3.18.83 \(\int \frac {(a+b x)^3 (e+f x)^{3/2}}{c+d x} \, dx\) [1783]

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

[Out]

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

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Rubi [A]
time = 0.12, antiderivative size = 244, normalized size of antiderivative = 1.00, 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 = {90, 52, 65, 214} \begin {gather*} \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 (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 \sqrt {e+f x} (b c-a d)^3 (d e-c f)}{d^5}-\frac {2 (e+f x)^{3/2} (b c-a d)^3}{3 d^4}+\frac {2 b^3 (e+f x)^{9/2}}{9 d f^3} \end {gather*}

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 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 90

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 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]

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 ) \text {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*}

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Mathematica [A]
time = 0.45, size = 313, normalized size = 1.28 \begin {gather*} \frac {2 \sqrt {e+f x} \left (105 a^3 d^3 f^3 (4 d e-3 c f+d f x)+63 a^2 b d^2 f^2 \left (15 c^2 f^2+3 d^2 (e+f x)^2-5 c d f (4 e+f x)\right )-9 a b^2 d f \left (105 c^3 f^3+21 c d^2 f (e+f x)^2+3 d^3 (2 e-5 f x) (e+f x)^2-35 c^2 d f^2 (4 e+f x)\right )+b^3 \left (315 c^4 f^4+63 c^2 d^2 f^2 (e+f x)^2+9 c d^3 f (2 e-5 f x) (e+f x)^2-105 c^3 d f^3 (4 e+f x)+d^4 (e+f x)^2 \left (8 e^2-20 e f x+35 f^2 x^2\right )\right )\right )}{315 d^5 f^3}+\frac {2 (-b c+a d)^3 (-d e+c f)^{3/2} \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{d^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[e + f*x]*(105*a^3*d^3*f^3*(4*d*e - 3*c*f + d*f*x) + 63*a^2*b*d^2*f^2*(15*c^2*f^2 + 3*d^2*(e + f*x)^2 -
5*c*d*f*(4*e + f*x)) - 9*a*b^2*d*f*(105*c^3*f^3 + 21*c*d^2*f*(e + f*x)^2 + 3*d^3*(2*e - 5*f*x)*(e + f*x)^2 -
35*c^2*d*f^2*(4*e + f*x)) + b^3*(315*c^4*f^4 + 63*c^2*d^2*f^2*(e + f*x)^2 + 9*c*d^3*f*(2*e - 5*f*x)*(e + f*x)^
2 - 105*c^3*d*f^3*(4*e + f*x) + d^4*(e + f*x)^2*(8*e^2 - 20*e*f*x + 35*f^2*x^2))))/(315*d^5*f^3) + (2*(-(b*c)
+ a*d)^3*(-(d*e) + c*f)^(3/2)*ArcTan[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[-(d*e) + c*f]])/d^(11/2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(625\) vs. \(2(216)=432\).
time = 0.10, size = 626, normalized size = 2.57 Too large to display

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,method=_RETURNVERBOSE)

[Out]

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

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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)^(3/2)/(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-%e*d>0)', see `assume?` fo
r more detai

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 579 vs. \(2 (227) = 454\).
time = 1.05, size = 1173, normalized size = 4.81 \begin {gather*} \left [\frac {315 \, {\left ({\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3}\right )} f^{4} - {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} f^{3} e\right )} \sqrt {-\frac {c f - d e}{d}} \log \left (\frac {d f x - c f - 2 \, \sqrt {f x + e} d \sqrt {-\frac {c f - d e}{d}} + 2 \, d e}{d x + c}\right ) + 2 \, {\left (35 \, b^{3} d^{4} f^{4} x^{4} - 45 \, {\left (b^{3} c d^{3} - 3 \, a b^{2} d^{4}\right )} f^{4} x^{3} + 8 \, b^{3} d^{4} e^{4} + 63 \, {\left (b^{3} c^{2} d^{2} - 3 \, a b^{2} c d^{3} + 3 \, a^{2} b d^{4}\right )} f^{4} x^{2} - 105 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} f^{4} x + 315 \, {\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3}\right )} f^{4} - 2 \, {\left (2 \, b^{3} d^{4} f x - 9 \, {\left (b^{3} c d^{3} - 3 \, a b^{2} d^{4}\right )} f\right )} e^{3} + 3 \, {\left (b^{3} d^{4} f^{2} x^{2} - 3 \, {\left (b^{3} c d^{3} - 3 \, a b^{2} d^{4}\right )} f^{2} x + 21 \, {\left (b^{3} c^{2} d^{2} - 3 \, a b^{2} c d^{3} + 3 \, a^{2} b d^{4}\right )} f^{2}\right )} e^{2} + 2 \, {\left (25 \, b^{3} d^{4} f^{3} x^{3} - 36 \, {\left (b^{3} c d^{3} - 3 \, a b^{2} d^{4}\right )} f^{3} x^{2} + 63 \, {\left (b^{3} c^{2} d^{2} - 3 \, a b^{2} c d^{3} + 3 \, a^{2} b d^{4}\right )} f^{3} x - 210 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} f^{3}\right )} e\right )} \sqrt {f x + e}}{315 \, d^{5} f^{3}}, \frac {2 \, {\left (315 \, {\left ({\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3}\right )} f^{4} - {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} f^{3} e\right )} \sqrt {\frac {c f - d e}{d}} \arctan \left (-\frac {\sqrt {f x + e} d \sqrt {\frac {c f - d e}{d}}}{c f - d e}\right ) + {\left (35 \, b^{3} d^{4} f^{4} x^{4} - 45 \, {\left (b^{3} c d^{3} - 3 \, a b^{2} d^{4}\right )} f^{4} x^{3} + 8 \, b^{3} d^{4} e^{4} + 63 \, {\left (b^{3} c^{2} d^{2} - 3 \, a b^{2} c d^{3} + 3 \, a^{2} b d^{4}\right )} f^{4} x^{2} - 105 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} f^{4} x + 315 \, {\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3}\right )} f^{4} - 2 \, {\left (2 \, b^{3} d^{4} f x - 9 \, {\left (b^{3} c d^{3} - 3 \, a b^{2} d^{4}\right )} f\right )} e^{3} + 3 \, {\left (b^{3} d^{4} f^{2} x^{2} - 3 \, {\left (b^{3} c d^{3} - 3 \, a b^{2} d^{4}\right )} f^{2} x + 21 \, {\left (b^{3} c^{2} d^{2} - 3 \, a b^{2} c d^{3} + 3 \, a^{2} b d^{4}\right )} f^{2}\right )} e^{2} + 2 \, {\left (25 \, b^{3} d^{4} f^{3} x^{3} - 36 \, {\left (b^{3} c d^{3} - 3 \, a b^{2} d^{4}\right )} f^{3} x^{2} + 63 \, {\left (b^{3} c^{2} d^{2} - 3 \, a b^{2} c d^{3} + 3 \, a^{2} b d^{4}\right )} f^{3} x - 210 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} f^{3}\right )} e\right )} \sqrt {f x + e}\right )}}{315 \, d^{5} f^{3}}\right ] \end {gather*}

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^4 - 3*a*b^2*c^3*d + 3*a^2*b*c^2*d^2 - a^3*c*d^3)*f^4 - (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^3*e)*sqrt(-(c*f - d*e)/d)*log((d*f*x - c*f - 2*sqrt(f*x + e)*d*sqrt(-(c*f - d*e)/d) + 2
*d*e)/(d*x + c)) + 2*(35*b^3*d^4*f^4*x^4 - 45*(b^3*c*d^3 - 3*a*b^2*d^4)*f^4*x^3 + 8*b^3*d^4*e^4 + 63*(b^3*c^2*
d^2 - 3*a*b^2*c*d^3 + 3*a^2*b*d^4)*f^4*x^2 - 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
+ 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 - 2*(2*b^3*d^4*f*x - 9*(b^3*c*d^3 - 3*a*b^2
*d^4)*f)*e^3 + 3*(b^3*d^4*f^2*x^2 - 3*(b^3*c*d^3 - 3*a*b^2*d^4)*f^2*x + 21*(b^3*c^2*d^2 - 3*a*b^2*c*d^3 + 3*a^
2*b*d^4)*f^2)*e^2 + 2*(25*b^3*d^4*f^3*x^3 - 36*(b^3*c*d^3 - 3*a*b^2*d^4)*f^3*x^2 + 63*(b^3*c^2*d^2 - 3*a*b^2*c
*d^3 + 3*a^2*b*d^4)*f^3*x - 210*(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^3)*e)*sqrt(f*x + e))
/(d^5*f^3), 2/315*(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 - (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^3*e)*sqrt((c*f - d*e)/d)*arctan(-sqrt(f*x + e)*d*sqrt((c*f - d*e)/d)/(c*f -
d*e)) + (35*b^3*d^4*f^4*x^4 - 45*(b^3*c*d^3 - 3*a*b^2*d^4)*f^4*x^3 + 8*b^3*d^4*e^4 + 63*(b^3*c^2*d^2 - 3*a*b^2
*c*d^3 + 3*a^2*b*d^4)*f^4*x^2 - 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 + 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 - 2*(2*b^3*d^4*f*x - 9*(b^3*c*d^3 - 3*a*b^2*d^4)*f)*e^3
+ 3*(b^3*d^4*f^2*x^2 - 3*(b^3*c*d^3 - 3*a*b^2*d^4)*f^2*x + 21*(b^3*c^2*d^2 - 3*a*b^2*c*d^3 + 3*a^2*b*d^4)*f^2)
*e^2 + 2*(25*b^3*d^4*f^3*x^3 - 36*(b^3*c*d^3 - 3*a*b^2*d^4)*f^3*x^2 + 63*(b^3*c^2*d^2 - 3*a*b^2*c*d^3 + 3*a^2*
b*d^4)*f^3*x - 210*(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^3)*e)*sqrt(f*x + e))/(d^5*f^3)]

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Sympy [A]
time = 63.72, size = 381, normalized size = 1.56 \begin {gather*} \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}} \cdot \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}} \cdot \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}} \cdot \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 683 vs. \(2 (227) = 454\).
time = 0.57, size = 683, normalized size = 2.80 \begin {gather*} -\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 {gather*}

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)

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Mupad [B]
time = 1.28, size = 672, normalized size = 2.75 \begin {gather*} {\left (e+f\,x\right )}^{5/2}\,\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 )}{5\,d\,f^3}+\frac {6\,b\,{\left (a\,f-b\,e\right )}^2}{5\,d\,f^3}\right )-{\left (e+f\,x\right )}^{7/2}\,\left (\frac {6\,b^3\,e-6\,a\,b^2\,f}{7\,d\,f^3}+\frac {2\,b^3\,\left (c\,f^4-d\,e\,f^3\right )}{7\,d^2\,f^6}\right )+{\left (e+f\,x\right )}^{3/2}\,\left (\frac {2\,{\left (a\,f-b\,e\right )}^3}{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 )}{3\,d\,f^3}\right )+\frac {2\,b^3\,{\left (e+f\,x\right )}^{9/2}}{9\,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 )}^{3/2}}{a^3\,c^2\,d^3\,f^2-2\,a^3\,c\,d^4\,e\,f+a^3\,d^5\,e^2-3\,a^2\,b\,c^3\,d^2\,f^2+6\,a^2\,b\,c^2\,d^3\,e\,f-3\,a^2\,b\,c\,d^4\,e^2+3\,a\,b^2\,c^4\,d\,f^2-6\,a\,b^2\,c^3\,d^2\,e\,f+3\,a\,b^2\,c^2\,d^3\,e^2-b^3\,c^5\,f^2+2\,b^3\,c^4\,d\,e\,f-b^3\,c^3\,d^2\,e^2}\right )\,{\left (a\,d-b\,c\right )}^3\,{\left (c\,f-d\,e\right )}^{3/2}}{d^{11/2}}-\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 )}{d\,f^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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