3.1783 \(\int \frac{(a+b x)^3 (e+f x)^{3/2}}{c+d x} \, dx\)

Optimal. Leaf size=244 \[ \frac{2 b (e+f x)^{5/2} \left (3 a^2 d^2 f^2-3 a b d f (c f+d e)+b^2 \left (c^2 f^2+c d e f+d^2 e^2\right )\right )}{5 d^3 f^3}-\frac{2 b^2 (e+f x)^{7/2} (-3 a d f+b c f+2 b d e)}{7 d^2 f^3}-\frac{2 (e+f x)^{3/2} (b c-a d)^3}{3 d^4}-\frac{2 \sqrt{e+f x} (b c-a d)^3 (d e-c f)}{d^5}+\frac{2 (b c-a d)^3 (d e-c f)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{11/2}}+\frac{2 b^3 (e+f x)^{9/2}}{9 d f^3} \]

[Out]

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

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Rubi [A]  time = 0.218008, antiderivative size = 244, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {88, 50, 63, 208} \[ \frac{2 b (e+f x)^{5/2} \left (3 a^2 d^2 f^2-3 a b d f (c f+d e)+b^2 \left (c^2 f^2+c d e f+d^2 e^2\right )\right )}{5 d^3 f^3}-\frac{2 b^2 (e+f x)^{7/2} (-3 a d f+b c f+2 b d e)}{7 d^2 f^3}-\frac{2 (e+f x)^{3/2} (b c-a d)^3}{3 d^4}-\frac{2 \sqrt{e+f x} (b c-a d)^3 (d e-c f)}{d^5}+\frac{2 (b c-a d)^3 (d e-c f)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{11/2}}+\frac{2 b^3 (e+f x)^{9/2}}{9 d f^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.366962, size = 232, normalized size = 0.95 \[ \frac{2 \left (\frac{63 b d (e+f x)^{5/2} \left (3 a^2 d^2 f^2-3 a b d f (c f+d e)+b^2 \left (c^2 f^2+c d e f+d^2 e^2\right )\right )}{f^3}-\frac{45 b^2 d^2 (e+f x)^{7/2} (-3 a d f+b c f+2 b d e)}{f^3}+\frac{315 (a d-b c)^3 (d e-c f) \left (\sqrt{d} \sqrt{e+f x}-\sqrt{d e-c f} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )\right )}{d^{3/2}}-105 (e+f x)^{3/2} (b c-a d)^3+\frac{35 b^3 d^3 (e+f x)^{9/2}}{f^3}\right )}{315 d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-105*(b*c - a*d)^3*(e + f*x)^(3/2) + (63*b*d*(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)^(5/2))/f^3 - (45*b^2*d^2*(2*b*d*e + b*c*f - 3*a*d*f)*(e + f*x)^(7/2))/f^3 + (35*b^3*d
^3*(e + f*x)^(9/2))/f^3 + (315*(-(b*c) + a*d)^3*(d*e - c*f)*(Sqrt[d]*Sqrt[e + f*x] - Sqrt[d*e - c*f]*ArcTanh[(
Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f]]))/d^(3/2)))/(315*d^4)

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Maple [B]  time = 0.013, size = 984, normalized size = 4. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.47808, size = 2376, normalized size = 9.74 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/315*(315*((b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*e*f^3 - (b^3*c^4 - 3*a*b^2*c^3*d + 3*a^2*
b*c^2*d^2 - a^3*c*d^3)*f^4)*sqrt((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*(35*b^3*d^4*f^4*x^4 + 8*b^3*d^4*e^4 + 18*(b^3*c*d^3 - 3*a*b^2*d^4)*e^3*f + 63*(b^3*c^2*d^2
- 3*a*b^2*c*d^3 + 3*a^2*b*d^4)*e^2*f^2 - 420*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*e*f^3 + 3
15*(b^3*c^4 - 3*a*b^2*c^3*d + 3*a^2*b*c^2*d^2 - a^3*c*d^3)*f^4 + 5*(10*b^3*d^4*e*f^3 - 9*(b^3*c*d^3 - 3*a*b^2*
d^4)*f^4)*x^3 + 3*(b^3*d^4*e^2*f^2 - 24*(b^3*c*d^3 - 3*a*b^2*d^4)*e*f^3 + 21*(b^3*c^2*d^2 - 3*a*b^2*c*d^3 + 3*
a^2*b*d^4)*f^4)*x^2 - (4*b^3*d^4*e^3*f + 9*(b^3*c*d^3 - 3*a*b^2*d^4)*e^2*f^2 - 126*(b^3*c^2*d^2 - 3*a*b^2*c*d^
3 + 3*a^2*b*d^4)*e*f^3 + 105*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*f^4)*x)*sqrt(f*x + e))/(d
^5*f^3), 2/315*(315*((b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*e*f^3 - (b^3*c^4 - 3*a*b^2*c^3*d
+ 3*a^2*b*c^2*d^2 - a^3*c*d^3)*f^4)*sqrt(-(d*e - c*f)/d)*arctan(-sqrt(f*x + e)*d*sqrt(-(d*e - c*f)/d)/(d*e - c
*f)) + (35*b^3*d^4*f^4*x^4 + 8*b^3*d^4*e^4 + 18*(b^3*c*d^3 - 3*a*b^2*d^4)*e^3*f + 63*(b^3*c^2*d^2 - 3*a*b^2*c*
d^3 + 3*a^2*b*d^4)*e^2*f^2 - 420*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*e*f^3 + 315*(b^3*c^4
- 3*a*b^2*c^3*d + 3*a^2*b*c^2*d^2 - a^3*c*d^3)*f^4 + 5*(10*b^3*d^4*e*f^3 - 9*(b^3*c*d^3 - 3*a*b^2*d^4)*f^4)*x^
3 + 3*(b^3*d^4*e^2*f^2 - 24*(b^3*c*d^3 - 3*a*b^2*d^4)*e*f^3 + 21*(b^3*c^2*d^2 - 3*a*b^2*c*d^3 + 3*a^2*b*d^4)*f
^4)*x^2 - (4*b^3*d^4*e^3*f + 9*(b^3*c*d^3 - 3*a*b^2*d^4)*e^2*f^2 - 126*(b^3*c^2*d^2 - 3*a*b^2*c*d^3 + 3*a^2*b*
d^4)*e*f^3 + 105*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*f^4)*x)*sqrt(f*x + e))/(d^5*f^3)]

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Sympy [A]  time = 86.4346, size = 381, normalized size = 1.56 \begin{align*} \frac{2 b^{3} \left (e + f x\right )^{\frac{9}{2}}}{9 d f^{3}} + \frac{\left (e + f x\right )^{\frac{7}{2}} \left (6 a b^{2} d f - 2 b^{3} c f - 4 b^{3} d e\right )}{7 d^{2} f^{3}} + \frac{\left (e + f x\right )^{\frac{5}{2}} \left (6 a^{2} b d^{2} f^{2} - 6 a b^{2} c d f^{2} - 6 a b^{2} d^{2} e f + 2 b^{3} c^{2} f^{2} + 2 b^{3} c d e f + 2 b^{3} d^{2} e^{2}\right )}{5 d^{3} f^{3}} + \frac{\left (e + f x\right )^{\frac{3}{2}} \left (2 a^{3} d^{3} - 6 a^{2} b c d^{2} + 6 a b^{2} c^{2} d - 2 b^{3} c^{3}\right )}{3 d^{4}} + \frac{\sqrt{e + f x} \left (- 2 a^{3} c d^{3} f + 2 a^{3} d^{4} e + 6 a^{2} b c^{2} d^{2} f - 6 a^{2} b c d^{3} e - 6 a b^{2} c^{3} d f + 6 a b^{2} c^{2} d^{2} e + 2 b^{3} c^{4} f - 2 b^{3} c^{3} d e\right )}{d^{5}} + \frac{2 \left (a d - b c\right )^{3} \left (c f - d e\right )^{2} \operatorname{atan}{\left (\frac{\sqrt{e + f x}}{\sqrt{\frac{c f - d e}{d}}} \right )}}{d^{6} \sqrt{\frac{c f - d e}{d}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 2.13617, size = 922, normalized size = 3.78 \begin{align*} -\frac{2 \,{\left (b^{3} c^{5} f^{2} - 3 \, a b^{2} c^{4} d f^{2} + 3 \, a^{2} b c^{3} d^{2} f^{2} - a^{3} c^{2} d^{3} f^{2} - 2 \, b^{3} c^{4} d f e + 6 \, a b^{2} c^{3} d^{2} f e - 6 \, a^{2} b c^{2} d^{3} f e + 2 \, a^{3} c d^{4} f e + b^{3} c^{3} d^{2} e^{2} - 3 \, a b^{2} c^{2} d^{3} e^{2} + 3 \, a^{2} b c d^{4} e^{2} - a^{3} d^{5} e^{2}\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^{5}} + \frac{2 \,{\left (35 \,{\left (f x + e\right )}^{\frac{9}{2}} b^{3} d^{8} f^{24} - 45 \,{\left (f x + e\right )}^{\frac{7}{2}} b^{3} c d^{7} f^{25} + 135 \,{\left (f x + e\right )}^{\frac{7}{2}} a b^{2} d^{8} f^{25} + 63 \,{\left (f x + e\right )}^{\frac{5}{2}} b^{3} c^{2} d^{6} f^{26} - 189 \,{\left (f x + e\right )}^{\frac{5}{2}} a b^{2} c d^{7} f^{26} + 189 \,{\left (f x + e\right )}^{\frac{5}{2}} a^{2} b d^{8} f^{26} - 105 \,{\left (f x + e\right )}^{\frac{3}{2}} b^{3} c^{3} d^{5} f^{27} + 315 \,{\left (f x + e\right )}^{\frac{3}{2}} a b^{2} c^{2} d^{6} f^{27} - 315 \,{\left (f x + e\right )}^{\frac{3}{2}} a^{2} b c d^{7} f^{27} + 105 \,{\left (f x + e\right )}^{\frac{3}{2}} a^{3} d^{8} f^{27} + 315 \, \sqrt{f x + e} b^{3} c^{4} d^{4} f^{28} - 945 \, \sqrt{f x + e} a b^{2} c^{3} d^{5} f^{28} + 945 \, \sqrt{f x + e} a^{2} b c^{2} d^{6} f^{28} - 315 \, \sqrt{f x + e} a^{3} c d^{7} f^{28} - 90 \,{\left (f x + e\right )}^{\frac{7}{2}} b^{3} d^{8} f^{24} e + 63 \,{\left (f x + e\right )}^{\frac{5}{2}} b^{3} c d^{7} f^{25} e - 189 \,{\left (f x + e\right )}^{\frac{5}{2}} a b^{2} d^{8} f^{25} e - 315 \, \sqrt{f x + e} b^{3} c^{3} d^{5} f^{27} e + 945 \, \sqrt{f x + e} a b^{2} c^{2} d^{6} f^{27} e - 945 \, \sqrt{f x + e} a^{2} b c d^{7} f^{27} e + 315 \, \sqrt{f x + e} a^{3} d^{8} f^{27} e + 63 \,{\left (f x + e\right )}^{\frac{5}{2}} b^{3} d^{8} f^{24} e^{2}\right )}}{315 \, d^{9} f^{27}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(b^3*c^5*f^2 - 3*a*b^2*c^4*d*f^2 + 3*a^2*b*c^3*d^2*f^2 - a^3*c^2*d^3*f^2 - 2*b^3*c^4*d*f*e + 6*a*b^2*c^3*d^
2*f*e - 6*a^2*b*c^2*d^3*f*e + 2*a^3*c*d^4*f*e + b^3*c^3*d^2*e^2 - 3*a*b^2*c^2*d^3*e^2 + 3*a^2*b*c*d^4*e^2 - a^
3*d^5*e^2)*arctan(sqrt(f*x + e)*d/sqrt(c*d*f - d^2*e))/(sqrt(c*d*f - d^2*e)*d^5) + 2/315*(35*(f*x + e)^(9/2)*b
^3*d^8*f^24 - 45*(f*x + e)^(7/2)*b^3*c*d^7*f^25 + 135*(f*x + e)^(7/2)*a*b^2*d^8*f^25 + 63*(f*x + e)^(5/2)*b^3*
c^2*d^6*f^26 - 189*(f*x + e)^(5/2)*a*b^2*c*d^7*f^26 + 189*(f*x + e)^(5/2)*a^2*b*d^8*f^26 - 105*(f*x + e)^(3/2)
*b^3*c^3*d^5*f^27 + 315*(f*x + e)^(3/2)*a*b^2*c^2*d^6*f^27 - 315*(f*x + e)^(3/2)*a^2*b*c*d^7*f^27 + 105*(f*x +
e)^(3/2)*a^3*d^8*f^27 + 315*sqrt(f*x + e)*b^3*c^4*d^4*f^28 - 945*sqrt(f*x + e)*a*b^2*c^3*d^5*f^28 + 945*sqrt(
f*x + e)*a^2*b*c^2*d^6*f^28 - 315*sqrt(f*x + e)*a^3*c*d^7*f^28 - 90*(f*x + e)^(7/2)*b^3*d^8*f^24*e + 63*(f*x +
e)^(5/2)*b^3*c*d^7*f^25*e - 189*(f*x + e)^(5/2)*a*b^2*d^8*f^25*e - 315*sqrt(f*x + e)*b^3*c^3*d^5*f^27*e + 945
*sqrt(f*x + e)*a*b^2*c^2*d^6*f^27*e - 945*sqrt(f*x + e)*a^2*b*c*d^7*f^27*e + 315*sqrt(f*x + e)*a^3*d^8*f^27*e
+ 63*(f*x + e)^(5/2)*b^3*d^8*f^24*e^2)/(d^9*f^27)