3.18.40 \(\int \frac {(a+b x)^3 (A+B x)}{(d+e x)^{5/2}} \, dx\) [1740]

Optimal. Leaf size=169 \[ -\frac {2 (b d-a e)^3 (B d-A e)}{3 e^5 (d+e x)^{3/2}}+\frac {2 (b d-a e)^2 (4 b B d-3 A b e-a B e)}{e^5 \sqrt {d+e x}}+\frac {6 b (b d-a e) (2 b B d-A b e-a B e) \sqrt {d+e x}}{e^5}-\frac {2 b^2 (4 b B d-A b e-3 a B e) (d+e x)^{3/2}}{3 e^5}+\frac {2 b^3 B (d+e x)^{5/2}}{5 e^5} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 169, 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)^{3/2} (-3 a B e-A b e+4 b B d)}{3 e^5}+\frac {6 b \sqrt {d+e x} (b d-a e) (-a B e-A b e+2 b B d)}{e^5}+\frac {2 (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{e^5 \sqrt {d+e x}}-\frac {2 (b d-a e)^3 (B d-A e)}{3 e^5 (d+e x)^{3/2}}+\frac {2 b^3 B (d+e x)^{5/2}}{5 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]
time = 0.20, size = 223, normalized size = 1.32 \begin {gather*} \frac {2 \left (-5 a^3 e^3 (2 B d+A e+3 B e x)+15 a^2 b e^2 \left (-A e (2 d+3 e x)+B \left (8 d^2+12 d e x+3 e^2 x^2\right )\right )+15 a b^2 e \left (A e \left (8 d^2+12 d e x+3 e^2 x^2\right )+B \left (-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3\right )\right )+b^3 \left (5 A e \left (-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3\right )+B \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )\right )\right )}{15 e^5 (d+e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-5*a^3*e^3*(2*B*d + A*e + 3*B*e*x) + 15*a^2*b*e^2*(-(A*e*(2*d + 3*e*x)) + B*(8*d^2 + 12*d*e*x + 3*e^2*x^2)
) + 15*a*b^2*e*(A*e*(8*d^2 + 12*d*e*x + 3*e^2*x^2) + B*(-16*d^3 - 24*d^2*e*x - 6*d*e^2*x^2 + e^3*x^3)) + b^3*(
5*A*e*(-16*d^3 - 24*d^2*e*x - 6*d*e^2*x^2 + e^3*x^3) + B*(128*d^4 + 192*d^3*e*x + 48*d^2*e^2*x^2 - 8*d*e^3*x^3
+ 3*e^4*x^4))))/(15*e^5*(d + e*x)^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(316\) vs. \(2(153)=306\).
time = 0.09, size = 317, normalized size = 1.88

method	result	size
risch	\(\frac {2 b \left (3 b^{2} B \,x^{2} e^{2}+5 A \,b^{2} e^{2} x +15 B a b \,e^{2} x -14 B \,b^{2} d e x +45 A a b \,e^{2}-40 A \,b^{2} d e +45 B \,a^{2} e^{2}-120 B a b d e +73 b^{2} B \,d^{2}\right ) \sqrt {e x +d}}{15 e^{5}}-\frac {2 \left (9 A b \,e^{2} x +3 B a \,e^{2} x -12 B b d e x +A a \,e^{2}+8 A b d e +2 B a d e -11 B b \,d^{2}\right ) \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{3 e^{5} \left (e x +d \right )^{\frac {3}{2}}}\)	\(179\)
gosper	\(-\frac {2 \left (-3 b^{3} B \,x^{4} e^{4}-5 A \,b^{3} e^{4} x^{3}-15 B a \,b^{2} e^{4} x^{3}+8 B \,b^{3} d \,e^{3} x^{3}-45 A a \,b^{2} e^{4} x^{2}+30 A \,b^{3} d \,e^{3} x^{2}-45 B \,a^{2} b \,e^{4} x^{2}+90 B a \,b^{2} d \,e^{3} x^{2}-48 B \,b^{3} d^{2} e^{2} x^{2}+45 A \,a^{2} b \,e^{4} x -180 A a \,b^{2} d \,e^{3} x +120 A \,b^{3} d^{2} e^{2} x +15 B \,a^{3} e^{4} x -180 B \,a^{2} b d \,e^{3} x +360 B a \,b^{2} d^{2} e^{2} x -192 B \,b^{3} d^{3} e x +5 a^{3} A \,e^{4}+30 A \,a^{2} b d \,e^{3}-120 A a \,b^{2} d^{2} e^{2}+80 A \,b^{3} d^{3} e +10 B \,a^{3} d \,e^{3}-120 B \,a^{2} b \,d^{2} e^{2}+240 B a \,b^{2} d^{3} e -128 b^{3} B \,d^{4}\right )}{15 \left (e x +d \right )^{\frac {3}{2}} e^{5}}\)	\(301\)
trager	\(-\frac {2 \left (-3 b^{3} B \,x^{4} e^{4}-5 A \,b^{3} e^{4} x^{3}-15 B a \,b^{2} e^{4} x^{3}+8 B \,b^{3} d \,e^{3} x^{3}-45 A a \,b^{2} e^{4} x^{2}+30 A \,b^{3} d \,e^{3} x^{2}-45 B \,a^{2} b \,e^{4} x^{2}+90 B a \,b^{2} d \,e^{3} x^{2}-48 B \,b^{3} d^{2} e^{2} x^{2}+45 A \,a^{2} b \,e^{4} x -180 A a \,b^{2} d \,e^{3} x +120 A \,b^{3} d^{2} e^{2} x +15 B \,a^{3} e^{4} x -180 B \,a^{2} b d \,e^{3} x +360 B a \,b^{2} d^{2} e^{2} x -192 B \,b^{3} d^{3} e x +5 a^{3} A \,e^{4}+30 A \,a^{2} b d \,e^{3}-120 A a \,b^{2} d^{2} e^{2}+80 A \,b^{3} d^{3} e +10 B \,a^{3} d \,e^{3}-120 B \,a^{2} b \,d^{2} e^{2}+240 B a \,b^{2} d^{3} e -128 b^{3} B \,d^{4}\right )}{15 \left (e x +d \right )^{\frac {3}{2}} e^{5}}\)	\(301\)
derivativedivides	\(\frac {\frac {2 b^{3} B \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 A \,b^{3} e \left (e x +d \right )^{\frac {3}{2}}}{3}+2 B a \,b^{2} e \left (e x +d \right )^{\frac {3}{2}}-\frac {8 B \,b^{3} d \left (e x +d \right )^{\frac {3}{2}}}{3}+6 A a \,b^{2} e^{2} \sqrt {e x +d}-6 A \,b^{3} d e \sqrt {e x +d}+6 B \,a^{2} b \,e^{2} \sqrt {e x +d}-18 B a \,b^{2} d e \sqrt {e x +d}+12 B \,b^{3} d^{2} \sqrt {e x +d}-\frac {2 \left (a^{3} A \,e^{4}-3 A \,a^{2} b d \,e^{3}+3 A a \,b^{2} d^{2} e^{2}-A \,b^{3} d^{3} e -B \,a^{3} d \,e^{3}+3 B \,a^{2} b \,d^{2} e^{2}-3 B a \,b^{2} d^{3} e +b^{3} B \,d^{4}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +B \,a^{3} e^{3}-6 B \,a^{2} b d \,e^{2}+9 B a \,b^{2} d^{2} e -4 b^{3} B \,d^{3}\right )}{\sqrt {e x +d}}}{e^{5}}\)	\(317\)
default	\(\frac {\frac {2 b^{3} B \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 A \,b^{3} e \left (e x +d \right )^{\frac {3}{2}}}{3}+2 B a \,b^{2} e \left (e x +d \right )^{\frac {3}{2}}-\frac {8 B \,b^{3} d \left (e x +d \right )^{\frac {3}{2}}}{3}+6 A a \,b^{2} e^{2} \sqrt {e x +d}-6 A \,b^{3} d e \sqrt {e x +d}+6 B \,a^{2} b \,e^{2} \sqrt {e x +d}-18 B a \,b^{2} d e \sqrt {e x +d}+12 B \,b^{3} d^{2} \sqrt {e x +d}-\frac {2 \left (a^{3} A \,e^{4}-3 A \,a^{2} b d \,e^{3}+3 A a \,b^{2} d^{2} e^{2}-A \,b^{3} d^{3} e -B \,a^{3} d \,e^{3}+3 B \,a^{2} b \,d^{2} e^{2}-3 B a \,b^{2} d^{3} e +b^{3} B \,d^{4}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +B \,a^{3} e^{3}-6 B \,a^{2} b d \,e^{2}+9 B a \,b^{2} d^{2} e -4 b^{3} B \,d^{3}\right )}{\sqrt {e x +d}}}{e^{5}}\)	\(317\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3*(B*x+A)/(e*x+d)^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.32, size = 283, normalized size = 1.67 \begin {gather*} \frac {2}{15} \, {\left ({\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{3} - 5 \, {\left (4 \, B b^{3} d - 3 \, B a b^{2} e - A b^{3} e\right )} {\left (x e + d\right )}^{\frac {3}{2}} + 45 \, {\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 )} \sqrt {x e + d}\right )} e^{\left (-4\right )} - \frac {5 \, {\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} - 3 \, {\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 )} - {\left (B a^{3} e^{3} + 3 \, A a^{2} b e^{3}\right )} d\right )} e^{\left (-4\right )}}{{\left (x e + d\right )}^{\frac {3}{2}}}\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)^(5/2),x, algorithm="maxima")

[Out]

2/15*((3*(x*e + d)^(5/2)*B*b^3 - 5*(4*B*b^3*d - 3*B*a*b^2*e - A*b^3*e)*(x*e + d)^(3/2) + 45*(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)*sqrt(x*e + d))*e^(-4) - 5*(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 - 3*(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) - (B*a^3*e^3 + 3*A*a^2*b*e^3)*d)*e^(-4)/(
x*e + d)^(3/2))*e^(-1)

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Fricas [A]
time = 1.38, size = 266, normalized size = 1.57 \begin {gather*} \frac {2 \, {\left (128 \, B b^{3} d^{4} + {\left (3 \, B b^{3} x^{4} - 5 \, A a^{3} + 5 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 45 \, {\left (B a^{2} b + A a b^{2}\right )} x^{2} - 15 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )} e^{4} - 2 \, {\left (4 \, B b^{3} d x^{3} + 15 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d x^{2} - 90 \, {\left (B a^{2} b + A a b^{2}\right )} d x + 5 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d\right )} e^{3} + 24 \, {\left (2 \, B b^{3} d^{2} x^{2} - 5 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} x + 5 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2}\right )} e^{2} + 16 \, {\left (12 \, B b^{3} d^{3} x - 5 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3}\right )} e\right )} \sqrt {x e + d}}{15 \, {\left (x^{2} e^{7} + 2 \, d x e^{6} + d^{2} e^{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)^(5/2),x, algorithm="fricas")

[Out]

2/15*(128*B*b^3*d^4 + (3*B*b^3*x^4 - 5*A*a^3 + 5*(3*B*a*b^2 + A*b^3)*x^3 + 45*(B*a^2*b + A*a*b^2)*x^2 - 15*(B*
a^3 + 3*A*a^2*b)*x)*e^4 - 2*(4*B*b^3*d*x^3 + 15*(3*B*a*b^2 + A*b^3)*d*x^2 - 90*(B*a^2*b + A*a*b^2)*d*x + 5*(B*
a^3 + 3*A*a^2*b)*d)*e^3 + 24*(2*B*b^3*d^2*x^2 - 5*(3*B*a*b^2 + A*b^3)*d^2*x + 5*(B*a^2*b + A*a*b^2)*d^2)*e^2 +
16*(12*B*b^3*d^3*x - 5*(3*B*a*b^2 + A*b^3)*d^3)*e)*sqrt(x*e + d)/(x^2*e^7 + 2*d*x*e^6 + d^2*e^5)

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Sympy [A]
time = 35.01, size = 199, normalized size = 1.18 \begin {gather*} \frac {2 B b^{3} \left (d + e x\right )^{\frac {5}{2}}}{5 e^{5}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (2 A b^{3} e + 6 B a b^{2} e - 8 B b^{3} d\right )}{3 e^{5}} + \frac {\sqrt {d + e x} \left (6 A a b^{2} e^{2} - 6 A b^{3} d e + 6 B a^{2} b e^{2} - 18 B a b^{2} d e + 12 B b^{3} d^{2}\right )}{e^{5}} - \frac {2 \left (a e - b d\right )^{2} \cdot \left (3 A b e + B a e - 4 B b d\right )}{e^{5} \sqrt {d + e x}} + \frac {2 \left (- A e + B d\right ) \left (a e - b d\right )^{3}}{3 e^{5} \left (d + e x\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3*(B*x+A)/(e*x+d)**(5/2),x)

[Out]

2*B*b**3*(d + e*x)**(5/2)/(5*e**5) + (d + e*x)**(3/2)*(2*A*b**3*e + 6*B*a*b**2*e - 8*B*b**3*d)/(3*e**5) + sqrt
(d + e*x)*(6*A*a*b**2*e**2 - 6*A*b**3*d*e + 6*B*a**2*b*e**2 - 18*B*a*b**2*d*e + 12*B*b**3*d**2)/e**5 - 2*(a*e
- b*d)**2*(3*A*b*e + B*a*e - 4*B*b*d)/(e**5*sqrt(d + e*x)) + 2*(-A*e + B*d)*(a*e - b*d)**3/(3*e**5*(d + e*x)**
(3/2))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 365 vs. \(2 (163) = 326\).
time = 0.81, size = 365, normalized size = 2.16 \begin {gather*} \frac {2}{15} \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{3} e^{20} - 20 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{3} d e^{20} + 90 \, \sqrt {x e + d} B b^{3} d^{2} e^{20} + 15 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{2} e^{21} + 5 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{3} e^{21} - 135 \, \sqrt {x e + d} B a b^{2} d e^{21} - 45 \, \sqrt {x e + d} A b^{3} d e^{21} + 45 \, \sqrt {x e + d} B a^{2} b e^{22} + 45 \, \sqrt {x e + d} A a b^{2} e^{22}\right )} e^{\left (-25\right )} + \frac {2 \, {\left (12 \, {\left (x e + d\right )} B b^{3} d^{3} - B b^{3} d^{4} - 27 \, {\left (x e + d\right )} B a b^{2} d^{2} e - 9 \, {\left (x e + d\right )} A b^{3} d^{2} e + 3 \, B a b^{2} d^{3} e + A b^{3} d^{3} e + 18 \, {\left (x e + d\right )} B a^{2} b d e^{2} + 18 \, {\left (x e + d\right )} A a b^{2} d e^{2} - 3 \, B a^{2} b d^{2} e^{2} - 3 \, A a b^{2} d^{2} e^{2} - 3 \, {\left (x e + d\right )} B a^{3} e^{3} - 9 \, {\left (x e + d\right )} A a^{2} b e^{3} + B a^{3} d e^{3} + 3 \, A a^{2} b d e^{3} - A a^{3} e^{4}\right )} e^{\left (-5\right )}}{3 \, {\left (x e + d\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(3*(x*e + d)^(5/2)*B*b^3*e^20 - 20*(x*e + d)^(3/2)*B*b^3*d*e^20 + 90*sqrt(x*e + d)*B*b^3*d^2*e^20 + 15*(x
*e + d)^(3/2)*B*a*b^2*e^21 + 5*(x*e + d)^(3/2)*A*b^3*e^21 - 135*sqrt(x*e + d)*B*a*b^2*d*e^21 - 45*sqrt(x*e + d
)*A*b^3*d*e^21 + 45*sqrt(x*e + d)*B*a^2*b*e^22 + 45*sqrt(x*e + d)*A*a*b^2*e^22)*e^(-25) + 2/3*(12*(x*e + d)*B*
b^3*d^3 - B*b^3*d^4 - 27*(x*e + d)*B*a*b^2*d^2*e - 9*(x*e + d)*A*b^3*d^2*e + 3*B*a*b^2*d^3*e + A*b^3*d^3*e + 1
8*(x*e + d)*B*a^2*b*d*e^2 + 18*(x*e + d)*A*a*b^2*d*e^2 - 3*B*a^2*b*d^2*e^2 - 3*A*a*b^2*d^2*e^2 - 3*(x*e + d)*B
*a^3*e^3 - 9*(x*e + d)*A*a^2*b*e^3 + B*a^3*d*e^3 + 3*A*a^2*b*d*e^3 - A*a^3*e^4)*e^(-5)/(x*e + d)^(3/2)

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Mupad [B]
time = 1.27, size = 264, normalized size = 1.56 \begin {gather*} \frac {{\left (d+e\,x\right )}^{3/2}\,\left (2\,A\,b^3\,e-8\,B\,b^3\,d+6\,B\,a\,b^2\,e\right )}{3\,e^5}-\frac {\left (d+e\,x\right )\,\left (2\,B\,a^3\,e^3-12\,B\,a^2\,b\,d\,e^2+6\,A\,a^2\,b\,e^3+18\,B\,a\,b^2\,d^2\,e-12\,A\,a\,b^2\,d\,e^2-8\,B\,b^3\,d^3+6\,A\,b^3\,d^2\,e\right )+\frac {2\,A\,a^3\,e^4}{3}+\frac {2\,B\,b^3\,d^4}{3}-\frac {2\,A\,b^3\,d^3\,e}{3}-\frac {2\,B\,a^3\,d\,e^3}{3}+2\,A\,a\,b^2\,d^2\,e^2+2\,B\,a^2\,b\,d^2\,e^2-2\,A\,a^2\,b\,d\,e^3-2\,B\,a\,b^2\,d^3\,e}{e^5\,{\left (d+e\,x\right )}^{3/2}}+\frac {2\,B\,b^3\,{\left (d+e\,x\right )}^{5/2}}{5\,e^5}+\frac {6\,b\,\left (a\,e-b\,d\right )\,\sqrt {d+e\,x}\,\left (A\,b\,e+B\,a\,e-2\,B\,b\,d\right )}{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)^(5/2),x)

[Out]

((d + e*x)^(3/2)*(2*A*b^3*e - 8*B*b^3*d + 6*B*a*b^2*e))/(3*e^5) - ((d + e*x)*(2*B*a^3*e^3 - 8*B*b^3*d^3 + 6*A*
a^2*b*e^3 + 6*A*b^3*d^2*e - 12*A*a*b^2*d*e^2 + 18*B*a*b^2*d^2*e - 12*B*a^2*b*d*e^2) + (2*A*a^3*e^4)/3 + (2*B*b
^3*d^4)/3 - (2*A*b^3*d^3*e)/3 - (2*B*a^3*d*e^3)/3 + 2*A*a*b^2*d^2*e^2 + 2*B*a^2*b*d^2*e^2 - 2*A*a^2*b*d*e^3 -
2*B*a*b^2*d^3*e)/(e^5*(d + e*x)^(3/2)) + (2*B*b^3*(d + e*x)^(5/2))/(5*e^5) + (6*b*(a*e - b*d)*(d + e*x)^(1/2)*
(A*b*e + B*a*e - 2*B*b*d))/e^5

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