3.18.35 \(\int (a+b x)^3 (A+B x) (d+e x)^{5/2} \, dx\) [1735]

Optimal. Leaf size=173 \[ \frac {2 (b d-a e)^3 (B d-A e) (d+e x)^{7/2}}{7 e^5}-\frac {2 (b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^{9/2}}{9 e^5}+\frac {6 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^{11/2}}{11 e^5}-\frac {2 b^2 (4 b B d-A b e-3 a B e) (d+e x)^{13/2}}{13 e^5}+\frac {2 b^3 B (d+e x)^{15/2}}{15 e^5} \]

[Out]

2/7*(-a*e+b*d)^3*(-A*e+B*d)*(e*x+d)^(7/2)/e^5-2/9*(-a*e+b*d)^2*(-3*A*b*e-B*a*e+4*B*b*d)*(e*x+d)^(9/2)/e^5+6/11
*b*(-a*e+b*d)*(-A*b*e-B*a*e+2*B*b*d)*(e*x+d)^(11/2)/e^5-2/13*b^2*(-A*b*e-3*B*a*e+4*B*b*d)*(e*x+d)^(13/2)/e^5+2
/15*b^3*B*(e*x+d)^(15/2)/e^5

________________________________________________________________________________________

Rubi [A]
time = 0.05, antiderivative size = 173, 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)^{13/2} (-3 a B e-A b e+4 b B d)}{13 e^5}+\frac {6 b (d+e x)^{11/2} (b d-a e) (-a B e-A b e+2 b B d)}{11 e^5}-\frac {2 (d+e x)^{9/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{9 e^5}+\frac {2 (d+e x)^{7/2} (b d-a e)^3 (B d-A e)}{7 e^5}+\frac {2 b^3 B (d+e x)^{15/2}}{15 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)*(d + e*x)^(7/2))/(7*e^5) - (2*(b*d - a*e)^2*(4*b*B*d - 3*A*b*e - a*B*e)*(d + e*x)
^(9/2))/(9*e^5) + (6*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(11/2))/(11*e^5) - (2*b^2*(4*b*B*d - A*
b*e - 3*a*B*e)*(d + e*x)^(13/2))/(13*e^5) + (2*b^3*B*(d + e*x)^(15/2))/(15*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 (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) (d+e x)^{5/2}}{e^4}+\frac {(-b d+a e)^2 (-4 b B d+3 A b e+a B e) (d+e x)^{7/2}}{e^4}-\frac {3 b (b d-a e) (-2 b B d+A b e+a B e) (d+e x)^{9/2}}{e^4}+\frac {b^2 (-4 b B d+A b e+3 a B e) (d+e x)^{11/2}}{e^4}+\frac {b^3 B (d+e x)^{13/2}}{e^4}\right ) \, dx\\ &=\frac {2 (b d-a e)^3 (B d-A e) (d+e x)^{7/2}}{7 e^5}-\frac {2 (b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^{9/2}}{9 e^5}+\frac {6 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^{11/2}}{11 e^5}-\frac {2 b^2 (4 b B d-A b e-3 a B e) (d+e x)^{13/2}}{13 e^5}+\frac {2 b^3 B (d+e x)^{15/2}}{15 e^5}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.19, size = 228, normalized size = 1.32 \begin {gather*} \frac {2 (d+e x)^{7/2} \left (715 a^3 e^3 (-2 B d+9 A e+7 B e x)+195 a^2 b e^2 \left (11 A e (-2 d+7 e x)+B \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )-15 a b^2 e \left (-13 A e \left (8 d^2-28 d e x+63 e^2 x^2\right )+3 B \left (16 d^3-56 d^2 e x+126 d e^2 x^2-231 e^3 x^3\right )\right )+b^3 \left (15 A e \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+B \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )\right )\right )}{45045 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(7/2)*(715*a^3*e^3*(-2*B*d + 9*A*e + 7*B*e*x) + 195*a^2*b*e^2*(11*A*e*(-2*d + 7*e*x) + B*(8*d^2 -
28*d*e*x + 63*e^2*x^2)) - 15*a*b^2*e*(-13*A*e*(8*d^2 - 28*d*e*x + 63*e^2*x^2) + 3*B*(16*d^3 - 56*d^2*e*x + 12
6*d*e^2*x^2 - 231*e^3*x^3)) + b^3*(15*A*e*(-16*d^3 + 56*d^2*e*x - 126*d*e^2*x^2 + 231*e^3*x^3) + B*(128*d^4 -
448*d^3*e*x + 1008*d^2*e^2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4))))/(45045*e^5)

________________________________________________________________________________________

Maple [A]
time = 0.09, size = 171, normalized size = 0.99 Too large to display

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*(1/15*b^3*B*(e*x+d)^(15/2)+1/13*(3*(a*e-b*d)*b^2*B+b^3*(A*e-B*d))*(e*x+d)^(13/2)+1/11*(3*(a*e-b*d)^2*b*B
+3*(a*e-b*d)*b^2*(A*e-B*d))*(e*x+d)^(11/2)+1/9*((a*e-b*d)^3*B+3*(a*e-b*d)^2*b*(A*e-B*d))*(e*x+d)^(9/2)+1/7*(a*
e-b*d)^3*(A*e-B*d)*(e*x+d)^(7/2))

________________________________________________________________________________________

Maxima [A]
time = 0.29, size = 279, normalized size = 1.61 \begin {gather*} \frac {2}{45045} \, {\left (3003 \, {\left (x e + d\right )}^{\frac {15}{2}} B b^{3} - 3465 \, {\left (4 \, B b^{3} d - 3 \, B a b^{2} e - A b^{3} e\right )} {\left (x e + d\right )}^{\frac {13}{2}} + 12285 \, {\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 {11}{2}} - 5005 \, {\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 {9}{2}} + 6435 \, {\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 )} {\left (x e + d\right )}^{\frac {7}{2}}\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)^(5/2),x, algorithm="maxima")

[Out]

2/45045*(3003*(x*e + d)^(15/2)*B*b^3 - 3465*(4*B*b^3*d - 3*B*a*b^2*e - A*b^3*e)*(x*e + d)^(13/2) + 12285*(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)^(11/2) - 5005*(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)^(9/2) + 6435*
(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)*(x*e + d)^(7/2))*e^(-5)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 518 vs. \(2 (163) = 326\).
time = 0.85, size = 518, normalized size = 2.99 \begin {gather*} \frac {2}{45045} \, {\left (128 \, B b^{3} d^{7} + {\left (3003 \, B b^{3} x^{7} + 6435 \, A a^{3} x^{3} + 3465 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{6} + 12285 \, {\left (B a^{2} b + A a b^{2}\right )} x^{5} + 5005 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{4}\right )} e^{7} + {\left (7161 \, B b^{3} d x^{6} + 19305 \, A a^{3} d x^{2} + 8505 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d x^{5} + 31395 \, {\left (B a^{2} b + A a b^{2}\right )} d x^{4} + 13585 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d x^{3}\right )} e^{6} + 3 \, {\left (1491 \, B b^{3} d^{2} x^{5} + 6435 \, A a^{3} d^{2} x + 1855 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} x^{4} + 7345 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} x^{3} + 3575 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} x^{2}\right )} e^{5} + 5 \, {\left (7 \, B b^{3} d^{3} x^{4} + 1287 \, A a^{3} d^{3} + 15 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} x^{3} + 117 \, {\left (B a^{2} b + A a b^{2}\right )} d^{3} x^{2} + 143 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} x\right )} e^{4} - 10 \, {\left (4 \, B b^{3} d^{4} x^{3} + 9 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} x^{2} + 78 \, {\left (B a^{2} b + A a b^{2}\right )} d^{4} x + 143 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{4}\right )} e^{3} + 24 \, {\left (2 \, B b^{3} d^{5} x^{2} + 5 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{5} x + 65 \, {\left (B a^{2} b + A a b^{2}\right )} d^{5}\right )} e^{2} - 16 \, {\left (4 \, B b^{3} d^{6} x + 15 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{6}\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)^(5/2),x, algorithm="fricas")

[Out]

2/45045*(128*B*b^3*d^7 + (3003*B*b^3*x^7 + 6435*A*a^3*x^3 + 3465*(3*B*a*b^2 + A*b^3)*x^6 + 12285*(B*a^2*b + A*
a*b^2)*x^5 + 5005*(B*a^3 + 3*A*a^2*b)*x^4)*e^7 + (7161*B*b^3*d*x^6 + 19305*A*a^3*d*x^2 + 8505*(3*B*a*b^2 + A*b
^3)*d*x^5 + 31395*(B*a^2*b + A*a*b^2)*d*x^4 + 13585*(B*a^3 + 3*A*a^2*b)*d*x^3)*e^6 + 3*(1491*B*b^3*d^2*x^5 + 6
435*A*a^3*d^2*x + 1855*(3*B*a*b^2 + A*b^3)*d^2*x^4 + 7345*(B*a^2*b + A*a*b^2)*d^2*x^3 + 3575*(B*a^3 + 3*A*a^2*
b)*d^2*x^2)*e^5 + 5*(7*B*b^3*d^3*x^4 + 1287*A*a^3*d^3 + 15*(3*B*a*b^2 + A*b^3)*d^3*x^3 + 117*(B*a^2*b + A*a*b^
2)*d^3*x^2 + 143*(B*a^3 + 3*A*a^2*b)*d^3*x)*e^4 - 10*(4*B*b^3*d^4*x^3 + 9*(3*B*a*b^2 + A*b^3)*d^4*x^2 + 78*(B*
a^2*b + A*a*b^2)*d^4*x + 143*(B*a^3 + 3*A*a^2*b)*d^4)*e^3 + 24*(2*B*b^3*d^5*x^2 + 5*(3*B*a*b^2 + A*b^3)*d^5*x
+ 65*(B*a^2*b + A*a*b^2)*d^5)*e^2 - 16*(4*B*b^3*d^6*x + 15*(3*B*a*b^2 + A*b^3)*d^6)*e)*sqrt(x*e + d)*e^(-5)

________________________________________________________________________________________

Sympy [A]
time = 33.67, size = 1564, normalized size = 9.04 \begin {gather*} \text {Too large to display} \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]

A*a**3*d**2*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 4*A*a**3*d*(-d*(d + e*x)**(3/
2)/3 + (d + e*x)**(5/2)/5)/e + 2*A*a**3*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7
)/e + 6*A*a**2*b*d**2*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 12*A*a**2*b*d*(d**2*(d + e*x)**(3/2)
/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 6*A*a**2*b*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*
x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**2 + 6*A*a*b**2*d**2*(d**2*(d + e*x)**(3/2)/3 - 2
*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 12*A*a*b**2*d*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)*
*(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**3 + 6*A*a*b**2*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d
+ e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**3 + 2*A*b**3*
d**2*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**4
+ 4*A*b**3*d*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)
**(9/2)/9 + (d + e*x)**(11/2)/11)/e**4 + 2*A*b**3*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*
(d + e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**4 + 2*B*
a**3*d**2*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 4*B*a**3*d*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e
*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 2*B*a**3*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d
*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**2 + 6*B*a**2*b*d**2*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/
2)/5 + (d + e*x)**(7/2)/7)/e**3 + 12*B*a**2*b*d*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d
+ e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**3 + 6*B*a**2*b*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5
+ 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**3 + 6*B*a*b**2*d**2*(-d**3*(d
+ e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**4 + 12*B*a*b**2*
d*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 +
(d + e*x)**(11/2)/11)/e**4 + 6*B*a*b**2*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)*
*(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**4 + 2*B*b**3*d**2*
(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d
+ e*x)**(11/2)/11)/e**5 + 4*B*b**3*d*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(
7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**5 + 2*B*b**3*(d**6*(
d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20*d**3*(d + e*x)**(9/2)/9 + 15*d
**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**5

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2062 vs. \(2 (163) = 326\).
time = 1.17, size = 2062, normalized size = 11.92 \begin {gather*} \text {Too large to display} \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]

2/45045*(15015*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a^3*d^3*e^(-1) + 45045*((x*e + d)^(3/2) - 3*sqrt(x*e +
d)*d)*A*a^2*b*d^3*e^(-1) + 9009*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*B*a^2*b*d^3*
e^(-2) + 9009*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*a*b^2*d^3*e^(-2) + 3861*(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*d^3*e^(-3) + 12
87*(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*d^3*e^(-3)
+ 143*(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*s
qrt(x*e + d)*d^4)*B*b^3*d^3*e^(-4) + 9009*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*B*
a^3*d^2*e^(-1) + 27027*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*a^2*b*d^2*e^(-1) +
11583*(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^2*b*d^2*e
^(-2) + 11583*(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*a*b
^2*d^2*e^(-2) + 1287*(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*a*b^2*d^2*e^(-3) + 429*(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)*A*b^3*d^2*e^(-3) + 195*(63*(x*e + d)^(11/2
) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 69
3*sqrt(x*e + d)*d^5)*B*b^3*d^2*e^(-4) + 45045*sqrt(x*e + d)*A*a^3*d^3 + 45045*((x*e + d)^(3/2) - 3*sqrt(x*e +
d)*d)*A*a^3*d^2 + 3861*(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^3*d*e^(-1) + 11583*(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*a^2*b*d*e^(-1) + 1287*(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*a^2*b*d*e^(-2) + 1287*(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)*A*a*b^2*d*e^(-2) + 585*(63*(x*e
+ d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2
)*d^4 - 693*sqrt(x*e + d)*d^5)*B*a*b^2*d*e^(-3) + 195*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e
+ d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*A*b^3*d*e^(-3) +
45*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 90
09*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*B*b^3*d*e^(-4) + 9009*(3*(x*e + d)
^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*a^3*d + 143*(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*a^3*e^(-1) + 429*(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)*A
*a^2*b*e^(-1) + 195*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5
/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*B*a^2*b*e^(-2) + 195*(63*(x*e + d)^(11/2) - 385*(x
*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e
+ d)*d^5)*A*a*b^2*e^(-2) + 45*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 85
80*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*B*a*b^2
*e^(-3) + 15*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)
*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*A*b^3*e^(-3) + 7*(429*(x*
e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/2)*d^3 + 32175*(x*e
+ d)^(7/2)*d^4 - 27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*B*b^3*e^(-4)
+ 1287*(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*a^3)*e^(-1
)

________________________________________________________________________________________

Mupad [B]
time = 1.23, size = 154, normalized size = 0.89 \begin {gather*} \frac {{\left (d+e\,x\right )}^{13/2}\,\left (2\,A\,b^3\,e-8\,B\,b^3\,d+6\,B\,a\,b^2\,e\right )}{13\,e^5}+\frac {2\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{9/2}\,\left (3\,A\,b\,e+B\,a\,e-4\,B\,b\,d\right )}{9\,e^5}+\frac {2\,B\,b^3\,{\left (d+e\,x\right )}^{15/2}}{15\,e^5}+\frac {2\,\left (A\,e-B\,d\right )\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{7/2}}{7\,e^5}+\frac {6\,b\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{11/2}\,\left (A\,b\,e+B\,a\,e-2\,B\,b\,d\right )}{11\,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)^(13/2)*(2*A*b^3*e - 8*B*b^3*d + 6*B*a*b^2*e))/(13*e^5) + (2*(a*e - b*d)^2*(d + e*x)^(9/2)*(3*A*b*e
+ B*a*e - 4*B*b*d))/(9*e^5) + (2*B*b^3*(d + e*x)^(15/2))/(15*e^5) + (2*(A*e - B*d)*(a*e - b*d)^3*(d + e*x)^(7/
2))/(7*e^5) + (6*b*(a*e - b*d)*(d + e*x)^(11/2)*(A*b*e + B*a*e - 2*B*b*d))/(11*e^5)

________________________________________________________________________________________