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

Optimal. Leaf size=134 \[ \frac{(a d+3 b c) \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{7/4} c^{5/4}}+\frac{(a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{7/4} c^{5/4}}-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{a c x} \]

[Out]

-(((a + b*x)^(1/4)*(c + d*x)^(3/4))/(a*c*x)) + ((3*b*c + a*d)*ArcTan[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d
*x)^(1/4))])/(2*a^(7/4)*c^(5/4)) + ((3*b*c + a*d)*ArcTanh[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))]
)/(2*a^(7/4)*c^(5/4))

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Rubi [A]  time = 0.0598347, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {96, 93, 212, 208, 205} \[ \frac{(a d+3 b c) \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{7/4} c^{5/4}}+\frac{(a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{7/4} c^{5/4}}-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{a c x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((a + b*x)^(1/4)*(c + d*x)^(3/4))/(a*c*x)) + ((3*b*c + a*d)*ArcTan[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d
*x)^(1/4))])/(2*a^(7/4)*c^(5/4)) + ((3*b*c + a*d)*ArcTanh[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))]
)/(2*a^(7/4)*c^(5/4))

Rule 96

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*
x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*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}{x^2 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx &=-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{a c x}-\frac{\left (\frac{3 b c}{4}+\frac{a d}{4}\right ) \int \frac{1}{x (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx}{a c}\\ &=-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{a c x}-\frac{\left (4 \left (\frac{3 b c}{4}+\frac{a d}{4}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-a+c x^4} \, dx,x,\frac{\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{a c}\\ &=-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{a c x}+\frac{(3 b c+a d) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a}-\sqrt{c} x^2} \, dx,x,\frac{\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{2 a^{3/2} c}+\frac{(3 b c+a d) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a}+\sqrt{c} x^2} \, dx,x,\frac{\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{2 a^{3/2} c}\\ &=-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{a c x}+\frac{(3 b c+a d) \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{7/4} c^{5/4}}+\frac{(3 b c+a d) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{7/4} c^{5/4}}\\ \end{align*}

Mathematica [C]  time = 0.0198898, size = 72, normalized size = 0.54 \[ \frac{\sqrt [4]{a+b x} \left (x (a d+3 b c) \, _2F_1\left (\frac{1}{4},1;\frac{5}{4};\frac{c (a+b x)}{a (c+d x)}\right )-a (c+d x)\right )}{a^2 c x \sqrt [4]{c+d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.026, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ( bx+a \right ) ^{-{\frac{3}{4}}}{\frac{1}{\sqrt [4]{dx+c}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.05707, size = 1883, normalized size = 14.05 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(4*a*c*x*((81*b^4*c^4 + 108*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 + a^4*d^4)/(a^7*c^5))^(1/4)
*arctan(-((3*a^5*b*c^5 + a^6*c^4*d)*(b*x + a)^(1/4)*(d*x + c)^(3/4)*((81*b^4*c^4 + 108*a*b^3*c^3*d + 54*a^2*b^
2*c^2*d^2 + 12*a^3*b*c*d^3 + a^4*d^4)/(a^7*c^5))^(3/4) - (a^5*c^4*d*x + a^5*c^5)*sqrt(((9*b^2*c^2 + 6*a*b*c*d
+ a^2*d^2)*sqrt(b*x + a)*sqrt(d*x + c) + (a^4*c^2*d*x + a^4*c^3)*sqrt((81*b^4*c^4 + 108*a*b^3*c^3*d + 54*a^2*b
^2*c^2*d^2 + 12*a^3*b*c*d^3 + a^4*d^4)/(a^7*c^5)))/(d*x + c))*((81*b^4*c^4 + 108*a*b^3*c^3*d + 54*a^2*b^2*c^2*
d^2 + 12*a^3*b*c*d^3 + a^4*d^4)/(a^7*c^5))^(3/4))/(81*b^4*c^5 + 108*a*b^3*c^4*d + 54*a^2*b^2*c^3*d^2 + 12*a^3*
b*c^2*d^3 + a^4*c*d^4 + (81*b^4*c^4*d + 108*a*b^3*c^3*d^2 + 54*a^2*b^2*c^2*d^3 + 12*a^3*b*c*d^4 + a^4*d^5)*x))
- a*c*x*((81*b^4*c^4 + 108*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 + a^4*d^4)/(a^7*c^5))^(1/4)*log(
((3*b*c + a*d)*(b*x + a)^(1/4)*(d*x + c)^(3/4) + (a^2*c*d*x + a^2*c^2)*((81*b^4*c^4 + 108*a*b^3*c^3*d + 54*a^2
*b^2*c^2*d^2 + 12*a^3*b*c*d^3 + a^4*d^4)/(a^7*c^5))^(1/4))/(d*x + c)) + a*c*x*((81*b^4*c^4 + 108*a*b^3*c^3*d +
54*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 + a^4*d^4)/(a^7*c^5))^(1/4)*log(((3*b*c + a*d)*(b*x + a)^(1/4)*(d*x + c)^
(3/4) - (a^2*c*d*x + a^2*c^2)*((81*b^4*c^4 + 108*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 + a^4*d^4)/
(a^7*c^5))^(1/4))/(d*x + c)) + 4*(b*x + a)^(1/4)*(d*x + c)^(3/4))/(a*c*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out