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

Optimal. Leaf size=340 \[ \frac{(a+b x)^{5/4} (c+d x)^{3/4} \left (77 a^2 d^2-8 b d x (11 a d+13 b c)+94 a b c d+117 b^2 c^2\right )}{384 b^3 d^3}-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4} \left (105 a^2 b c d^2+77 a^3 d^3+135 a b^2 c^2 d+195 b^3 c^3\right )}{512 b^3 d^4}+\frac{(b c-a d) \left (105 a^2 b c d^2+77 a^3 d^3+135 a b^2 c^2 d+195 b^3 c^3\right ) \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{1024 b^{15/4} d^{17/4}}+\frac{(b c-a d) \left (105 a^2 b c d^2+77 a^3 d^3+135 a b^2 c^2 d+195 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{1024 b^{15/4} d^{17/4}}+\frac{x^2 (a+b x)^{5/4} (c+d x)^{3/4}}{4 b d} \]

[Out]

-((195*b^3*c^3 + 135*a*b^2*c^2*d + 105*a^2*b*c*d^2 + 77*a^3*d^3)*(a + b*x)^(1/4)*(c + d*x)^(3/4))/(512*b^3*d^4
) + (x^2*(a + b*x)^(5/4)*(c + d*x)^(3/4))/(4*b*d) + ((a + b*x)^(5/4)*(c + d*x)^(3/4)*(117*b^2*c^2 + 94*a*b*c*d
+ 77*a^2*d^2 - 8*b*d*(13*b*c + 11*a*d)*x))/(384*b^3*d^3) + ((b*c - a*d)*(195*b^3*c^3 + 135*a*b^2*c^2*d + 105*
a^2*b*c*d^2 + 77*a^3*d^3)*ArcTan[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4))])/(1024*b^(15/4)*d^(17/4)
) + ((b*c - a*d)*(195*b^3*c^3 + 135*a*b^2*c^2*d + 105*a^2*b*c*d^2 + 77*a^3*d^3)*ArcTanh[(d^(1/4)*(a + b*x)^(1/
4))/(b^(1/4)*(c + d*x)^(1/4))])/(1024*b^(15/4)*d^(17/4))

________________________________________________________________________________________

Rubi [A]  time = 0.275761, antiderivative size = 340, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {100, 147, 50, 63, 240, 212, 208, 205} \[ \frac{(a+b x)^{5/4} (c+d x)^{3/4} \left (77 a^2 d^2-8 b d x (11 a d+13 b c)+94 a b c d+117 b^2 c^2\right )}{384 b^3 d^3}-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4} \left (105 a^2 b c d^2+77 a^3 d^3+135 a b^2 c^2 d+195 b^3 c^3\right )}{512 b^3 d^4}+\frac{(b c-a d) \left (105 a^2 b c d^2+77 a^3 d^3+135 a b^2 c^2 d+195 b^3 c^3\right ) \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{1024 b^{15/4} d^{17/4}}+\frac{(b c-a d) \left (105 a^2 b c d^2+77 a^3 d^3+135 a b^2 c^2 d+195 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{1024 b^{15/4} d^{17/4}}+\frac{x^2 (a+b x)^{5/4} (c+d x)^{3/4}}{4 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((195*b^3*c^3 + 135*a*b^2*c^2*d + 105*a^2*b*c*d^2 + 77*a^3*d^3)*(a + b*x)^(1/4)*(c + d*x)^(3/4))/(512*b^3*d^4
) + (x^2*(a + b*x)^(5/4)*(c + d*x)^(3/4))/(4*b*d) + ((a + b*x)^(5/4)*(c + d*x)^(3/4)*(117*b^2*c^2 + 94*a*b*c*d
+ 77*a^2*d^2 - 8*b*d*(13*b*c + 11*a*d)*x))/(384*b^3*d^3) + ((b*c - a*d)*(195*b^3*c^3 + 135*a*b^2*c^2*d + 105*
a^2*b*c*d^2 + 77*a^3*d^3)*ArcTan[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4))])/(1024*b^(15/4)*d^(17/4)
) + ((b*c - a*d)*(195*b^3*c^3 + 135*a*b^2*c^2*d + 105*a^2*b*c*d^2 + 77*a^3*d^3)*ArcTanh[(d^(1/4)*(a + b*x)^(1/
4))/(b^(1/4)*(c + d*x)^(1/4))])/(1024*b^(15/4)*d^(17/4))

Rule 100

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))/(d*f*(m + n + p + 1)), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]

Rule 147

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> -Simp[((a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x)*(a + b*x)^
(m + 1)*(c + d*x)^(n + 1))/(b^2*d^2*(m + n + 2)*(m + n + 3)), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(
n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*
(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)^n
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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^3 \sqrt [4]{a+b x}}{\sqrt [4]{c+d x}} \, dx &=\frac{x^2 (a+b x)^{5/4} (c+d x)^{3/4}}{4 b d}+\frac{\int \frac{x \sqrt [4]{a+b x} \left (-2 a c+\frac{1}{4} (-13 b c-11 a d) x\right )}{\sqrt [4]{c+d x}} \, dx}{4 b d}\\ &=\frac{x^2 (a+b x)^{5/4} (c+d x)^{3/4}}{4 b d}+\frac{(a+b x)^{5/4} (c+d x)^{3/4} \left (117 b^2 c^2+94 a b c d+77 a^2 d^2-8 b d (13 b c+11 a d) x\right )}{384 b^3 d^3}-\frac{\left (195 b^3 c^3+135 a b^2 c^2 d+105 a^2 b c d^2+77 a^3 d^3\right ) \int \frac{\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}} \, dx}{512 b^3 d^3}\\ &=-\frac{\left (195 b^3 c^3+135 a b^2 c^2 d+105 a^2 b c d^2+77 a^3 d^3\right ) \sqrt [4]{a+b x} (c+d x)^{3/4}}{512 b^3 d^4}+\frac{x^2 (a+b x)^{5/4} (c+d x)^{3/4}}{4 b d}+\frac{(a+b x)^{5/4} (c+d x)^{3/4} \left (117 b^2 c^2+94 a b c d+77 a^2 d^2-8 b d (13 b c+11 a d) x\right )}{384 b^3 d^3}+\frac{\left ((b c-a d) \left (195 b^3 c^3+135 a b^2 c^2 d+105 a^2 b c d^2+77 a^3 d^3\right )\right ) \int \frac{1}{(a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx}{2048 b^3 d^4}\\ &=-\frac{\left (195 b^3 c^3+135 a b^2 c^2 d+105 a^2 b c d^2+77 a^3 d^3\right ) \sqrt [4]{a+b x} (c+d x)^{3/4}}{512 b^3 d^4}+\frac{x^2 (a+b x)^{5/4} (c+d x)^{3/4}}{4 b d}+\frac{(a+b x)^{5/4} (c+d x)^{3/4} \left (117 b^2 c^2+94 a b c d+77 a^2 d^2-8 b d (13 b c+11 a d) x\right )}{384 b^3 d^3}+\frac{\left ((b c-a d) \left (195 b^3 c^3+135 a b^2 c^2 d+105 a^2 b c d^2+77 a^3 d^3\right )\right ) \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 )}{512 b^4 d^4}\\ &=-\frac{\left (195 b^3 c^3+135 a b^2 c^2 d+105 a^2 b c d^2+77 a^3 d^3\right ) \sqrt [4]{a+b x} (c+d x)^{3/4}}{512 b^3 d^4}+\frac{x^2 (a+b x)^{5/4} (c+d x)^{3/4}}{4 b d}+\frac{(a+b x)^{5/4} (c+d x)^{3/4} \left (117 b^2 c^2+94 a b c d+77 a^2 d^2-8 b d (13 b c+11 a d) x\right )}{384 b^3 d^3}+\frac{\left ((b c-a d) \left (195 b^3 c^3+135 a b^2 c^2 d+105 a^2 b c d^2+77 a^3 d^3\right )\right ) \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 )}{512 b^4 d^4}\\ &=-\frac{\left (195 b^3 c^3+135 a b^2 c^2 d+105 a^2 b c d^2+77 a^3 d^3\right ) \sqrt [4]{a+b x} (c+d x)^{3/4}}{512 b^3 d^4}+\frac{x^2 (a+b x)^{5/4} (c+d x)^{3/4}}{4 b d}+\frac{(a+b x)^{5/4} (c+d x)^{3/4} \left (117 b^2 c^2+94 a b c d+77 a^2 d^2-8 b d (13 b c+11 a d) x\right )}{384 b^3 d^3}+\frac{\left ((b c-a d) \left (195 b^3 c^3+135 a b^2 c^2 d+105 a^2 b c d^2+77 a^3 d^3\right )\right ) \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 )}{1024 b^{7/2} d^4}+\frac{\left ((b c-a d) \left (195 b^3 c^3+135 a b^2 c^2 d+105 a^2 b c d^2+77 a^3 d^3\right )\right ) \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 )}{1024 b^{7/2} d^4}\\ &=-\frac{\left (195 b^3 c^3+135 a b^2 c^2 d+105 a^2 b c d^2+77 a^3 d^3\right ) \sqrt [4]{a+b x} (c+d x)^{3/4}}{512 b^3 d^4}+\frac{x^2 (a+b x)^{5/4} (c+d x)^{3/4}}{4 b d}+\frac{(a+b x)^{5/4} (c+d x)^{3/4} \left (117 b^2 c^2+94 a b c d+77 a^2 d^2-8 b d (13 b c+11 a d) x\right )}{384 b^3 d^3}+\frac{(b c-a d) \left (195 b^3 c^3+135 a b^2 c^2 d+105 a^2 b c d^2+77 a^3 d^3\right ) \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{1024 b^{15/4} d^{17/4}}+\frac{(b c-a d) \left (195 b^3 c^3+135 a b^2 c^2 d+105 a^2 b c d^2+77 a^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{1024 b^{15/4} d^{17/4}}\\ \end{align*}

Mathematica [C]  time = 0.646723, size = 233, normalized size = 0.69 \[ \frac{(a+b x)^{5/4} \sqrt [4]{\frac{b (c+d x)}{b c-a d}} \left (b \left (2 c \left (-7 a^2 d^2-6 a b c d+13 b^2 c^2\right ) \, _2F_1\left (-\frac{3}{4},\frac{5}{4};\frac{9}{4};\frac{d (a+b x)}{a d-b c}\right )+b \left (\frac{5 b d^2 x^2 (c+d x)}{\sqrt [4]{\frac{b (c+d x)}{b c-a d}}}-c^2 (3 a d+13 b c) \, _2F_1\left (\frac{1}{4},\frac{5}{4};\frac{9}{4};\frac{d (a+b x)}{a d-b c}\right )\right )\right )-(b c-a d)^2 (11 a d+13 b c) \, _2F_1\left (-\frac{7}{4},\frac{5}{4};\frac{9}{4};\frac{d (a+b x)}{a d-b c}\right )\right )}{20 b^4 d^3 \sqrt [4]{c+d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^(5/4)*((b*(c + d*x))/(b*c - a*d))^(1/4)*(-((b*c - a*d)^2*(13*b*c + 11*a*d)*Hypergeometric2F1[-7/4,
5/4, 9/4, (d*(a + b*x))/(-(b*c) + a*d)]) + b*(2*c*(13*b^2*c^2 - 6*a*b*c*d - 7*a^2*d^2)*Hypergeometric2F1[-3/4,
5/4, 9/4, (d*(a + b*x))/(-(b*c) + a*d)] + b*((5*b*d^2*x^2*(c + d*x))/((b*(c + d*x))/(b*c - a*d))^(1/4) - c^2*
(13*b*c + 3*a*d)*Hypergeometric2F1[1/4, 5/4, 9/4, (d*(a + b*x))/(-(b*c) + a*d)]))))/(20*b^4*d^3*(c + d*x)^(1/4
))

________________________________________________________________________________________

Maple [F]  time = 0.029, size = 0, normalized size = 0. \begin{align*} \int{{x}^{3}\sqrt [4]{bx+a}{\frac{1}{\sqrt [4]{dx+c}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 5.36081, size = 8077, normalized size = 23.76 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6144*(12*b^3*d^4*((1445900625*b^16*c^16 - 1779570000*a*b^15*c^15*d - 68445000*a^2*b^14*c^14*d^2 - 177606000*
a^3*b^13*c^13*d^3 - 1551622500*a^4*b^12*c^12*d^4 + 2155086000*a^5*b^11*c^11*d^5 + 370974600*a^6*b^10*c^10*d^6
+ 275389200*a^7*b^9*c^9*d^7 + 665778150*a^8*b^8*c^8*d^8 - 989262960*a^9*b^7*c^7*d^9 - 318453240*a^10*b^6*c^6*d
^10 - 191017680*a^11*b^5*c^5*d^11 - 182203364*a^12*b^4*c^4*d^12 + 176093456*a^13*b^3*c^3*d^13 + 82673976*a^14*
b^2*c^2*d^14 + 51131696*a^15*b*c*d^15 + 35153041*a^16*d^16)/(b^15*d^17))^(1/4)*arctan(((195*b^15*c^4*d^13 - 60
*a*b^14*c^3*d^14 - 30*a^2*b^13*c^2*d^15 - 28*a^3*b^12*c*d^16 - 77*a^4*b^11*d^17)*(b*x + a)^(1/4)*(d*x + c)^(3/
4)*((1445900625*b^16*c^16 - 1779570000*a*b^15*c^15*d - 68445000*a^2*b^14*c^14*d^2 - 177606000*a^3*b^13*c^13*d^
3 - 1551622500*a^4*b^12*c^12*d^4 + 2155086000*a^5*b^11*c^11*d^5 + 370974600*a^6*b^10*c^10*d^6 + 275389200*a^7*
b^9*c^9*d^7 + 665778150*a^8*b^8*c^8*d^8 - 989262960*a^9*b^7*c^7*d^9 - 318453240*a^10*b^6*c^6*d^10 - 191017680*
a^11*b^5*c^5*d^11 - 182203364*a^12*b^4*c^4*d^12 + 176093456*a^13*b^3*c^3*d^13 + 82673976*a^14*b^2*c^2*d^14 + 5
1131696*a^15*b*c*d^15 + 35153041*a^16*d^16)/(b^15*d^17))^(3/4) + (b^11*d^14*x + b^11*c*d^13)*sqrt(((38025*b^8*
c^8 - 23400*a*b^7*c^7*d - 8100*a^2*b^6*c^6*d^2 - 7320*a^3*b^5*c^5*d^3 - 25770*a^4*b^4*c^4*d^4 + 10920*a^5*b^3*
c^3*d^5 + 5404*a^6*b^2*c^2*d^6 + 4312*a^7*b*c*d^7 + 5929*a^8*d^8)*sqrt(b*x + a)*sqrt(d*x + c) + (b^8*d^9*x + b
^8*c*d^8)*sqrt((1445900625*b^16*c^16 - 1779570000*a*b^15*c^15*d - 68445000*a^2*b^14*c^14*d^2 - 177606000*a^3*b
^13*c^13*d^3 - 1551622500*a^4*b^12*c^12*d^4 + 2155086000*a^5*b^11*c^11*d^5 + 370974600*a^6*b^10*c^10*d^6 + 275
389200*a^7*b^9*c^9*d^7 + 665778150*a^8*b^8*c^8*d^8 - 989262960*a^9*b^7*c^7*d^9 - 318453240*a^10*b^6*c^6*d^10 -
191017680*a^11*b^5*c^5*d^11 - 182203364*a^12*b^4*c^4*d^12 + 176093456*a^13*b^3*c^3*d^13 + 82673976*a^14*b^2*c
^2*d^14 + 51131696*a^15*b*c*d^15 + 35153041*a^16*d^16)/(b^15*d^17)))/(d*x + c))*((1445900625*b^16*c^16 - 17795
70000*a*b^15*c^15*d - 68445000*a^2*b^14*c^14*d^2 - 177606000*a^3*b^13*c^13*d^3 - 1551622500*a^4*b^12*c^12*d^4
+ 2155086000*a^5*b^11*c^11*d^5 + 370974600*a^6*b^10*c^10*d^6 + 275389200*a^7*b^9*c^9*d^7 + 665778150*a^8*b^8*c
^8*d^8 - 989262960*a^9*b^7*c^7*d^9 - 318453240*a^10*b^6*c^6*d^10 - 191017680*a^11*b^5*c^5*d^11 - 182203364*a^1
2*b^4*c^4*d^12 + 176093456*a^13*b^3*c^3*d^13 + 82673976*a^14*b^2*c^2*d^14 + 51131696*a^15*b*c*d^15 + 35153041*
a^16*d^16)/(b^15*d^17))^(3/4))/(1445900625*b^16*c^17 - 1779570000*a*b^15*c^16*d - 68445000*a^2*b^14*c^15*d^2 -
177606000*a^3*b^13*c^14*d^3 - 1551622500*a^4*b^12*c^13*d^4 + 2155086000*a^5*b^11*c^12*d^5 + 370974600*a^6*b^1
0*c^11*d^6 + 275389200*a^7*b^9*c^10*d^7 + 665778150*a^8*b^8*c^9*d^8 - 989262960*a^9*b^7*c^8*d^9 - 318453240*a^
10*b^6*c^7*d^10 - 191017680*a^11*b^5*c^6*d^11 - 182203364*a^12*b^4*c^5*d^12 + 176093456*a^13*b^3*c^4*d^13 + 82
673976*a^14*b^2*c^3*d^14 + 51131696*a^15*b*c^2*d^15 + 35153041*a^16*c*d^16 + (1445900625*b^16*c^16*d - 1779570
000*a*b^15*c^15*d^2 - 68445000*a^2*b^14*c^14*d^3 - 177606000*a^3*b^13*c^13*d^4 - 1551622500*a^4*b^12*c^12*d^5
+ 2155086000*a^5*b^11*c^11*d^6 + 370974600*a^6*b^10*c^10*d^7 + 275389200*a^7*b^9*c^9*d^8 + 665778150*a^8*b^8*c
^8*d^9 - 989262960*a^9*b^7*c^7*d^10 - 318453240*a^10*b^6*c^6*d^11 - 191017680*a^11*b^5*c^5*d^12 - 182203364*a^
12*b^4*c^4*d^13 + 176093456*a^13*b^3*c^3*d^14 + 82673976*a^14*b^2*c^2*d^15 + 51131696*a^15*b*c*d^16 + 35153041
*a^16*d^17)*x)) + 3*b^3*d^4*((1445900625*b^16*c^16 - 1779570000*a*b^15*c^15*d - 68445000*a^2*b^14*c^14*d^2 - 1
77606000*a^3*b^13*c^13*d^3 - 1551622500*a^4*b^12*c^12*d^4 + 2155086000*a^5*b^11*c^11*d^5 + 370974600*a^6*b^10*
c^10*d^6 + 275389200*a^7*b^9*c^9*d^7 + 665778150*a^8*b^8*c^8*d^8 - 989262960*a^9*b^7*c^7*d^9 - 318453240*a^10*
b^6*c^6*d^10 - 191017680*a^11*b^5*c^5*d^11 - 182203364*a^12*b^4*c^4*d^12 + 176093456*a^13*b^3*c^3*d^13 + 82673
976*a^14*b^2*c^2*d^14 + 51131696*a^15*b*c*d^15 + 35153041*a^16*d^16)/(b^15*d^17))^(1/4)*log(-((195*b^4*c^4 - 6
0*a*b^3*c^3*d - 30*a^2*b^2*c^2*d^2 - 28*a^3*b*c*d^3 - 77*a^4*d^4)*(b*x + a)^(1/4)*(d*x + c)^(3/4) + (b^4*d^5*x
+ b^4*c*d^4)*((1445900625*b^16*c^16 - 1779570000*a*b^15*c^15*d - 68445000*a^2*b^14*c^14*d^2 - 177606000*a^3*b
^13*c^13*d^3 - 1551622500*a^4*b^12*c^12*d^4 + 2155086000*a^5*b^11*c^11*d^5 + 370974600*a^6*b^10*c^10*d^6 + 275
389200*a^7*b^9*c^9*d^7 + 665778150*a^8*b^8*c^8*d^8 - 989262960*a^9*b^7*c^7*d^9 - 318453240*a^10*b^6*c^6*d^10 -
191017680*a^11*b^5*c^5*d^11 - 182203364*a^12*b^4*c^4*d^12 + 176093456*a^13*b^3*c^3*d^13 + 82673976*a^14*b^2*c
^2*d^14 + 51131696*a^15*b*c*d^15 + 35153041*a^16*d^16)/(b^15*d^17))^(1/4))/(d*x + c)) - 3*b^3*d^4*((1445900625
*b^16*c^16 - 1779570000*a*b^15*c^15*d - 68445000*a^2*b^14*c^14*d^2 - 177606000*a^3*b^13*c^13*d^3 - 1551622500*
a^4*b^12*c^12*d^4 + 2155086000*a^5*b^11*c^11*d^5 + 370974600*a^6*b^10*c^10*d^6 + 275389200*a^7*b^9*c^9*d^7 + 6
65778150*a^8*b^8*c^8*d^8 - 989262960*a^9*b^7*c^7*d^9 - 318453240*a^10*b^6*c^6*d^10 - 191017680*a^11*b^5*c^5*d^
11 - 182203364*a^12*b^4*c^4*d^12 + 176093456*a^13*b^3*c^3*d^13 + 82673976*a^14*b^2*c^2*d^14 + 51131696*a^15*b*
c*d^15 + 35153041*a^16*d^16)/(b^15*d^17))^(1/4)*log(-((195*b^4*c^4 - 60*a*b^3*c^3*d - 30*a^2*b^2*c^2*d^2 - 28*
a^3*b*c*d^3 - 77*a^4*d^4)*(b*x + a)^(1/4)*(d*x + c)^(3/4) - (b^4*d^5*x + b^4*c*d^4)*((1445900625*b^16*c^16 - 1
779570000*a*b^15*c^15*d - 68445000*a^2*b^14*c^14*d^2 - 177606000*a^3*b^13*c^13*d^3 - 1551622500*a^4*b^12*c^12*
d^4 + 2155086000*a^5*b^11*c^11*d^5 + 370974600*a^6*b^10*c^10*d^6 + 275389200*a^7*b^9*c^9*d^7 + 665778150*a^8*b
^8*c^8*d^8 - 989262960*a^9*b^7*c^7*d^9 - 318453240*a^10*b^6*c^6*d^10 - 191017680*a^11*b^5*c^5*d^11 - 182203364
*a^12*b^4*c^4*d^12 + 176093456*a^13*b^3*c^3*d^13 + 82673976*a^14*b^2*c^2*d^14 + 51131696*a^15*b*c*d^15 + 35153
041*a^16*d^16)/(b^15*d^17))^(1/4))/(d*x + c)) + 4*(384*b^3*d^3*x^3 - 585*b^3*c^3 + 63*a*b^2*c^2*d + 61*a^2*b*c
*d^2 + 77*a^3*d^3 - 32*(13*b^3*c*d^2 - a*b^2*d^3)*x^2 + 4*(117*b^3*c^2*d - 10*a*b^2*c*d^2 - 11*a^2*b*d^3)*x)*(
b*x + a)^(1/4)*(d*x + c)^(3/4))/(b^3*d^4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out