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

Optimal. Leaf size=130 \[ -\frac{(3 a d+b c) \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 b^{7/4} d^{5/4}}-\frac{(3 a d+b c) \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 b^{7/4} d^{5/4}}+\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{b d} \]

[Out]

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

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Rubi [A]  time = 0.0783379, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {80, 63, 240, 212, 208, 205} \[ -\frac{(3 a d+b c) \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 b^{7/4} d^{5/4}}-\frac{(3 a d+b c) \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 b^{7/4} d^{5/4}}+\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
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 240

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[1/(1 - b*x^n)^(p + 1/n + 1), x], x
, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p
+ 1/n]

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{x}{(a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx &=\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{b d}-\frac{(b c+3 a d) \int \frac{1}{(a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx}{4 b d}\\ &=\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{b d}-\frac{(b c+3 a d) \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{c-\frac{a d}{b}+\frac{d x^4}{b}}} \, dx,x,\sqrt [4]{a+b x}\right )}{b^2 d}\\ &=\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{b d}-\frac{(b c+3 a d) \operatorname{Subst}\left (\int \frac{1}{1-\frac{d x^4}{b}} \, dx,x,\frac{\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{b^2 d}\\ &=\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{b d}-\frac{(b c+3 a d) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b}-\sqrt{d} x^2} \, dx,x,\frac{\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{2 b^{3/2} d}-\frac{(b c+3 a d) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b}+\sqrt{d} x^2} \, dx,x,\frac{\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{2 b^{3/2} d}\\ &=\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{b d}-\frac{(b c+3 a d) \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 b^{7/4} d^{5/4}}-\frac{(b c+3 a d) \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 b^{7/4} d^{5/4}}\\ \end{align*}

Mathematica [C]  time = 0.0802098, size = 91, normalized size = 0.7 \[ \frac{\sqrt [4]{a+b x} \left (b (c+d x)-(3 a d+b c) \sqrt [4]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};\frac{d (a+b x)}{a d-b c}\right )\right )}{b^2 d \sqrt [4]{c+d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.021, size = 0, normalized size = 0. \begin{align*} \int{x \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(x/(b*x+a)^(3/4)/(d*x+c)^(1/4),x)

[Out]

int(x/(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{x}{{\left (b x + a\right )}^{\frac{3}{4}}{\left (d x + c\right )}^{\frac{1}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.02972, size = 1871, normalized size = 14.39 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

[Out]

Timed out