3.17.43 \(\int \frac {(a+b x)^3 (e+f x)^{5/2}}{c+d x} \, dx\)

Optimal. Leaf size=280 \[ \frac {2 b (e+f x)^{7/2} \left (3 a^2 d^2 f^2-3 a b d f (c f+d e)+b^2 \left (c^2 f^2+c d e f+d^2 e^2\right )\right )}{7 d^3 f^3}-\frac {2 b^2 (e+f x)^{9/2} (-3 a d f+b c f+2 b d e)}{9 d^2 f^3}+\frac {2 (b c-a d)^3 (d e-c f)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{13/2}}-\frac {2 \sqrt {e+f x} (b c-a d)^3 (d e-c f)^2}{d^6}-\frac {2 (e+f x)^{3/2} (b c-a d)^3 (d e-c f)}{3 d^5}-\frac {2 (e+f x)^{5/2} (b c-a d)^3}{5 d^4}+\frac {2 b^3 (e+f x)^{11/2}}{11 d f^3} \]

________________________________________________________________________________________

Rubi [A]  time = 0.30, antiderivative size = 280, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {88, 50, 63, 208} \begin {gather*} \frac {2 b (e+f x)^{7/2} \left (3 a^2 d^2 f^2-3 a b d f (c f+d e)+b^2 \left (c^2 f^2+c d e f+d^2 e^2\right )\right )}{7 d^3 f^3}-\frac {2 b^2 (e+f x)^{9/2} (-3 a d f+b c f+2 b d e)}{9 d^2 f^3}-\frac {2 (e+f x)^{5/2} (b c-a d)^3}{5 d^4}-\frac {2 (e+f x)^{3/2} (b c-a d)^3 (d e-c f)}{3 d^5}-\frac {2 \sqrt {e+f x} (b c-a d)^3 (d e-c f)^2}{d^6}+\frac {2 (b c-a d)^3 (d e-c f)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{13/2}}+\frac {2 b^3 (e+f x)^{11/2}}{11 d f^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^3*(e + f*x)^(5/2))/(c + d*x),x]

[Out]

(-2*(b*c - a*d)^3*(d*e - c*f)^2*Sqrt[e + f*x])/d^6 - (2*(b*c - a*d)^3*(d*e - c*f)*(e + f*x)^(3/2))/(3*d^5) - (
2*(b*c - a*d)^3*(e + f*x)^(5/2))/(5*d^4) + (2*b*(3*a^2*d^2*f^2 - 3*a*b*d*f*(d*e + c*f) + b^2*(d^2*e^2 + c*d*e*
f + c^2*f^2))*(e + f*x)^(7/2))/(7*d^3*f^3) - (2*b^2*(2*b*d*e + b*c*f - 3*a*d*f)*(e + f*x)^(9/2))/(9*d^2*f^3) +
(2*b^3*(e + f*x)^(11/2))/(11*d*f^3) + (2*(b*c - a*d)^3*(d*e - c*f)^(5/2)*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt
[d*e - c*f]])/d^(13/2)

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 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

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]

Rubi steps

\begin {align*} \int \frac {(a+b x)^3 (e+f x)^{5/2}}{c+d x} \, dx &=\int \left (\frac {b \left (3 a^2 d^2 f^2-3 a b d f (d e+c f)+b^2 \left (d^2 e^2+c d e f+c^2 f^2\right )\right ) (e+f x)^{5/2}}{d^3 f^2}+\frac {(-b c+a d)^3 (e+f x)^{5/2}}{d^3 (c+d x)}-\frac {b^2 (2 b d e+b c f-3 a d f) (e+f x)^{7/2}}{d^2 f^2}+\frac {b^3 (e+f x)^{9/2}}{d f^2}\right ) \, dx\\ &=\frac {2 b \left (3 a^2 d^2 f^2-3 a b d f (d e+c f)+b^2 \left (d^2 e^2+c d e f+c^2 f^2\right )\right ) (e+f x)^{7/2}}{7 d^3 f^3}-\frac {2 b^2 (2 b d e+b c f-3 a d f) (e+f x)^{9/2}}{9 d^2 f^3}+\frac {2 b^3 (e+f x)^{11/2}}{11 d f^3}-\frac {(b c-a d)^3 \int \frac {(e+f x)^{5/2}}{c+d x} \, dx}{d^3}\\ &=-\frac {2 (b c-a d)^3 (e+f x)^{5/2}}{5 d^4}+\frac {2 b \left (3 a^2 d^2 f^2-3 a b d f (d e+c f)+b^2 \left (d^2 e^2+c d e f+c^2 f^2\right )\right ) (e+f x)^{7/2}}{7 d^3 f^3}-\frac {2 b^2 (2 b d e+b c f-3 a d f) (e+f x)^{9/2}}{9 d^2 f^3}+\frac {2 b^3 (e+f x)^{11/2}}{11 d f^3}-\frac {\left ((b c-a d)^3 (d e-c f)\right ) \int \frac {(e+f x)^{3/2}}{c+d x} \, dx}{d^4}\\ &=-\frac {2 (b c-a d)^3 (d e-c f) (e+f x)^{3/2}}{3 d^5}-\frac {2 (b c-a d)^3 (e+f x)^{5/2}}{5 d^4}+\frac {2 b \left (3 a^2 d^2 f^2-3 a b d f (d e+c f)+b^2 \left (d^2 e^2+c d e f+c^2 f^2\right )\right ) (e+f x)^{7/2}}{7 d^3 f^3}-\frac {2 b^2 (2 b d e+b c f-3 a d f) (e+f x)^{9/2}}{9 d^2 f^3}+\frac {2 b^3 (e+f x)^{11/2}}{11 d f^3}-\frac {\left ((b c-a d)^3 (d e-c f)^2\right ) \int \frac {\sqrt {e+f x}}{c+d x} \, dx}{d^5}\\ &=-\frac {2 (b c-a d)^3 (d e-c f)^2 \sqrt {e+f x}}{d^6}-\frac {2 (b c-a d)^3 (d e-c f) (e+f x)^{3/2}}{3 d^5}-\frac {2 (b c-a d)^3 (e+f x)^{5/2}}{5 d^4}+\frac {2 b \left (3 a^2 d^2 f^2-3 a b d f (d e+c f)+b^2 \left (d^2 e^2+c d e f+c^2 f^2\right )\right ) (e+f x)^{7/2}}{7 d^3 f^3}-\frac {2 b^2 (2 b d e+b c f-3 a d f) (e+f x)^{9/2}}{9 d^2 f^3}+\frac {2 b^3 (e+f x)^{11/2}}{11 d f^3}-\frac {\left ((b c-a d)^3 (d e-c f)^3\right ) \int \frac {1}{(c+d x) \sqrt {e+f x}} \, dx}{d^6}\\ &=-\frac {2 (b c-a d)^3 (d e-c f)^2 \sqrt {e+f x}}{d^6}-\frac {2 (b c-a d)^3 (d e-c f) (e+f x)^{3/2}}{3 d^5}-\frac {2 (b c-a d)^3 (e+f x)^{5/2}}{5 d^4}+\frac {2 b \left (3 a^2 d^2 f^2-3 a b d f (d e+c f)+b^2 \left (d^2 e^2+c d e f+c^2 f^2\right )\right ) (e+f x)^{7/2}}{7 d^3 f^3}-\frac {2 b^2 (2 b d e+b c f-3 a d f) (e+f x)^{9/2}}{9 d^2 f^3}+\frac {2 b^3 (e+f x)^{11/2}}{11 d f^3}-\frac {\left (2 (b c-a d)^3 (d e-c f)^3\right ) \operatorname {Subst}\left (\int \frac {1}{c-\frac {d e}{f}+\frac {d x^2}{f}} \, dx,x,\sqrt {e+f x}\right )}{d^6 f}\\ &=-\frac {2 (b c-a d)^3 (d e-c f)^2 \sqrt {e+f x}}{d^6}-\frac {2 (b c-a d)^3 (d e-c f) (e+f x)^{3/2}}{3 d^5}-\frac {2 (b c-a d)^3 (e+f x)^{5/2}}{5 d^4}+\frac {2 b \left (3 a^2 d^2 f^2-3 a b d f (d e+c f)+b^2 \left (d^2 e^2+c d e f+c^2 f^2\right )\right ) (e+f x)^{7/2}}{7 d^3 f^3}-\frac {2 b^2 (2 b d e+b c f-3 a d f) (e+f x)^{9/2}}{9 d^2 f^3}+\frac {2 b^3 (e+f x)^{11/2}}{11 d f^3}+\frac {2 (b c-a d)^3 (d e-c f)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{13/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.91, size = 253, normalized size = 0.90 \begin {gather*} \frac {2 b (e+f x)^{7/2} \left (3 a^2 d^2 f^2-3 a b d f (c f+d e)+b^2 \left (c^2 f^2+c d e f+d^2 e^2\right )\right )}{7 d^3 f^3}-\frac {2 b^2 (e+f x)^{9/2} (-3 a d f+b c f+2 b d e)}{9 d^2 f^3}+\frac {2 (a d-b c)^3 (d e-c f) \left (\sqrt {d} \sqrt {e+f x} (-3 c f+4 d e+d f x)-3 (d e-c f)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )\right )}{3 d^{13/2}}+\frac {2 (e+f x)^{5/2} (a d-b c)^3}{5 d^4}+\frac {2 b^3 (e+f x)^{11/2}}{11 d f^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^3*(e + f*x)^(5/2))/(c + d*x),x]

[Out]

(2*(-(b*c) + a*d)^3*(e + f*x)^(5/2))/(5*d^4) + (2*b*(3*a^2*d^2*f^2 - 3*a*b*d*f*(d*e + c*f) + b^2*(d^2*e^2 + c*
d*e*f + c^2*f^2))*(e + f*x)^(7/2))/(7*d^3*f^3) - (2*b^2*(2*b*d*e + b*c*f - 3*a*d*f)*(e + f*x)^(9/2))/(9*d^2*f^
3) + (2*b^3*(e + f*x)^(11/2))/(11*d*f^3) + (2*(-(b*c) + a*d)^3*(d*e - c*f)*(Sqrt[d]*Sqrt[e + f*x]*(4*d*e - 3*c
*f + d*f*x) - 3*(d*e - c*f)^(3/2)*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f]]))/(3*d^(13/2))

________________________________________________________________________________________

IntegrateAlgebraic [B]  time = 0.44, size = 834, normalized size = 2.98 \begin {gather*} \frac {2 (a d-b c)^3 (c f-d e)^{5/2} \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {c f-d e} \sqrt {e+f x}}{d e-c f}\right )}{d^{13/2}}-\frac {2 \left (-315 b^3 d^5 (e+f x)^{11/2}+770 b^3 d^5 e (e+f x)^{9/2}-1155 a b^2 d^5 f (e+f x)^{9/2}+385 b^3 c d^4 f (e+f x)^{9/2}-495 b^3 d^5 e^2 (e+f x)^{7/2}-1485 a^2 b d^5 f^2 (e+f x)^{7/2}+1485 a b^2 c d^4 f^2 (e+f x)^{7/2}-495 b^3 c^2 d^3 f^2 (e+f x)^{7/2}+1485 a b^2 d^5 e f (e+f x)^{7/2}-495 b^3 c d^4 e f (e+f x)^{7/2}-693 a^3 d^5 f^3 (e+f x)^{5/2}+2079 a^2 b c d^4 f^3 (e+f x)^{5/2}-2079 a b^2 c^2 d^3 f^3 (e+f x)^{5/2}+693 b^3 c^3 d^2 f^3 (e+f x)^{5/2}+1155 a^3 c d^4 f^4 (e+f x)^{3/2}-3465 a^2 b c^2 d^3 f^4 (e+f x)^{3/2}+3465 a b^2 c^3 d^2 f^4 (e+f x)^{3/2}-1155 b^3 c^4 d f^4 (e+f x)^{3/2}-1155 a^3 d^5 e f^3 (e+f x)^{3/2}+3465 a^2 b c d^4 e f^3 (e+f x)^{3/2}-3465 a b^2 c^2 d^3 e f^3 (e+f x)^{3/2}+1155 b^3 c^3 d^2 e f^3 (e+f x)^{3/2}+3465 b^3 c^5 f^5 \sqrt {e+f x}-3465 a^3 c^2 d^3 f^5 \sqrt {e+f x}+10395 a^2 b c^3 d^2 f^5 \sqrt {e+f x}-10395 a b^2 c^4 d f^5 \sqrt {e+f x}+6930 a^3 c d^4 e f^4 \sqrt {e+f x}-20790 a^2 b c^2 d^3 e f^4 \sqrt {e+f x}+20790 a b^2 c^3 d^2 e f^4 \sqrt {e+f x}-6930 b^3 c^4 d e f^4 \sqrt {e+f x}-3465 a^3 d^5 e^2 f^3 \sqrt {e+f x}+10395 a^2 b c d^4 e^2 f^3 \sqrt {e+f x}-10395 a b^2 c^2 d^3 e^2 f^3 \sqrt {e+f x}+3465 b^3 c^3 d^2 e^2 f^3 \sqrt {e+f x}\right )}{3465 d^6 f^3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((a + b*x)^3*(e + f*x)^(5/2))/(c + d*x),x]

[Out]

(-2*(3465*b^3*c^3*d^2*e^2*f^3*Sqrt[e + f*x] - 10395*a*b^2*c^2*d^3*e^2*f^3*Sqrt[e + f*x] + 10395*a^2*b*c*d^4*e^
2*f^3*Sqrt[e + f*x] - 3465*a^3*d^5*e^2*f^3*Sqrt[e + f*x] - 6930*b^3*c^4*d*e*f^4*Sqrt[e + f*x] + 20790*a*b^2*c^
3*d^2*e*f^4*Sqrt[e + f*x] - 20790*a^2*b*c^2*d^3*e*f^4*Sqrt[e + f*x] + 6930*a^3*c*d^4*e*f^4*Sqrt[e + f*x] + 346
5*b^3*c^5*f^5*Sqrt[e + f*x] - 10395*a*b^2*c^4*d*f^5*Sqrt[e + f*x] + 10395*a^2*b*c^3*d^2*f^5*Sqrt[e + f*x] - 34
65*a^3*c^2*d^3*f^5*Sqrt[e + f*x] + 1155*b^3*c^3*d^2*e*f^3*(e + f*x)^(3/2) - 3465*a*b^2*c^2*d^3*e*f^3*(e + f*x)
^(3/2) + 3465*a^2*b*c*d^4*e*f^3*(e + f*x)^(3/2) - 1155*a^3*d^5*e*f^3*(e + f*x)^(3/2) - 1155*b^3*c^4*d*f^4*(e +
f*x)^(3/2) + 3465*a*b^2*c^3*d^2*f^4*(e + f*x)^(3/2) - 3465*a^2*b*c^2*d^3*f^4*(e + f*x)^(3/2) + 1155*a^3*c*d^4
*f^4*(e + f*x)^(3/2) + 693*b^3*c^3*d^2*f^3*(e + f*x)^(5/2) - 2079*a*b^2*c^2*d^3*f^3*(e + f*x)^(5/2) + 2079*a^2
*b*c*d^4*f^3*(e + f*x)^(5/2) - 693*a^3*d^5*f^3*(e + f*x)^(5/2) - 495*b^3*d^5*e^2*(e + f*x)^(7/2) - 495*b^3*c*d
^4*e*f*(e + f*x)^(7/2) + 1485*a*b^2*d^5*e*f*(e + f*x)^(7/2) - 495*b^3*c^2*d^3*f^2*(e + f*x)^(7/2) + 1485*a*b^2
*c*d^4*f^2*(e + f*x)^(7/2) - 1485*a^2*b*d^5*f^2*(e + f*x)^(7/2) + 770*b^3*d^5*e*(e + f*x)^(9/2) + 385*b^3*c*d^
4*f*(e + f*x)^(9/2) - 1155*a*b^2*d^5*f*(e + f*x)^(9/2) - 315*b^3*d^5*(e + f*x)^(11/2)))/(3465*d^6*f^3) + (2*(-
(b*c) + a*d)^3*(-(d*e) + c*f)^(5/2)*ArcTan[(Sqrt[d]*Sqrt[-(d*e) + c*f]*Sqrt[e + f*x])/(d*e - c*f)])/d^(13/2)

________________________________________________________________________________________

fricas [B]  time = 1.77, size = 1741, normalized size = 6.22

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(f*x+e)^(5/2)/(d*x+c),x, algorithm="fricas")

[Out]

[-1/3465*(3465*((b^3*c^3*d^2 - 3*a*b^2*c^2*d^3 + 3*a^2*b*c*d^4 - a^3*d^5)*e^2*f^3 - 2*(b^3*c^4*d - 3*a*b^2*c^3
*d^2 + 3*a^2*b*c^2*d^3 - a^3*c*d^4)*e*f^4 + (b^3*c^5 - 3*a*b^2*c^4*d + 3*a^2*b*c^3*d^2 - a^3*c^2*d^3)*f^5)*sqr
t((d*e - c*f)/d)*log((d*f*x + 2*d*e - c*f - 2*sqrt(f*x + e)*d*sqrt((d*e - c*f)/d))/(d*x + c)) - 2*(315*b^3*d^5
*f^5*x^5 + 40*b^3*d^5*e^5 + 110*(b^3*c*d^4 - 3*a*b^2*d^5)*e^4*f + 495*(b^3*c^2*d^3 - 3*a*b^2*c*d^4 + 3*a^2*b*d
^5)*e^3*f^2 - 5313*(b^3*c^3*d^2 - 3*a*b^2*c^2*d^3 + 3*a^2*b*c*d^4 - a^3*d^5)*e^2*f^3 + 8085*(b^3*c^4*d - 3*a*b
^2*c^3*d^2 + 3*a^2*b*c^2*d^3 - a^3*c*d^4)*e*f^4 - 3465*(b^3*c^5 - 3*a*b^2*c^4*d + 3*a^2*b*c^3*d^2 - a^3*c^2*d^
3)*f^5 + 35*(23*b^3*d^5*e*f^4 - 11*(b^3*c*d^4 - 3*a*b^2*d^5)*f^5)*x^4 + 5*(113*b^3*d^5*e^2*f^3 - 209*(b^3*c*d^
4 - 3*a*b^2*d^5)*e*f^4 + 99*(b^3*c^2*d^3 - 3*a*b^2*c*d^4 + 3*a^2*b*d^5)*f^5)*x^3 + 3*(5*b^3*d^5*e^3*f^2 - 275*
(b^3*c*d^4 - 3*a*b^2*d^5)*e^2*f^3 + 495*(b^3*c^2*d^3 - 3*a*b^2*c*d^4 + 3*a^2*b*d^5)*e*f^4 - 231*(b^3*c^3*d^2 -
3*a*b^2*c^2*d^3 + 3*a^2*b*c*d^4 - a^3*d^5)*f^5)*x^2 - (20*b^3*d^5*e^4*f + 55*(b^3*c*d^4 - 3*a*b^2*d^5)*e^3*f^
2 - 1485*(b^3*c^2*d^3 - 3*a*b^2*c*d^4 + 3*a^2*b*d^5)*e^2*f^3 + 2541*(b^3*c^3*d^2 - 3*a*b^2*c^2*d^3 + 3*a^2*b*c
*d^4 - a^3*d^5)*e*f^4 - 1155*(b^3*c^4*d - 3*a*b^2*c^3*d^2 + 3*a^2*b*c^2*d^3 - a^3*c*d^4)*f^5)*x)*sqrt(f*x + e)
)/(d^6*f^3), 2/3465*(3465*((b^3*c^3*d^2 - 3*a*b^2*c^2*d^3 + 3*a^2*b*c*d^4 - a^3*d^5)*e^2*f^3 - 2*(b^3*c^4*d -
3*a*b^2*c^3*d^2 + 3*a^2*b*c^2*d^3 - a^3*c*d^4)*e*f^4 + (b^3*c^5 - 3*a*b^2*c^4*d + 3*a^2*b*c^3*d^2 - a^3*c^2*d^
3)*f^5)*sqrt(-(d*e - c*f)/d)*arctan(-sqrt(f*x + e)*d*sqrt(-(d*e - c*f)/d)/(d*e - c*f)) + (315*b^3*d^5*f^5*x^5
+ 40*b^3*d^5*e^5 + 110*(b^3*c*d^4 - 3*a*b^2*d^5)*e^4*f + 495*(b^3*c^2*d^3 - 3*a*b^2*c*d^4 + 3*a^2*b*d^5)*e^3*f
^2 - 5313*(b^3*c^3*d^2 - 3*a*b^2*c^2*d^3 + 3*a^2*b*c*d^4 - a^3*d^5)*e^2*f^3 + 8085*(b^3*c^4*d - 3*a*b^2*c^3*d^
2 + 3*a^2*b*c^2*d^3 - a^3*c*d^4)*e*f^4 - 3465*(b^3*c^5 - 3*a*b^2*c^4*d + 3*a^2*b*c^3*d^2 - a^3*c^2*d^3)*f^5 +
35*(23*b^3*d^5*e*f^4 - 11*(b^3*c*d^4 - 3*a*b^2*d^5)*f^5)*x^4 + 5*(113*b^3*d^5*e^2*f^3 - 209*(b^3*c*d^4 - 3*a*b
^2*d^5)*e*f^4 + 99*(b^3*c^2*d^3 - 3*a*b^2*c*d^4 + 3*a^2*b*d^5)*f^5)*x^3 + 3*(5*b^3*d^5*e^3*f^2 - 275*(b^3*c*d^
4 - 3*a*b^2*d^5)*e^2*f^3 + 495*(b^3*c^2*d^3 - 3*a*b^2*c*d^4 + 3*a^2*b*d^5)*e*f^4 - 231*(b^3*c^3*d^2 - 3*a*b^2*
c^2*d^3 + 3*a^2*b*c*d^4 - a^3*d^5)*f^5)*x^2 - (20*b^3*d^5*e^4*f + 55*(b^3*c*d^4 - 3*a*b^2*d^5)*e^3*f^2 - 1485*
(b^3*c^2*d^3 - 3*a*b^2*c*d^4 + 3*a^2*b*d^5)*e^2*f^3 + 2541*(b^3*c^3*d^2 - 3*a*b^2*c^2*d^3 + 3*a^2*b*c*d^4 - a^
3*d^5)*e*f^4 - 1155*(b^3*c^4*d - 3*a*b^2*c^3*d^2 + 3*a^2*b*c^2*d^3 - a^3*c*d^4)*f^5)*x)*sqrt(f*x + e))/(d^6*f^
3)]

________________________________________________________________________________________

giac [B]  time = 1.47, size = 1028, normalized size = 3.67 \begin {gather*} \frac {2 \, {\left (b^{3} c^{6} f^{3} - 3 \, a b^{2} c^{5} d f^{3} + 3 \, a^{2} b c^{4} d^{2} f^{3} - a^{3} c^{3} d^{3} f^{3} - 3 \, b^{3} c^{5} d f^{2} e + 9 \, a b^{2} c^{4} d^{2} f^{2} e - 9 \, a^{2} b c^{3} d^{3} f^{2} e + 3 \, a^{3} c^{2} d^{4} f^{2} e + 3 \, b^{3} c^{4} d^{2} f e^{2} - 9 \, a b^{2} c^{3} d^{3} f e^{2} + 9 \, a^{2} b c^{2} d^{4} f e^{2} - 3 \, a^{3} c d^{5} f e^{2} - b^{3} c^{3} d^{3} e^{3} + 3 \, a b^{2} c^{2} d^{4} e^{3} - 3 \, a^{2} b c d^{5} e^{3} + a^{3} d^{6} e^{3}\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {c d f - d^{2} e}}\right )}{\sqrt {c d f - d^{2} e} d^{6}} + \frac {2 \, {\left (315 \, {\left (f x + e\right )}^{\frac {11}{2}} b^{3} d^{10} f^{30} - 385 \, {\left (f x + e\right )}^{\frac {9}{2}} b^{3} c d^{9} f^{31} + 1155 \, {\left (f x + e\right )}^{\frac {9}{2}} a b^{2} d^{10} f^{31} + 495 \, {\left (f x + e\right )}^{\frac {7}{2}} b^{3} c^{2} d^{8} f^{32} - 1485 \, {\left (f x + e\right )}^{\frac {7}{2}} a b^{2} c d^{9} f^{32} + 1485 \, {\left (f x + e\right )}^{\frac {7}{2}} a^{2} b d^{10} f^{32} - 693 \, {\left (f x + e\right )}^{\frac {5}{2}} b^{3} c^{3} d^{7} f^{33} + 2079 \, {\left (f x + e\right )}^{\frac {5}{2}} a b^{2} c^{2} d^{8} f^{33} - 2079 \, {\left (f x + e\right )}^{\frac {5}{2}} a^{2} b c d^{9} f^{33} + 693 \, {\left (f x + e\right )}^{\frac {5}{2}} a^{3} d^{10} f^{33} + 1155 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{3} c^{4} d^{6} f^{34} - 3465 \, {\left (f x + e\right )}^{\frac {3}{2}} a b^{2} c^{3} d^{7} f^{34} + 3465 \, {\left (f x + e\right )}^{\frac {3}{2}} a^{2} b c^{2} d^{8} f^{34} - 1155 \, {\left (f x + e\right )}^{\frac {3}{2}} a^{3} c d^{9} f^{34} - 3465 \, \sqrt {f x + e} b^{3} c^{5} d^{5} f^{35} + 10395 \, \sqrt {f x + e} a b^{2} c^{4} d^{6} f^{35} - 10395 \, \sqrt {f x + e} a^{2} b c^{3} d^{7} f^{35} + 3465 \, \sqrt {f x + e} a^{3} c^{2} d^{8} f^{35} - 770 \, {\left (f x + e\right )}^{\frac {9}{2}} b^{3} d^{10} f^{30} e + 495 \, {\left (f x + e\right )}^{\frac {7}{2}} b^{3} c d^{9} f^{31} e - 1485 \, {\left (f x + e\right )}^{\frac {7}{2}} a b^{2} d^{10} f^{31} e - 1155 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{3} c^{3} d^{7} f^{33} e + 3465 \, {\left (f x + e\right )}^{\frac {3}{2}} a b^{2} c^{2} d^{8} f^{33} e - 3465 \, {\left (f x + e\right )}^{\frac {3}{2}} a^{2} b c d^{9} f^{33} e + 1155 \, {\left (f x + e\right )}^{\frac {3}{2}} a^{3} d^{10} f^{33} e + 6930 \, \sqrt {f x + e} b^{3} c^{4} d^{6} f^{34} e - 20790 \, \sqrt {f x + e} a b^{2} c^{3} d^{7} f^{34} e + 20790 \, \sqrt {f x + e} a^{2} b c^{2} d^{8} f^{34} e - 6930 \, \sqrt {f x + e} a^{3} c d^{9} f^{34} e + 495 \, {\left (f x + e\right )}^{\frac {7}{2}} b^{3} d^{10} f^{30} e^{2} - 3465 \, \sqrt {f x + e} b^{3} c^{3} d^{7} f^{33} e^{2} + 10395 \, \sqrt {f x + e} a b^{2} c^{2} d^{8} f^{33} e^{2} - 10395 \, \sqrt {f x + e} a^{2} b c d^{9} f^{33} e^{2} + 3465 \, \sqrt {f x + e} a^{3} d^{10} f^{33} e^{2}\right )}}{3465 \, d^{11} f^{33}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(f*x+e)^(5/2)/(d*x+c),x, algorithm="giac")

[Out]

2*(b^3*c^6*f^3 - 3*a*b^2*c^5*d*f^3 + 3*a^2*b*c^4*d^2*f^3 - a^3*c^3*d^3*f^3 - 3*b^3*c^5*d*f^2*e + 9*a*b^2*c^4*d
^2*f^2*e - 9*a^2*b*c^3*d^3*f^2*e + 3*a^3*c^2*d^4*f^2*e + 3*b^3*c^4*d^2*f*e^2 - 9*a*b^2*c^3*d^3*f*e^2 + 9*a^2*b
*c^2*d^4*f*e^2 - 3*a^3*c*d^5*f*e^2 - b^3*c^3*d^3*e^3 + 3*a*b^2*c^2*d^4*e^3 - 3*a^2*b*c*d^5*e^3 + a^3*d^6*e^3)*
arctan(sqrt(f*x + e)*d/sqrt(c*d*f - d^2*e))/(sqrt(c*d*f - d^2*e)*d^6) + 2/3465*(315*(f*x + e)^(11/2)*b^3*d^10*
f^30 - 385*(f*x + e)^(9/2)*b^3*c*d^9*f^31 + 1155*(f*x + e)^(9/2)*a*b^2*d^10*f^31 + 495*(f*x + e)^(7/2)*b^3*c^2
*d^8*f^32 - 1485*(f*x + e)^(7/2)*a*b^2*c*d^9*f^32 + 1485*(f*x + e)^(7/2)*a^2*b*d^10*f^32 - 693*(f*x + e)^(5/2)
*b^3*c^3*d^7*f^33 + 2079*(f*x + e)^(5/2)*a*b^2*c^2*d^8*f^33 - 2079*(f*x + e)^(5/2)*a^2*b*c*d^9*f^33 + 693*(f*x
+ e)^(5/2)*a^3*d^10*f^33 + 1155*(f*x + e)^(3/2)*b^3*c^4*d^6*f^34 - 3465*(f*x + e)^(3/2)*a*b^2*c^3*d^7*f^34 +
3465*(f*x + e)^(3/2)*a^2*b*c^2*d^8*f^34 - 1155*(f*x + e)^(3/2)*a^3*c*d^9*f^34 - 3465*sqrt(f*x + e)*b^3*c^5*d^5
*f^35 + 10395*sqrt(f*x + e)*a*b^2*c^4*d^6*f^35 - 10395*sqrt(f*x + e)*a^2*b*c^3*d^7*f^35 + 3465*sqrt(f*x + e)*a
^3*c^2*d^8*f^35 - 770*(f*x + e)^(9/2)*b^3*d^10*f^30*e + 495*(f*x + e)^(7/2)*b^3*c*d^9*f^31*e - 1485*(f*x + e)^
(7/2)*a*b^2*d^10*f^31*e - 1155*(f*x + e)^(3/2)*b^3*c^3*d^7*f^33*e + 3465*(f*x + e)^(3/2)*a*b^2*c^2*d^8*f^33*e
- 3465*(f*x + e)^(3/2)*a^2*b*c*d^9*f^33*e + 1155*(f*x + e)^(3/2)*a^3*d^10*f^33*e + 6930*sqrt(f*x + e)*b^3*c^4*
d^6*f^34*e - 20790*sqrt(f*x + e)*a*b^2*c^3*d^7*f^34*e + 20790*sqrt(f*x + e)*a^2*b*c^2*d^8*f^34*e - 6930*sqrt(f
*x + e)*a^3*c*d^9*f^34*e + 495*(f*x + e)^(7/2)*b^3*d^10*f^30*e^2 - 3465*sqrt(f*x + e)*b^3*c^3*d^7*f^33*e^2 + 1
0395*sqrt(f*x + e)*a*b^2*c^2*d^8*f^33*e^2 - 10395*sqrt(f*x + e)*a^2*b*c*d^9*f^33*e^2 + 3465*sqrt(f*x + e)*a^3*
d^10*f^33*e^2)/(d^11*f^33)

________________________________________________________________________________________

maple [B]  time = 0.02, size = 1437, normalized size = 5.13

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3*(f*x+e)^(5/2)/(d*x+c),x)

[Out]

-2/d^3/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2)*d)*b^3*c^3*e^3-2/d^2*(f*x+e)^(3/2)*a^2*b*c
*e-2*f^3/d^3/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2)*d)*a^3*c^3+2*f^3/d^6/((c*f-d*e)*d)^(
1/2)*arctan((f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2)*d)*b^3*c^6+2/3/d*(f*x+e)^(3/2)*a^3*e-2/5/d^4*(f*x+e)^(5/2)*b^3*c
^3+2/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2)*d)*a^3*e^3+2/d*a^3*e^2*(f*x+e)^(1/2)+12*f/d^
3*a^2*b*c^2*e*(f*x+e)^(1/2)-18*f^2/d^3/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2)*d)*a^2*b*c
^3*e+18*f/d^2/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2)*d)*a^2*b*c^2*e^2+18*f^2/d^4/((c*f-d
*e)*d)^(1/2)*arctan((f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2)*d)*a*b^2*c^4*e-18*f/d^3/((c*f-d*e)*d)^(1/2)*arctan((f*x+
e)^(1/2)/((c*f-d*e)*d)^(1/2)*d)*a*b^2*c^3*e^2+2/11*b^3*(f*x+e)^(11/2)/d/f^3+6/7/f/d*(f*x+e)^(7/2)*a^2*b+2/5/d*
(f*x+e)^(5/2)*a^3-2/9/f^2/d^2*(f*x+e)^(9/2)*b^3*c+2/7/f/d^3*(f*x+e)^(7/2)*b^3*c^2+2*f^2/d^3*a^3*c^2*(f*x+e)^(1
/2)-2/d^4*b^3*c^3*e^2*(f*x+e)^(1/2)-6/5/d^2*(f*x+e)^(5/2)*a^2*b*c+6/5/d^3*(f*x+e)^(5/2)*a*b^2*c^2-2/3/d^4*(f*x
+e)^(3/2)*b^3*c^3*e-4/9/f^3/d*(f*x+e)^(9/2)*b^3*e+2/7/f^3/d*(f*x+e)^(7/2)*b^3*e^2-2*f^2/d^6*b^3*c^5*(f*x+e)^(1
/2)+2/3/f^2/d*(f*x+e)^(9/2)*a*b^2-2/3*f/d^2*(f*x+e)^(3/2)*a^3*c+2/3*f/d^5*(f*x+e)^(3/2)*b^3*c^4-6/7/f/d^2*(f*x
+e)^(7/2)*a*b^2*c-6/7/f^2/d*(f*x+e)^(7/2)*a*b^2*e+2/7/f^2/d^2*(f*x+e)^(7/2)*b^3*c*e+2*f/d^3*(f*x+e)^(3/2)*a^2*
b*c^2-2*f/d^4*(f*x+e)^(3/2)*a*b^2*c^3-4*f/d^2*a^3*c*e*(f*x+e)^(1/2)-6*f^2/d^4*a^2*b*c^3*(f*x+e)^(1/2)+6*f^2/d^
5*a*b^2*c^4*(f*x+e)^(1/2)+4*f/d^5*b^3*c^4*e*(f*x+e)^(1/2)+2/d^3*(f*x+e)^(3/2)*a*b^2*c^2*e+6/d^3*a*b^2*c^2*e^2*
(f*x+e)^(1/2)-6/d^2*a^2*b*c*e^2*(f*x+e)^(1/2)-12*f/d^4*a*b^2*c^3*e*(f*x+e)^(1/2)-6/d/((c*f-d*e)*d)^(1/2)*arcta
n((f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2)*d)*a^2*b*c*e^3+6/d^2/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)/((c*f-d*e)*d
)^(1/2)*d)*a*b^2*c^2*e^3+6*f^2/d^2/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2)*d)*a^3*c^2*e-6
*f/d/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2)*d)*a^3*c*e^2+6*f^3/d^4/((c*f-d*e)*d)^(1/2)*a
rctan((f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2)*d)*a^2*b*c^4-6*f^3/d^5/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)/((c*f-
d*e)*d)^(1/2)*d)*a*b^2*c^5-6*f^2/d^5/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2)*d)*b^3*c^5*e
+6*f/d^4/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2)*d)*b^3*c^4*e^2

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(f*x+e)^(5/2)/(d*x+c),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(c*f-d*e>0)', see `assume?` for
more details)Is c*f-d*e positive or negative?

________________________________________________________________________________________

mupad [B]  time = 0.16, size = 897, normalized size = 3.20 \begin {gather*} {\left (e+f\,x\right )}^{7/2}\,\left (\frac {\left (\frac {6\,b^3\,e-6\,a\,b^2\,f}{d\,f^3}+\frac {2\,b^3\,\left (c\,f^4-d\,e\,f^3\right )}{d^2\,f^6}\right )\,\left (c\,f^4-d\,e\,f^3\right )}{7\,d\,f^3}+\frac {6\,b\,{\left (a\,f-b\,e\right )}^2}{7\,d\,f^3}\right )-{\left (e+f\,x\right )}^{9/2}\,\left (\frac {6\,b^3\,e-6\,a\,b^2\,f}{9\,d\,f^3}+\frac {2\,b^3\,\left (c\,f^4-d\,e\,f^3\right )}{9\,d^2\,f^6}\right )+{\left (e+f\,x\right )}^{5/2}\,\left (\frac {2\,{\left (a\,f-b\,e\right )}^3}{5\,d\,f^3}-\frac {\left (\frac {\left (\frac {6\,b^3\,e-6\,a\,b^2\,f}{d\,f^3}+\frac {2\,b^3\,\left (c\,f^4-d\,e\,f^3\right )}{d^2\,f^6}\right )\,\left (c\,f^4-d\,e\,f^3\right )}{d\,f^3}+\frac {6\,b\,{\left (a\,f-b\,e\right )}^2}{d\,f^3}\right )\,\left (c\,f^4-d\,e\,f^3\right )}{5\,d\,f^3}\right )+\frac {2\,b^3\,{\left (e+f\,x\right )}^{11/2}}{11\,d\,f^3}+\frac {2\,\mathrm {atan}\left (\frac {\sqrt {d}\,\sqrt {e+f\,x}\,{\left (a\,d-b\,c\right )}^3\,{\left (c\,f-d\,e\right )}^{5/2}}{-a^3\,c^3\,d^3\,f^3+3\,a^3\,c^2\,d^4\,e\,f^2-3\,a^3\,c\,d^5\,e^2\,f+a^3\,d^6\,e^3+3\,a^2\,b\,c^4\,d^2\,f^3-9\,a^2\,b\,c^3\,d^3\,e\,f^2+9\,a^2\,b\,c^2\,d^4\,e^2\,f-3\,a^2\,b\,c\,d^5\,e^3-3\,a\,b^2\,c^5\,d\,f^3+9\,a\,b^2\,c^4\,d^2\,e\,f^2-9\,a\,b^2\,c^3\,d^3\,e^2\,f+3\,a\,b^2\,c^2\,d^4\,e^3+b^3\,c^6\,f^3-3\,b^3\,c^5\,d\,e\,f^2+3\,b^3\,c^4\,d^2\,e^2\,f-b^3\,c^3\,d^3\,e^3}\right )\,{\left (a\,d-b\,c\right )}^3\,{\left (c\,f-d\,e\right )}^{5/2}}{d^{13/2}}-\frac {{\left (e+f\,x\right )}^{3/2}\,\left (\frac {2\,{\left (a\,f-b\,e\right )}^3}{d\,f^3}-\frac {\left (\frac {\left (\frac {6\,b^3\,e-6\,a\,b^2\,f}{d\,f^3}+\frac {2\,b^3\,\left (c\,f^4-d\,e\,f^3\right )}{d^2\,f^6}\right )\,\left (c\,f^4-d\,e\,f^3\right )}{d\,f^3}+\frac {6\,b\,{\left (a\,f-b\,e\right )}^2}{d\,f^3}\right )\,\left (c\,f^4-d\,e\,f^3\right )}{d\,f^3}\right )\,\left (c\,f^4-d\,e\,f^3\right )}{3\,d\,f^3}+\frac {\sqrt {e+f\,x}\,\left (\frac {2\,{\left (a\,f-b\,e\right )}^3}{d\,f^3}-\frac {\left (\frac {\left (\frac {6\,b^3\,e-6\,a\,b^2\,f}{d\,f^3}+\frac {2\,b^3\,\left (c\,f^4-d\,e\,f^3\right )}{d^2\,f^6}\right )\,\left (c\,f^4-d\,e\,f^3\right )}{d\,f^3}+\frac {6\,b\,{\left (a\,f-b\,e\right )}^2}{d\,f^3}\right )\,\left (c\,f^4-d\,e\,f^3\right )}{d\,f^3}\right )\,{\left (c\,f^4-d\,e\,f^3\right )}^2}{d^2\,f^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((e + f*x)^(5/2)*(a + b*x)^3)/(c + d*x),x)

[Out]

(e + f*x)^(7/2)*((((6*b^3*e - 6*a*b^2*f)/(d*f^3) + (2*b^3*(c*f^4 - d*e*f^3))/(d^2*f^6))*(c*f^4 - d*e*f^3))/(7*
d*f^3) + (6*b*(a*f - b*e)^2)/(7*d*f^3)) - (e + f*x)^(9/2)*((6*b^3*e - 6*a*b^2*f)/(9*d*f^3) + (2*b^3*(c*f^4 - d
*e*f^3))/(9*d^2*f^6)) + (e + f*x)^(5/2)*((2*(a*f - b*e)^3)/(5*d*f^3) - (((((6*b^3*e - 6*a*b^2*f)/(d*f^3) + (2*
b^3*(c*f^4 - d*e*f^3))/(d^2*f^6))*(c*f^4 - d*e*f^3))/(d*f^3) + (6*b*(a*f - b*e)^2)/(d*f^3))*(c*f^4 - d*e*f^3))
/(5*d*f^3)) + (2*b^3*(e + f*x)^(11/2))/(11*d*f^3) + (2*atan((d^(1/2)*(e + f*x)^(1/2)*(a*d - b*c)^3*(c*f - d*e)
^(5/2))/(a^3*d^6*e^3 + b^3*c^6*f^3 - a^3*c^3*d^3*f^3 - b^3*c^3*d^3*e^3 - 3*a^2*b*c*d^5*e^3 - 3*a*b^2*c^5*d*f^3
- 3*a^3*c*d^5*e^2*f - 3*b^3*c^5*d*e*f^2 + 3*a*b^2*c^2*d^4*e^3 + 3*a^2*b*c^4*d^2*f^3 + 3*a^3*c^2*d^4*e*f^2 + 3
*b^3*c^4*d^2*e^2*f - 9*a*b^2*c^3*d^3*e^2*f + 9*a*b^2*c^4*d^2*e*f^2 + 9*a^2*b*c^2*d^4*e^2*f - 9*a^2*b*c^3*d^3*e
*f^2))*(a*d - b*c)^3*(c*f - d*e)^(5/2))/d^(13/2) - ((e + f*x)^(3/2)*((2*(a*f - b*e)^3)/(d*f^3) - (((((6*b^3*e
- 6*a*b^2*f)/(d*f^3) + (2*b^3*(c*f^4 - d*e*f^3))/(d^2*f^6))*(c*f^4 - d*e*f^3))/(d*f^3) + (6*b*(a*f - b*e)^2)/(
d*f^3))*(c*f^4 - d*e*f^3))/(d*f^3))*(c*f^4 - d*e*f^3))/(3*d*f^3) + ((e + f*x)^(1/2)*((2*(a*f - b*e)^3)/(d*f^3)
- (((((6*b^3*e - 6*a*b^2*f)/(d*f^3) + (2*b^3*(c*f^4 - d*e*f^3))/(d^2*f^6))*(c*f^4 - d*e*f^3))/(d*f^3) + (6*b*
(a*f - b*e)^2)/(d*f^3))*(c*f^4 - d*e*f^3))/(d*f^3))*(c*f^4 - d*e*f^3)^2)/(d^2*f^6)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3*(f*x+e)**(5/2)/(d*x+c),x)

[Out]

Timed out

________________________________________________________________________________________