3.1782 \(\int \frac{(a+b x)^3 (e+f x)^{5/2}}{c+d x} \, dx\)

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

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.299293, antiderivative size = 280, normalized size of antiderivative = 1., 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} \[ \frac{2 b (e+f x)^{7/2} \left (3 a^2 d^2 f^2-3 a b d f (c f+d e)+b^2 \left (c^2 f^2+c d e f+d^2 e^2\right )\right )}{7 d^3 f^3}-\frac{2 b^2 (e+f x)^{9/2} (-3 a d f+b c f+2 b d e)}{9 d^2 f^3}-\frac{2 (e+f x)^{5/2} (b c-a d)^3}{5 d^4}-\frac{2 (e+f x)^{3/2} (b c-a d)^3 (d e-c f)}{3 d^5}-\frac{2 \sqrt{e+f x} (b c-a d)^3 (d e-c f)^2}{d^6}+\frac{2 (b c-a d)^3 (d e-c f)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{13/2}}+\frac{2 b^3 (e+f x)^{11/2}}{11 d f^3} \]

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 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)^{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 = 0.733092, size = 253, normalized size = 0.9 \[ \frac{2 b (e+f x)^{7/2} \left (3 a^2 d^2 f^2-3 a b d f (c f+d e)+b^2 \left (c^2 f^2+c d e f+d^2 e^2\right )\right )}{7 d^3 f^3}-\frac{2 b^2 (e+f x)^{9/2} (-3 a d f+b c f+2 b d e)}{9 d^2 f^3}+\frac{2 (e+f x)^{5/2} (a d-b c)^3}{5 d^4}+\frac{2 (a d-b c)^3 (d e-c f) \left (\sqrt{d} \sqrt{e+f x} (-3 c f+4 d e+d f x)-3 (d e-c f)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )\right )}{3 d^{13/2}}+\frac{2 b^3 (e+f x)^{11/2}}{11 d f^3} \]

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))

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Maple [B]  time = 0.016, size = 1437, normalized size = 5.1 \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)^(5/2)/(d*x+c),x)

[Out]

-6/7/f^2/d*(f*x+e)^(7/2)*a*b^2*e-2*f^3/d^3/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)^(1/2))*a^3
*c^3+2*f^3/d^6/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)^(1/2))*b^3*c^6-2*f/d^4*(f*x+e)^(3/2)*a
*b^2*c^3-6/7/f/d^2*(f*x+e)^(7/2)*a*b^2*c+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*f
/d^3*(f*x+e)^(3/2)*a^2*b*c^2-2/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^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/d^2*a^2*b*c*e^2*(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*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+6*f^3/d^4/((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^4-6/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^3+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^3+6*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+12*f/d^3*a^2*b*c^2*e*(f*x+e)^(1/2)-12*f/d^4*a*b^2*c^3*e*(f*x+e)^(1/2)+2/5/d*(f*x+e)^(
5/2)*a^3-6*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*e+6*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*e-6*f^3/d^5/((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^5+2/7/f^2/d^2*(f*x+e)^(7/2)*b^3*c*e-2/d^2*(f*x+e)^(3/2)*a^2*b*c*e-18*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*e+18*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^2+18*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*e-18*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+2/d*a^3*e^2*(f*x+e)^(1/2)+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
^3+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*f^2/d^3*a^3*c^2*(f*x+e)^(1/2)-6/5/d^2*(f*x+e)^(5/
2)*a^2*b*c-2/3/d^4*(f*x+e)^(3/2)*b^3*c^3*e+6/5/d^3*(f*x+e)^(5/2)*a*b^2*c^2-4/9/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+2/7/f/d^3*(f*x+e)^(7/2)*b^3*c^2+2/3/f^2/d*(f*x+e)^(9/2)*a*b^2-2/9/f^2/d^2*(f*x+e)^(
9/2)*b^3*c+6/7/f/d*(f*x+e)^(7/2)*a^2*b-2/3*f/d^2*(f*x+e)^(3/2)*a^3*c-2/d^4*b^3*c^3*e^2*(f*x+e)^(1/2)+2/3*f/d^5
*(f*x+e)^(3/2)*b^3*c^4-2*f^2/d^6*b^3*c^5*(f*x+e)^(1/2)+2/11*b^3*(f*x+e)^(11/2)/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)^(5/2)/(d*x+c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.5961, size = 3573, normalized size = 12.76 \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)^(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)]

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Sympy [B]  time = 149.146, size = 575, normalized size = 2.05 \begin{align*} \frac{2 b^{3} \left (e + f x\right )^{\frac{11}{2}}}{11 d f^{3}} + \frac{\left (e + f x\right )^{\frac{9}{2}} \left (6 a b^{2} d f - 2 b^{3} c f - 4 b^{3} d e\right )}{9 d^{2} f^{3}} + \frac{\left (e + f x\right )^{\frac{7}{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 )}{7 d^{3} f^{3}} + \frac{\left (e + f x\right )^{\frac{5}{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 )}{5 d^{4}} + \frac{\left (e + f x\right )^{\frac{3}{2}} \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 )}{3 d^{5}} + \frac{\sqrt{e + f x} \left (2 a^{3} c^{2} d^{3} f^{2} - 4 a^{3} c d^{4} e f + 2 a^{3} d^{5} e^{2} - 6 a^{2} b c^{3} d^{2} f^{2} + 12 a^{2} b c^{2} d^{3} e f - 6 a^{2} b c d^{4} e^{2} + 6 a b^{2} c^{4} d f^{2} - 12 a b^{2} c^{3} d^{2} e f + 6 a b^{2} c^{2} d^{3} e^{2} - 2 b^{3} c^{5} f^{2} + 4 b^{3} c^{4} d e f - 2 b^{3} c^{3} d^{2} e^{2}\right )}{d^{6}} - \frac{2 \left (a d - b c\right )^{3} \left (c f - d e\right )^{3} \operatorname{atan}{\left (\frac{\sqrt{e + f x}}{\sqrt{\frac{c f - d e}{d}}} \right )}}{d^{7} \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)**(5/2)/(d*x+c),x)

[Out]

2*b**3*(e + f*x)**(11/2)/(11*d*f**3) + (e + f*x)**(9/2)*(6*a*b**2*d*f - 2*b**3*c*f - 4*b**3*d*e)/(9*d**2*f**3)
+ (e + f*x)**(7/2)*(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)/(7*d**3*f**3) + (e + f*x)**(5/2)*(2*a**3*d**3 - 6*a**2*b*c*d**2 + 6*a*b**2*c**2*d -
2*b**3*c**3)/(5*d**4) + (e + f*x)**(3/2)*(-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)/(3*d**5) + sqrt(e + f*x)
*(2*a**3*c**2*d**3*f**2 - 4*a**3*c*d**4*e*f + 2*a**3*d**5*e**2 - 6*a**2*b*c**3*d**2*f**2 + 12*a**2*b*c**2*d**3
*e*f - 6*a**2*b*c*d**4*e**2 + 6*a*b**2*c**4*d*f**2 - 12*a*b**2*c**3*d**2*e*f + 6*a*b**2*c**2*d**3*e**2 - 2*b**
3*c**5*f**2 + 4*b**3*c**4*d*e*f - 2*b**3*c**3*d**2*e**2)/d**6 - 2*(a*d - b*c)**3*(c*f - d*e)**3*atan(sqrt(e +
f*x)/sqrt((c*f - d*e)/d))/(d**7*sqrt((c*f - d*e)/d))

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Giac [B]  time = 1.86432, size = 1388, normalized size = 4.96 \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)^(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)