3.9.83 \(\int \frac {x^3 \sqrt [4]{a+b x}}{\sqrt [4]{c+d x}} \, dx\) [883]

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

[Out]

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

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Rubi [A]
time = 0.18, antiderivative size = 340, normalized size of antiderivative = 1.00, 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 = {102, 152, 52, 65, 246, 218, 214, 211} \begin {gather*} \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 {(b c-a d) \left (77 a^3 d^3+105 a^2 b c d^2+135 a b^2 c^2 d+195 b^3 c^3\right ) \text {ArcTan}\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 {\sqrt [4]{a+b x} (c+d x)^{3/4} \left (77 a^3 d^3+105 a^2 b c d^2+135 a b^2 c^2 d+195 b^3 c^3\right )}{512 b^3 d^4}+\frac {(b c-a d) \left (77 a^3 d^3+105 a^2 b c d^2+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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/512*((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))/(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 + 10
5*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 52

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 65

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 102

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 152

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 211

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

Rule 214

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

Rule 218

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] &&  !Gt
Q[a/b, 0]

Rule 246

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]

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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 10.27, size = 221, normalized size = 0.65 \begin {gather*} \frac {(c+d x)^{3/4} \left (d (a+b x) \left (77 a^3 d^3+a^2 b d^2 (61 c-44 d x)+a b^2 d \left (63 c^2-40 c d x+32 d^2 x^2\right )+b^3 \left (-585 c^3+468 c^2 d x-416 c d^2 x^2+384 d^3 x^3\right )\right )-\left (-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\right ) \left (\frac {d (a+b x)}{-b c+a d}\right )^{3/4} \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};\frac {b (c+d x)}{b c-a d}\right )\right )}{1536 b^3 d^5 (a+b x)^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((c + d*x)^(3/4)*(d*(a + b*x)*(77*a^3*d^3 + a^2*b*d^2*(61*c - 44*d*x) + a*b^2*d*(63*c^2 - 40*c*d*x + 32*d^2*x^
2) + b^3*(-585*c^3 + 468*c^2*d*x - 416*c*d^2*x^2 + 384*d^3*x^3)) - (-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)*((d*(a + b*x))/(-(b*c) + a*d))^(3/4)*Hypergeometric2F1[3/4, 3/4, 7/4,
(b*(c + d*x))/(b*c - a*d)]))/(1536*b^3*d^5*(a + b*x)^(3/4))

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {x^{3} \left (b x +a \right )^{\frac {1}{4}}}{\left (d x +c \right )^{\frac {1}{4}}}\, dx\]

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)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2864 vs. \(2 (294) = 588\).
time = 0.93, size = 2864, normalized size = 8.42 \begin {gather*} \text {Too large to display} \end {gather*}

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^...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3} \sqrt [4]{a + b x}}{\sqrt [4]{c + d x}}\, dx \end {gather*}

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^3\,{\left (a+b\,x\right )}^{1/4}}{{\left (c+d\,x\right )}^{1/4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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