3.18.38 \(\int \frac {(a+b x)^3 (A+B x)}{\sqrt {d+e x}} \, dx\) [1738]

Optimal. Leaf size=171 \[ \frac {2 (b d-a e)^3 (B d-A e) \sqrt {d+e x}}{e^5}-\frac {2 (b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^{3/2}}{3 e^5}+\frac {6 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^{5/2}}{5 e^5}-\frac {2 b^2 (4 b B d-A b e-3 a B e) (d+e x)^{7/2}}{7 e^5}+\frac {2 b^3 B (d+e x)^{9/2}}{9 e^5} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {78} \begin {gather*} -\frac {2 b^2 (d+e x)^{7/2} (-3 a B e-A b e+4 b B d)}{7 e^5}+\frac {6 b (d+e x)^{5/2} (b d-a e) (-a B e-A b e+2 b B d)}{5 e^5}-\frac {2 (d+e x)^{3/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{3 e^5}+\frac {2 \sqrt {d+e x} (b d-a e)^3 (B d-A e)}{e^5}+\frac {2 b^3 B (d+e x)^{9/2}}{9 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^3*(A + B*x))/Sqrt[d + e*x],x]

[Out]

(2*(b*d - a*e)^3*(B*d - A*e)*Sqrt[d + e*x])/e^5 - (2*(b*d - a*e)^2*(4*b*B*d - 3*A*b*e - a*B*e)*(d + e*x)^(3/2)
)/(3*e^5) + (6*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(5/2))/(5*e^5) - (2*b^2*(4*b*B*d - A*b*e - 3*
a*B*e)*(d + e*x)^(7/2))/(7*e^5) + (2*b^3*B*(d + e*x)^(9/2))/(9*e^5)

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin {align*} \int \frac {(a+b x)^3 (A+B x)}{\sqrt {d+e x}} \, dx &=\int \left (\frac {(-b d+a e)^3 (-B d+A e)}{e^4 \sqrt {d+e x}}+\frac {(-b d+a e)^2 (-4 b B d+3 A b e+a B e) \sqrt {d+e x}}{e^4}-\frac {3 b (b d-a e) (-2 b B d+A b e+a B e) (d+e x)^{3/2}}{e^4}+\frac {b^2 (-4 b B d+A b e+3 a B e) (d+e x)^{5/2}}{e^4}+\frac {b^3 B (d+e x)^{7/2}}{e^4}\right ) \, dx\\ &=\frac {2 (b d-a e)^3 (B d-A e) \sqrt {d+e x}}{e^5}-\frac {2 (b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^{3/2}}{3 e^5}+\frac {6 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^{5/2}}{5 e^5}-\frac {2 b^2 (4 b B d-A b e-3 a B e) (d+e x)^{7/2}}{7 e^5}+\frac {2 b^3 B (d+e x)^{9/2}}{9 e^5}\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 226, normalized size = 1.32 \begin {gather*} \frac {2 \sqrt {d+e x} \left (105 a^3 e^3 (-2 B d+3 A e+B e x)+63 a^2 b e^2 \left (5 A e (-2 d+e x)+B \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )-9 a b^2 e \left (-7 A e \left (8 d^2-4 d e x+3 e^2 x^2\right )+3 B \left (16 d^3-8 d^2 e x+6 d e^2 x^2-5 e^3 x^3\right )\right )+b^3 \left (9 A e \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+B \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )\right )\right )}{315 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^3*(A + B*x))/Sqrt[d + e*x],x]

[Out]

(2*Sqrt[d + e*x]*(105*a^3*e^3*(-2*B*d + 3*A*e + B*e*x) + 63*a^2*b*e^2*(5*A*e*(-2*d + e*x) + B*(8*d^2 - 4*d*e*x
+ 3*e^2*x^2)) - 9*a*b^2*e*(-7*A*e*(8*d^2 - 4*d*e*x + 3*e^2*x^2) + 3*B*(16*d^3 - 8*d^2*e*x + 6*d*e^2*x^2 - 5*e
^3*x^3)) + b^3*(9*A*e*(-16*d^3 + 8*d^2*e*x - 6*d*e^2*x^2 + 5*e^3*x^3) + B*(128*d^4 - 64*d^3*e*x + 48*d^2*e^2*x
^2 - 40*d*e^3*x^3 + 35*e^4*x^4))))/(315*e^5)

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Maple [A]
time = 0.08, size = 170, normalized size = 0.99

method	result	size
derivativedivides	\(\frac {\frac {2 b^{3} B \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (3 \left (a e -b d \right ) b^{2} B +b^{3} \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (3 \left (a e -b d \right )^{2} b B +3 \left (a e -b d \right ) b^{2} \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (\left (a e -b d \right )^{3} B +3 \left (a e -b d \right )^{2} b \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {3}{2}}}{3}+2 \left (a e -b d \right )^{3} \left (A e -B d \right ) \sqrt {e x +d}}{e^{5}}\)	\(170\)
default	\(\frac {\frac {2 b^{3} B \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (3 \left (a e -b d \right ) b^{2} B +b^{3} \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (3 \left (a e -b d \right )^{2} b B +3 \left (a e -b d \right ) b^{2} \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (\left (a e -b d \right )^{3} B +3 \left (a e -b d \right )^{2} b \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {3}{2}}}{3}+2 \left (a e -b d \right )^{3} \left (A e -B d \right ) \sqrt {e x +d}}{e^{5}}\)	\(170\)
gosper	\(\frac {2 \sqrt {e x +d}\, \left (35 b^{3} B \,x^{4} e^{4}+45 A \,b^{3} e^{4} x^{3}+135 B a \,b^{2} e^{4} x^{3}-40 B \,b^{3} d \,e^{3} x^{3}+189 A a \,b^{2} e^{4} x^{2}-54 A \,b^{3} d \,e^{3} x^{2}+189 B \,a^{2} b \,e^{4} x^{2}-162 B a \,b^{2} d \,e^{3} x^{2}+48 B \,b^{3} d^{2} e^{2} x^{2}+315 A \,a^{2} b \,e^{4} x -252 A a \,b^{2} d \,e^{3} x +72 A \,b^{3} d^{2} e^{2} x +105 B \,a^{3} e^{4} x -252 B \,a^{2} b d \,e^{3} x +216 B a \,b^{2} d^{2} e^{2} x -64 B \,b^{3} d^{3} e x +315 a^{3} A \,e^{4}-630 A \,a^{2} b d \,e^{3}+504 A a \,b^{2} d^{2} e^{2}-144 A \,b^{3} d^{3} e -210 B \,a^{3} d \,e^{3}+504 B \,a^{2} b \,d^{2} e^{2}-432 B a \,b^{2} d^{3} e +128 b^{3} B \,d^{4}\right )}{315 e^{5}}\)	\(301\)
trager	\(\frac {2 \sqrt {e x +d}\, \left (35 b^{3} B \,x^{4} e^{4}+45 A \,b^{3} e^{4} x^{3}+135 B a \,b^{2} e^{4} x^{3}-40 B \,b^{3} d \,e^{3} x^{3}+189 A a \,b^{2} e^{4} x^{2}-54 A \,b^{3} d \,e^{3} x^{2}+189 B \,a^{2} b \,e^{4} x^{2}-162 B a \,b^{2} d \,e^{3} x^{2}+48 B \,b^{3} d^{2} e^{2} x^{2}+315 A \,a^{2} b \,e^{4} x -252 A a \,b^{2} d \,e^{3} x +72 A \,b^{3} d^{2} e^{2} x +105 B \,a^{3} e^{4} x -252 B \,a^{2} b d \,e^{3} x +216 B a \,b^{2} d^{2} e^{2} x -64 B \,b^{3} d^{3} e x +315 a^{3} A \,e^{4}-630 A \,a^{2} b d \,e^{3}+504 A a \,b^{2} d^{2} e^{2}-144 A \,b^{3} d^{3} e -210 B \,a^{3} d \,e^{3}+504 B \,a^{2} b \,d^{2} e^{2}-432 B a \,b^{2} d^{3} e +128 b^{3} B \,d^{4}\right )}{315 e^{5}}\)	\(301\)
risch	\(\frac {2 \sqrt {e x +d}\, \left (35 b^{3} B \,x^{4} e^{4}+45 A \,b^{3} e^{4} x^{3}+135 B a \,b^{2} e^{4} x^{3}-40 B \,b^{3} d \,e^{3} x^{3}+189 A a \,b^{2} e^{4} x^{2}-54 A \,b^{3} d \,e^{3} x^{2}+189 B \,a^{2} b \,e^{4} x^{2}-162 B a \,b^{2} d \,e^{3} x^{2}+48 B \,b^{3} d^{2} e^{2} x^{2}+315 A \,a^{2} b \,e^{4} x -252 A a \,b^{2} d \,e^{3} x +72 A \,b^{3} d^{2} e^{2} x +105 B \,a^{3} e^{4} x -252 B \,a^{2} b d \,e^{3} x +216 B a \,b^{2} d^{2} e^{2} x -64 B \,b^{3} d^{3} e x +315 a^{3} A \,e^{4}-630 A \,a^{2} b d \,e^{3}+504 A a \,b^{2} d^{2} e^{2}-144 A \,b^{3} d^{3} e -210 B \,a^{3} d \,e^{3}+504 B \,a^{2} b \,d^{2} e^{2}-432 B a \,b^{2} d^{3} e +128 b^{3} B \,d^{4}\right )}{315 e^{5}}\)	\(301\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3*(B*x+A)/(e*x+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/e^5*(1/9*b^3*B*(e*x+d)^(9/2)+1/7*(3*(a*e-b*d)*b^2*B+b^3*(A*e-B*d))*(e*x+d)^(7/2)+1/5*(3*(a*e-b*d)^2*b*B+3*(a
*e-b*d)*b^2*(A*e-B*d))*(e*x+d)^(5/2)+1/3*((a*e-b*d)^3*B+3*(a*e-b*d)^2*b*(A*e-B*d))*(e*x+d)^(3/2)+(a*e-b*d)^3*(
A*e-B*d)*(e*x+d)^(1/2))

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Maxima [A]
time = 0.31, size = 279, normalized size = 1.63 \begin {gather*} \frac {2}{315} \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} B b^{3} - 45 \, {\left (4 \, B b^{3} d - 3 \, B a b^{2} e - A b^{3} e\right )} {\left (x e + d\right )}^{\frac {7}{2}} + 189 \, {\left (2 \, B b^{3} d^{2} + B a^{2} b e^{2} + A a b^{2} e^{2} - {\left (3 \, B a b^{2} e + A b^{3} e\right )} d\right )} {\left (x e + d\right )}^{\frac {5}{2}} - 105 \, {\left (4 \, B b^{3} d^{3} - B a^{3} e^{3} - 3 \, A a^{2} b e^{3} - 3 \, {\left (3 \, B a b^{2} e + A b^{3} e\right )} d^{2} + 6 \, {\left (B a^{2} b e^{2} + A a b^{2} e^{2}\right )} d\right )} {\left (x e + d\right )}^{\frac {3}{2}} + 315 \, {\left (B b^{3} d^{4} + A a^{3} e^{4} - {\left (3 \, B a b^{2} e + A b^{3} e\right )} d^{3} + 3 \, {\left (B a^{2} b e^{2} + A a b^{2} e^{2}\right )} d^{2} - {\left (B a^{3} e^{3} + 3 \, A a^{2} b e^{3}\right )} d\right )} \sqrt {x e + d}\right )} e^{\left (-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/315*(35*(x*e + d)^(9/2)*B*b^3 - 45*(4*B*b^3*d - 3*B*a*b^2*e - A*b^3*e)*(x*e + d)^(7/2) + 189*(2*B*b^3*d^2 +
B*a^2*b*e^2 + A*a*b^2*e^2 - (3*B*a*b^2*e + A*b^3*e)*d)*(x*e + d)^(5/2) - 105*(4*B*b^3*d^3 - B*a^3*e^3 - 3*A*a^
2*b*e^3 - 3*(3*B*a*b^2*e + A*b^3*e)*d^2 + 6*(B*a^2*b*e^2 + A*a*b^2*e^2)*d)*(x*e + d)^(3/2) + 315*(B*b^3*d^4 +
A*a^3*e^4 - (3*B*a*b^2*e + A*b^3*e)*d^3 + 3*(B*a^2*b*e^2 + A*a*b^2*e^2)*d^2 - (B*a^3*e^3 + 3*A*a^2*b*e^3)*d)*s
qrt(x*e + d))*e^(-5)

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Fricas [A]
time = 1.07, size = 247, normalized size = 1.44 \begin {gather*} \frac {2}{315} \, {\left (128 \, B b^{3} d^{4} + {\left (35 \, B b^{3} x^{4} + 315 \, A a^{3} + 45 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 189 \, {\left (B a^{2} b + A a b^{2}\right )} x^{2} + 105 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )} e^{4} - 2 \, {\left (20 \, B b^{3} d x^{3} + 27 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d x^{2} + 126 \, {\left (B a^{2} b + A a b^{2}\right )} d x + 105 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d\right )} e^{3} + 24 \, {\left (2 \, B b^{3} d^{2} x^{2} + 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} x + 21 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2}\right )} e^{2} - 16 \, {\left (4 \, B b^{3} d^{3} x + 9 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3}\right )} e\right )} \sqrt {x e + d} e^{\left (-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/315*(128*B*b^3*d^4 + (35*B*b^3*x^4 + 315*A*a^3 + 45*(3*B*a*b^2 + A*b^3)*x^3 + 189*(B*a^2*b + A*a*b^2)*x^2 +
105*(B*a^3 + 3*A*a^2*b)*x)*e^4 - 2*(20*B*b^3*d*x^3 + 27*(3*B*a*b^2 + A*b^3)*d*x^2 + 126*(B*a^2*b + A*a*b^2)*d*
x + 105*(B*a^3 + 3*A*a^2*b)*d)*e^3 + 24*(2*B*b^3*d^2*x^2 + 3*(3*B*a*b^2 + A*b^3)*d^2*x + 21*(B*a^2*b + A*a*b^2
)*d^2)*e^2 - 16*(4*B*b^3*d^3*x + 9*(3*B*a*b^2 + A*b^3)*d^3)*e)*sqrt(x*e + d)*e^(-5)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 916 vs. \(2 (168) = 336\).
time = 50.77, size = 916, normalized size = 5.36 \begin {gather*} \begin {cases} \frac {- \frac {2 A a^{3} d}{\sqrt {d + e x}} - 2 A a^{3} \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right ) - \frac {6 A a^{2} b d \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right )}{e} - \frac {6 A a^{2} b \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{e} - \frac {6 A a b^{2} d \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{e^{2}} - \frac {6 A a b^{2} \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{2}} - \frac {2 A b^{3} d \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{3}} - \frac {2 A b^{3} \left (\frac {d^{4}}{\sqrt {d + e x}} + 4 d^{3} \sqrt {d + e x} - 2 d^{2} \left (d + e x\right )^{\frac {3}{2}} + \frac {4 d \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{3}} - \frac {2 B a^{3} d \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right )}{e} - \frac {2 B a^{3} \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{e} - \frac {6 B a^{2} b d \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{e^{2}} - \frac {6 B a^{2} b \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{2}} - \frac {6 B a b^{2} d \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{3}} - \frac {6 B a b^{2} \left (\frac {d^{4}}{\sqrt {d + e x}} + 4 d^{3} \sqrt {d + e x} - 2 d^{2} \left (d + e x\right )^{\frac {3}{2}} + \frac {4 d \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{3}} - \frac {2 B b^{3} d \left (\frac {d^{4}}{\sqrt {d + e x}} + 4 d^{3} \sqrt {d + e x} - 2 d^{2} \left (d + e x\right )^{\frac {3}{2}} + \frac {4 d \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{4}} - \frac {2 B b^{3} \left (- \frac {d^{5}}{\sqrt {d + e x}} - 5 d^{4} \sqrt {d + e x} + \frac {10 d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} - 2 d^{2} \left (d + e x\right )^{\frac {5}{2}} + \frac {5 d \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{4}}}{e} & \text {for}\: e \neq 0 \\\frac {A a^{3} x + \frac {B b^{3} x^{5}}{5} + \frac {x^{4} \left (A b^{3} + 3 B a b^{2}\right )}{4} + \frac {x^{3} \cdot \left (3 A a b^{2} + 3 B a^{2} b\right )}{3} + \frac {x^{2} \cdot \left (3 A a^{2} b + B a^{3}\right )}{2}}{\sqrt {d}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3*(B*x+A)/(e*x+d)**(1/2),x)

[Out]

Piecewise(((-2*A*a**3*d/sqrt(d + e*x) - 2*A*a**3*(-d/sqrt(d + e*x) - sqrt(d + e*x)) - 6*A*a**2*b*d*(-d/sqrt(d
+ e*x) - sqrt(d + e*x))/e - 6*A*a**2*b*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e - 6*A*a
*b**2*d*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 - 6*A*a*b**2*(-d**3/sqrt(d + e*x) -
3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**2 - 2*A*b**3*d*(-d**3/sqrt(d + e*x) - 3*d*
*2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**3 - 2*A*b**3*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(
d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**3 - 2*B*a**3*d*(-d/sqrt(d
+ e*x) - sqrt(d + e*x))/e - 2*B*a**3*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e - 6*B*a*
*2*b*d*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 - 6*B*a**2*b*(-d**3/sqrt(d + e*x) -
3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**2 - 6*B*a*b**2*d*(-d**3/sqrt(d + e*x) - 3*d
**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**3 - 6*B*a*b**2*(d**4/sqrt(d + e*x) + 4*d**3*sq
rt(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**3 - 2*B*b**3*d*(d**4/s
qrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e
**4 - 2*B*b**3*(-d**5/sqrt(d + e*x) - 5*d**4*sqrt(d + e*x) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d**2*(d + e*x)**(5
/2) + 5*d*(d + e*x)**(7/2)/7 - (d + e*x)**(9/2)/9)/e**4)/e, Ne(e, 0)), ((A*a**3*x + B*b**3*x**5/5 + x**4*(A*b*
*3 + 3*B*a*b**2)/4 + x**3*(3*A*a*b**2 + 3*B*a**2*b)/3 + x**2*(3*A*a**2*b + B*a**3)/2)/sqrt(d), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 345 vs. \(2 (163) = 326\).
time = 0.91, size = 345, normalized size = 2.02 \begin {gather*} \frac {2}{315} \, {\left (105 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} B a^{3} e^{\left (-1\right )} + 315 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} A a^{2} b e^{\left (-1\right )} + 63 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} B a^{2} b e^{\left (-2\right )} + 63 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} A a b^{2} e^{\left (-2\right )} + 27 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} B a b^{2} e^{\left (-3\right )} + 9 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} A b^{3} e^{\left (-3\right )} + {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} B b^{3} e^{\left (-4\right )} + 315 \, \sqrt {x e + d} A a^{3}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/315*(105*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a^3*e^(-1) + 315*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*A*a^
2*b*e^(-1) + 63*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*B*a^2*b*e^(-2) + 63*(3*(x*e
+ d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*a*b^2*e^(-2) + 27*(5*(x*e + d)^(7/2) - 21*(x*e + d
)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*B*a*b^2*e^(-3) + 9*(5*(x*e + d)^(7/2) - 21*(x*e + d
)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*A*b^3*e^(-3) + (35*(x*e + d)^(9/2) - 180*(x*e + d)^
(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*B*b^3*e^(-4) + 315*sqrt(x
*e + d)*A*a^3)*e^(-1)

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Mupad [B]
time = 1.23, size = 154, normalized size = 0.90 \begin {gather*} \frac {{\left (d+e\,x\right )}^{7/2}\,\left (2\,A\,b^3\,e-8\,B\,b^3\,d+6\,B\,a\,b^2\,e\right )}{7\,e^5}+\frac {2\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{3/2}\,\left (3\,A\,b\,e+B\,a\,e-4\,B\,b\,d\right )}{3\,e^5}+\frac {2\,B\,b^3\,{\left (d+e\,x\right )}^{9/2}}{9\,e^5}+\frac {2\,\left (A\,e-B\,d\right )\,{\left (a\,e-b\,d\right )}^3\,\sqrt {d+e\,x}}{e^5}+\frac {6\,b\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{5/2}\,\left (A\,b\,e+B\,a\,e-2\,B\,b\,d\right )}{5\,e^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^3)/(d + e*x)^(1/2),x)

[Out]

((d + e*x)^(7/2)*(2*A*b^3*e - 8*B*b^3*d + 6*B*a*b^2*e))/(7*e^5) + (2*(a*e - b*d)^2*(d + e*x)^(3/2)*(3*A*b*e +
B*a*e - 4*B*b*d))/(3*e^5) + (2*B*b^3*(d + e*x)^(9/2))/(9*e^5) + (2*(A*e - B*d)*(a*e - b*d)^3*(d + e*x)^(1/2))/
e^5 + (6*b*(a*e - b*d)*(d + e*x)^(5/2)*(A*b*e + B*a*e - 2*B*b*d))/(5*e^5)

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