt[-((b*x)/a)]*(a + b*x)^(3/4)*(c + d*x)), x], x, x^2], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d,
0]

Rule 108

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(3/4)), x_Symbol] :> Dist[-4, Subst[
Int[1/((b*e - a*f - b*x^4)*Sqrt[c - (d*e)/f + (d*x^4)/f]), x], x, (e + f*x)^(1/4)], x] /; FreeQ[{a, b, c, d, e
, f}, x] && GtQ[-(f/(d*e - c*f)), 0]

Rule 409

Int[1/(Sqrt[(a_) + (b_.)*(x_)^4]*((c_) + (d_.)*(x_)^4)), x_Symbol] :> Dist[1/(2*c), Int[1/(Sqrt[a + b*x^4]*(1
- Rt[-(d/c), 2]*x^2)), x], x] + Dist[1/(2*c), Int[1/(Sqrt[a + b*x^4]*(1 + Rt[-(d/c), 2]*x^2)), x], x] /; FreeQ
[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 1218

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[-(c/a), 4]}, Simp[(1*Ellipt
icPi[-(e/(d*q^2)), ArcSin[q*x], -1])/(d*Sqrt[a]*q), x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

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 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

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

Mathematica [A]  time = 0.201889, size = 237, normalized size = 0.88 $-\frac{\sqrt{b} x \sqrt [4]{a^2 d+b^2 c} \left (\tan ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{c+d x^2}}{\sqrt [4]{a^2 d+b^2 c}}\right )+\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{c+d x^2}}{\sqrt [4]{a^2 d+b^2 c}}\right )\right )+a \sqrt [4]{c} \sqrt{-\frac{d x^2}{c}} \Pi \left (-\frac{b \sqrt{c}}{\sqrt{d a^2+b^2 c}};\left .-\sin ^{-1}\left (\frac{\sqrt [4]{d x^2+c}}{\sqrt [4]{c}}\right )\right |-1\right )+a \sqrt [4]{c} \sqrt{-\frac{d x^2}{c}} \Pi \left (\frac{b \sqrt{c}}{\sqrt{d a^2+b^2 c}};\left .-\sin ^{-1}\left (\frac{\sqrt [4]{d x^2+c}}{\sqrt [4]{c}}\right )\right |-1\right )}{x \left (a^2 d+b^2 c\right )}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-((Sqrt[b]*(b^2*c + a^2*d)^(1/4)*x*(ArcTan[(Sqrt[b]*(c + d*x^2)^(1/4))/(b^2*c + a^2*d)^(1/4)] + ArcTanh[(Sqrt[
b]*(c + d*x^2)^(1/4))/(b^2*c + a^2*d)^(1/4)]) + a*c^(1/4)*Sqrt[-((d*x^2)/c)]*EllipticPi[-((b*Sqrt[c])/Sqrt[b^2
*c + a^2*d]), -ArcSin[(c + d*x^2)^(1/4)/c^(1/4)], -1] + a*c^(1/4)*Sqrt[-((d*x^2)/c)]*EllipticPi[(b*Sqrt[c])/Sq
rt[b^2*c + a^2*d], -ArcSin[(c + d*x^2)^(1/4)/c^(1/4)], -1])/((b^2*c + a^2*d)*x))

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Maple [F]  time = 0.615, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{bx+a} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{4}}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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