3.23.20 \(\int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{13/2}} \, dx\) [2220]

Optimal. Leaf size=201 \[ -\frac {2 (B d-A e) (a+b x)^{5/2}}{11 e (b d-a e) (d+e x)^{11/2}}+\frac {2 (5 b B d+6 A b e-11 a B e) (a+b x)^{5/2}}{99 e (b d-a e)^2 (d+e x)^{9/2}}+\frac {8 b (5 b B d+6 A b e-11 a B e) (a+b x)^{5/2}}{693 e (b d-a e)^3 (d+e x)^{7/2}}+\frac {16 b^2 (5 b B d+6 A b e-11 a B e) (a+b x)^{5/2}}{3465 e (b d-a e)^4 (d+e x)^{5/2}} \]

[Out]

-2/11*(-A*e+B*d)*(b*x+a)^(5/2)/e/(-a*e+b*d)/(e*x+d)^(11/2)+2/99*(6*A*b*e-11*B*a*e+5*B*b*d)*(b*x+a)^(5/2)/e/(-a
*e+b*d)^2/(e*x+d)^(9/2)+8/693*b*(6*A*b*e-11*B*a*e+5*B*b*d)*(b*x+a)^(5/2)/e/(-a*e+b*d)^3/(e*x+d)^(7/2)+16/3465*
b^2*(6*A*b*e-11*B*a*e+5*B*b*d)*(b*x+a)^(5/2)/e/(-a*e+b*d)^4/(e*x+d)^(5/2)

________________________________________________________________________________________

Rubi [A]
time = 0.08, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {79, 47, 37} \begin {gather*} \frac {16 b^2 (a+b x)^{5/2} (-11 a B e+6 A b e+5 b B d)}{3465 e (d+e x)^{5/2} (b d-a e)^4}+\frac {8 b (a+b x)^{5/2} (-11 a B e+6 A b e+5 b B d)}{693 e (d+e x)^{7/2} (b d-a e)^3}+\frac {2 (a+b x)^{5/2} (-11 a B e+6 A b e+5 b B d)}{99 e (d+e x)^{9/2} (b d-a e)^2}-\frac {2 (a+b x)^{5/2} (B d-A e)}{11 e (d+e x)^{11/2} (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(3/2)*(A + B*x))/(d + e*x)^(13/2),x]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(5/2))/(11*e*(b*d - a*e)*(d + e*x)^(11/2)) + (2*(5*b*B*d + 6*A*b*e - 11*a*B*e)*(a +
b*x)^(5/2))/(99*e*(b*d - a*e)^2*(d + e*x)^(9/2)) + (8*b*(5*b*B*d + 6*A*b*e - 11*a*B*e)*(a + b*x)^(5/2))/(693*e
*(b*d - a*e)^3*(d + e*x)^(7/2)) + (16*b^2*(5*b*B*d + 6*A*b*e - 11*a*B*e)*(a + b*x)^(5/2))/(3465*e*(b*d - a*e)^
4*(d + e*x)^(5/2))

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rubi steps

\begin {align*} \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{13/2}} \, dx &=-\frac {2 (B d-A e) (a+b x)^{5/2}}{11 e (b d-a e) (d+e x)^{11/2}}+\frac {(5 b B d+6 A b e-11 a B e) \int \frac {(a+b x)^{3/2}}{(d+e x)^{11/2}} \, dx}{11 e (b d-a e)}\\ &=-\frac {2 (B d-A e) (a+b x)^{5/2}}{11 e (b d-a e) (d+e x)^{11/2}}+\frac {2 (5 b B d+6 A b e-11 a B e) (a+b x)^{5/2}}{99 e (b d-a e)^2 (d+e x)^{9/2}}+\frac {(4 b (5 b B d+6 A b e-11 a B e)) \int \frac {(a+b x)^{3/2}}{(d+e x)^{9/2}} \, dx}{99 e (b d-a e)^2}\\ &=-\frac {2 (B d-A e) (a+b x)^{5/2}}{11 e (b d-a e) (d+e x)^{11/2}}+\frac {2 (5 b B d+6 A b e-11 a B e) (a+b x)^{5/2}}{99 e (b d-a e)^2 (d+e x)^{9/2}}+\frac {8 b (5 b B d+6 A b e-11 a B e) (a+b x)^{5/2}}{693 e (b d-a e)^3 (d+e x)^{7/2}}+\frac {\left (8 b^2 (5 b B d+6 A b e-11 a B e)\right ) \int \frac {(a+b x)^{3/2}}{(d+e x)^{7/2}} \, dx}{693 e (b d-a e)^3}\\ &=-\frac {2 (B d-A e) (a+b x)^{5/2}}{11 e (b d-a e) (d+e x)^{11/2}}+\frac {2 (5 b B d+6 A b e-11 a B e) (a+b x)^{5/2}}{99 e (b d-a e)^2 (d+e x)^{9/2}}+\frac {8 b (5 b B d+6 A b e-11 a B e) (a+b x)^{5/2}}{693 e (b d-a e)^3 (d+e x)^{7/2}}+\frac {16 b^2 (5 b B d+6 A b e-11 a B e) (a+b x)^{5/2}}{3465 e (b d-a e)^4 (d+e x)^{5/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.34, size = 199, normalized size = 0.99 \begin {gather*} \frac {2 (a+b x)^{5/2} \left (315 B d e^2 (a+b x)^3-315 A e^3 (a+b x)^3-770 b B d e (a+b x)^2 (d+e x)+1155 A b e^2 (a+b x)^2 (d+e x)-385 a B e^2 (a+b x)^2 (d+e x)+495 b^2 B d (a+b x) (d+e x)^2-1485 A b^2 e (a+b x) (d+e x)^2+990 a b B e (a+b x) (d+e x)^2+693 A b^3 (d+e x)^3-693 a b^2 B (d+e x)^3\right )}{3465 (b d-a e)^4 (d+e x)^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(3/2)*(A + B*x))/(d + e*x)^(13/2),x]

[Out]

(2*(a + b*x)^(5/2)*(315*B*d*e^2*(a + b*x)^3 - 315*A*e^3*(a + b*x)^3 - 770*b*B*d*e*(a + b*x)^2*(d + e*x) + 1155
*A*b*e^2*(a + b*x)^2*(d + e*x) - 385*a*B*e^2*(a + b*x)^2*(d + e*x) + 495*b^2*B*d*(a + b*x)*(d + e*x)^2 - 1485*
A*b^2*e*(a + b*x)*(d + e*x)^2 + 990*a*b*B*e*(a + b*x)*(d + e*x)^2 + 693*A*b^3*(d + e*x)^3 - 693*a*b^2*B*(d + e
*x)^3))/(3465*(b*d - a*e)^4*(d + e*x)^(11/2))

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(400\) vs. \(2(177)=354\).
time = 0.09, size = 401, normalized size = 2.00

method	result	size
gosper	\(-\frac {2 \left (b x +a \right )^{\frac {5}{2}} \left (-48 A \,b^{3} e^{3} x^{3}+88 B a \,b^{2} e^{3} x^{3}-40 B \,b^{3} d \,e^{2} x^{3}+120 A a \,b^{2} e^{3} x^{2}-264 A \,b^{3} d \,e^{2} x^{2}-220 B \,a^{2} b \,e^{3} x^{2}+584 B a \,b^{2} d \,e^{2} x^{2}-220 B \,b^{3} d^{2} e \,x^{2}-210 A \,a^{2} b \,e^{3} x +660 A a \,b^{2} d \,e^{2} x -594 A \,b^{3} d^{2} e x +385 B \,a^{3} e^{3} x -1385 B \,a^{2} b d \,e^{2} x +1639 B a \,b^{2} d^{2} e x -495 B \,b^{3} d^{3} x +315 a^{3} A \,e^{3}-1155 A \,a^{2} b d \,e^{2}+1485 A a \,b^{2} d^{2} e -693 A \,b^{3} d^{3}+70 B \,a^{3} d \,e^{2}-220 B \,a^{2} b \,d^{2} e +198 B a \,b^{2} d^{3}\right )}{3465 \left (e x +d \right )^{\frac {11}{2}} \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}\)	\(322\)
default	\(-\frac {2 \left (-48 A \,b^{4} e^{3} x^{4}+88 B a \,b^{3} e^{3} x^{4}-40 B \,b^{4} d \,e^{2} x^{4}+72 A a \,b^{3} e^{3} x^{3}-264 A \,b^{4} d \,e^{2} x^{3}-132 B \,a^{2} b^{2} e^{3} x^{3}+544 B a \,b^{3} d \,e^{2} x^{3}-220 B \,b^{4} d^{2} e \,x^{3}-90 A \,a^{2} b^{2} e^{3} x^{2}+396 A a \,b^{3} d \,e^{2} x^{2}-594 A \,b^{4} d^{2} e \,x^{2}+165 B \,a^{3} b \,e^{3} x^{2}-801 B \,a^{2} b^{2} d \,e^{2} x^{2}+1419 B a \,b^{3} d^{2} e \,x^{2}-495 B \,b^{4} d^{3} x^{2}+105 A \,a^{3} b \,e^{3} x -495 A \,a^{2} b^{2} d \,e^{2} x +891 A a \,b^{3} d^{2} e x -693 A \,b^{4} d^{3} x +385 B \,a^{4} e^{3} x -1315 B \,a^{3} b d \,e^{2} x +1419 B \,a^{2} b^{2} d^{2} e x -297 B a \,b^{3} d^{3} x +315 A \,a^{4} e^{3}-1155 A \,a^{3} b d \,e^{2}+1485 A \,a^{2} b^{2} d^{2} e -693 A a \,b^{3} d^{3}+70 B \,a^{4} d \,e^{2}-220 B \,a^{3} b \,d^{2} e +198 B \,a^{2} b^{2} d^{3}\right ) \left (b x +a \right )^{\frac {3}{2}}}{3465 \left (e x +d \right )^{\frac {11}{2}} \left (a e -b d \right )^{4}}\)	\(401\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(3/2)*(B*x+A)/(e*x+d)^(13/2),x,method=_RETURNVERBOSE)

[Out]

-2/3465*(-48*A*b^4*e^3*x^4+88*B*a*b^3*e^3*x^4-40*B*b^4*d*e^2*x^4+72*A*a*b^3*e^3*x^3-264*A*b^4*d*e^2*x^3-132*B*
a^2*b^2*e^3*x^3+544*B*a*b^3*d*e^2*x^3-220*B*b^4*d^2*e*x^3-90*A*a^2*b^2*e^3*x^2+396*A*a*b^3*d*e^2*x^2-594*A*b^4
*d^2*e*x^2+165*B*a^3*b*e^3*x^2-801*B*a^2*b^2*d*e^2*x^2+1419*B*a*b^3*d^2*e*x^2-495*B*b^4*d^3*x^2+105*A*a^3*b*e^
3*x-495*A*a^2*b^2*d*e^2*x+891*A*a*b^3*d^2*e*x-693*A*b^4*d^3*x+385*B*a^4*e^3*x-1315*B*a^3*b*d*e^2*x+1419*B*a^2*
b^2*d^2*e*x-297*B*a*b^3*d^3*x+315*A*a^4*e^3-1155*A*a^3*b*d*e^2+1485*A*a^2*b^2*d^2*e-693*A*a*b^3*d^3+70*B*a^4*d
*e^2-220*B*a^3*b*d^2*e+198*B*a^2*b^2*d^3)*(b*x+a)^(3/2)/(e*x+d)^(11/2)/(a*e-b*d)^4

________________________________________________________________________________________

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)^(3/2)*(B*x+A)/(e*x+d)^(13/2),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(b*d-%e*a>0)', see `assume?` fo
r more detai

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 871 vs. \(2 (188) = 376\).
time = 99.57, size = 871, normalized size = 4.33 \begin {gather*} \frac {2 \, {\left (495 \, B b^{5} d^{3} x^{3} + 99 \, {\left (8 \, B a b^{4} + 7 \, A b^{5}\right )} d^{3} x^{2} + 99 \, {\left (B a^{2} b^{3} + 14 \, A a b^{4}\right )} d^{3} x - 99 \, {\left (2 \, B a^{3} b^{2} - 7 \, A a^{2} b^{3}\right )} d^{3} - {\left (315 \, A a^{5} + 8 \, {\left (11 \, B a b^{4} - 6 \, A b^{5}\right )} x^{5} - 4 \, {\left (11 \, B a^{2} b^{3} - 6 \, A a b^{4}\right )} x^{4} + 3 \, {\left (11 \, B a^{3} b^{2} - 6 \, A a^{2} b^{3}\right )} x^{3} + 5 \, {\left (110 \, B a^{4} b + 3 \, A a^{3} b^{2}\right )} x^{2} + 35 \, {\left (11 \, B a^{5} + 12 \, A a^{4} b\right )} x\right )} e^{3} + {\left (40 \, B b^{5} d x^{5} - 24 \, {\left (21 \, B a b^{4} - 11 \, A b^{5}\right )} d x^{4} + {\left (257 \, B a^{2} b^{3} - 132 \, A a b^{4}\right )} d x^{3} + {\left (2116 \, B a^{3} b^{2} + 99 \, A a^{2} b^{3}\right )} d x^{2} + 15 \, {\left (83 \, B a^{4} b + 110 \, A a^{3} b^{2}\right )} d x - 35 \, {\left (2 \, B a^{5} - 33 \, A a^{4} b\right )} d\right )} e^{2} + 11 \, {\left (20 \, B b^{5} d^{2} x^{4} - {\left (109 \, B a b^{4} - 54 \, A b^{5}\right )} d^{2} x^{3} - 3 \, {\left (86 \, B a^{2} b^{3} + 9 \, A a b^{4}\right )} d^{2} x^{2} - {\left (109 \, B a^{3} b^{2} + 216 \, A a^{2} b^{3}\right )} d^{2} x + 5 \, {\left (4 \, B a^{4} b - 27 \, A a^{3} b^{2}\right )} d^{2}\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d}}{3465 \, {\left (b^{4} d^{10} + a^{4} x^{6} e^{10} - 2 \, {\left (2 \, a^{3} b d x^{6} - 3 \, a^{4} d x^{5}\right )} e^{9} + 3 \, {\left (2 \, a^{2} b^{2} d^{2} x^{6} - 8 \, a^{3} b d^{2} x^{5} + 5 \, a^{4} d^{2} x^{4}\right )} e^{8} - 4 \, {\left (a b^{3} d^{3} x^{6} - 9 \, a^{2} b^{2} d^{3} x^{5} + 15 \, a^{3} b d^{3} x^{4} - 5 \, a^{4} d^{3} x^{3}\right )} e^{7} + {\left (b^{4} d^{4} x^{6} - 24 \, a b^{3} d^{4} x^{5} + 90 \, a^{2} b^{2} d^{4} x^{4} - 80 \, a^{3} b d^{4} x^{3} + 15 \, a^{4} d^{4} x^{2}\right )} e^{6} + 6 \, {\left (b^{4} d^{5} x^{5} - 10 \, a b^{3} d^{5} x^{4} + 20 \, a^{2} b^{2} d^{5} x^{3} - 10 \, a^{3} b d^{5} x^{2} + a^{4} d^{5} x\right )} e^{5} + {\left (15 \, b^{4} d^{6} x^{4} - 80 \, a b^{3} d^{6} x^{3} + 90 \, a^{2} b^{2} d^{6} x^{2} - 24 \, a^{3} b d^{6} x + a^{4} d^{6}\right )} e^{4} + 4 \, {\left (5 \, b^{4} d^{7} x^{3} - 15 \, a b^{3} d^{7} x^{2} + 9 \, a^{2} b^{2} d^{7} x - a^{3} b d^{7}\right )} e^{3} + 3 \, {\left (5 \, b^{4} d^{8} x^{2} - 8 \, a b^{3} d^{8} x + 2 \, a^{2} b^{2} d^{8}\right )} e^{2} + 2 \, {\left (3 \, b^{4} d^{9} x - 2 \, a b^{3} d^{9}\right )} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2)*(B*x+A)/(e*x+d)^(13/2),x, algorithm="fricas")

[Out]

2/3465*(495*B*b^5*d^3*x^3 + 99*(8*B*a*b^4 + 7*A*b^5)*d^3*x^2 + 99*(B*a^2*b^3 + 14*A*a*b^4)*d^3*x - 99*(2*B*a^3
*b^2 - 7*A*a^2*b^3)*d^3 - (315*A*a^5 + 8*(11*B*a*b^4 - 6*A*b^5)*x^5 - 4*(11*B*a^2*b^3 - 6*A*a*b^4)*x^4 + 3*(11
*B*a^3*b^2 - 6*A*a^2*b^3)*x^3 + 5*(110*B*a^4*b + 3*A*a^3*b^2)*x^2 + 35*(11*B*a^5 + 12*A*a^4*b)*x)*e^3 + (40*B*
b^5*d*x^5 - 24*(21*B*a*b^4 - 11*A*b^5)*d*x^4 + (257*B*a^2*b^3 - 132*A*a*b^4)*d*x^3 + (2116*B*a^3*b^2 + 99*A*a^
2*b^3)*d*x^2 + 15*(83*B*a^4*b + 110*A*a^3*b^2)*d*x - 35*(2*B*a^5 - 33*A*a^4*b)*d)*e^2 + 11*(20*B*b^5*d^2*x^4 -
(109*B*a*b^4 - 54*A*b^5)*d^2*x^3 - 3*(86*B*a^2*b^3 + 9*A*a*b^4)*d^2*x^2 - (109*B*a^3*b^2 + 216*A*a^2*b^3)*d^2
*x + 5*(4*B*a^4*b - 27*A*a^3*b^2)*d^2)*e)*sqrt(b*x + a)*sqrt(x*e + d)/(b^4*d^10 + a^4*x^6*e^10 - 2*(2*a^3*b*d*
x^6 - 3*a^4*d*x^5)*e^9 + 3*(2*a^2*b^2*d^2*x^6 - 8*a^3*b*d^2*x^5 + 5*a^4*d^2*x^4)*e^8 - 4*(a*b^3*d^3*x^6 - 9*a^
2*b^2*d^3*x^5 + 15*a^3*b*d^3*x^4 - 5*a^4*d^3*x^3)*e^7 + (b^4*d^4*x^6 - 24*a*b^3*d^4*x^5 + 90*a^2*b^2*d^4*x^4 -
80*a^3*b*d^4*x^3 + 15*a^4*d^4*x^2)*e^6 + 6*(b^4*d^5*x^5 - 10*a*b^3*d^5*x^4 + 20*a^2*b^2*d^5*x^3 - 10*a^3*b*d^
5*x^2 + a^4*d^5*x)*e^5 + (15*b^4*d^6*x^4 - 80*a*b^3*d^6*x^3 + 90*a^2*b^2*d^6*x^2 - 24*a^3*b*d^6*x + a^4*d^6)*e
^4 + 4*(5*b^4*d^7*x^3 - 15*a*b^3*d^7*x^2 + 9*a^2*b^2*d^7*x - a^3*b*d^7)*e^3 + 3*(5*b^4*d^8*x^2 - 8*a*b^3*d^8*x
+ 2*a^2*b^2*d^8)*e^2 + 2*(3*b^4*d^9*x - 2*a*b^3*d^9)*e)

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(3/2)*(B*x+A)/(e*x+d)**(13/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 762 vs. \(2 (188) = 376\).
time = 2.25, size = 762, normalized size = 3.79 \begin {gather*} \frac {2 \, {\left ({\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (5 \, B b^{13} d^{2} {\left | b \right |} e^{7} - 16 \, B a b^{12} d {\left | b \right |} e^{8} + 6 \, A b^{13} d {\left | b \right |} e^{8} + 11 \, B a^{2} b^{11} {\left | b \right |} e^{9} - 6 \, A a b^{12} {\left | b \right |} e^{9}\right )} {\left (b x + a\right )}}{b^{7} d^{5} e^{5} - 5 \, a b^{6} d^{4} e^{6} + 10 \, a^{2} b^{5} d^{3} e^{7} - 10 \, a^{3} b^{4} d^{2} e^{8} + 5 \, a^{4} b^{3} d e^{9} - a^{5} b^{2} e^{10}} + \frac {11 \, {\left (5 \, B b^{14} d^{3} {\left | b \right |} e^{6} - 21 \, B a b^{13} d^{2} {\left | b \right |} e^{7} + 6 \, A b^{14} d^{2} {\left | b \right |} e^{7} + 27 \, B a^{2} b^{12} d {\left | b \right |} e^{8} - 12 \, A a b^{13} d {\left | b \right |} e^{8} - 11 \, B a^{3} b^{11} {\left | b \right |} e^{9} + 6 \, A a^{2} b^{12} {\left | b \right |} e^{9}\right )}}{b^{7} d^{5} e^{5} - 5 \, a b^{6} d^{4} e^{6} + 10 \, a^{2} b^{5} d^{3} e^{7} - 10 \, a^{3} b^{4} d^{2} e^{8} + 5 \, a^{4} b^{3} d e^{9} - a^{5} b^{2} e^{10}}\right )} + \frac {99 \, {\left (5 \, B b^{15} d^{4} {\left | b \right |} e^{5} - 26 \, B a b^{14} d^{3} {\left | b \right |} e^{6} + 6 \, A b^{15} d^{3} {\left | b \right |} e^{6} + 48 \, B a^{2} b^{13} d^{2} {\left | b \right |} e^{7} - 18 \, A a b^{14} d^{2} {\left | b \right |} e^{7} - 38 \, B a^{3} b^{12} d {\left | b \right |} e^{8} + 18 \, A a^{2} b^{13} d {\left | b \right |} e^{8} + 11 \, B a^{4} b^{11} {\left | b \right |} e^{9} - 6 \, A a^{3} b^{12} {\left | b \right |} e^{9}\right )}}{b^{7} d^{5} e^{5} - 5 \, a b^{6} d^{4} e^{6} + 10 \, a^{2} b^{5} d^{3} e^{7} - 10 \, a^{3} b^{4} d^{2} e^{8} + 5 \, a^{4} b^{3} d e^{9} - a^{5} b^{2} e^{10}}\right )} {\left (b x + a\right )} - \frac {693 \, {\left (B a b^{15} d^{4} {\left | b \right |} e^{5} - A b^{16} d^{4} {\left | b \right |} e^{5} - 4 \, B a^{2} b^{14} d^{3} {\left | b \right |} e^{6} + 4 \, A a b^{15} d^{3} {\left | b \right |} e^{6} + 6 \, B a^{3} b^{13} d^{2} {\left | b \right |} e^{7} - 6 \, A a^{2} b^{14} d^{2} {\left | b \right |} e^{7} - 4 \, B a^{4} b^{12} d {\left | b \right |} e^{8} + 4 \, A a^{3} b^{13} d {\left | b \right |} e^{8} + B a^{5} b^{11} {\left | b \right |} e^{9} - A a^{4} b^{12} {\left | b \right |} e^{9}\right )}}{b^{7} d^{5} e^{5} - 5 \, a b^{6} d^{4} e^{6} + 10 \, a^{2} b^{5} d^{3} e^{7} - 10 \, a^{3} b^{4} d^{2} e^{8} + 5 \, a^{4} b^{3} d e^{9} - a^{5} b^{2} e^{10}}\right )} {\left (b x + a\right )}^{\frac {5}{2}}}{3465 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {11}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2)*(B*x+A)/(e*x+d)^(13/2),x, algorithm="giac")

[Out]

2/3465*((4*(b*x + a)*(2*(5*B*b^13*d^2*abs(b)*e^7 - 16*B*a*b^12*d*abs(b)*e^8 + 6*A*b^13*d*abs(b)*e^8 + 11*B*a^2
*b^11*abs(b)*e^9 - 6*A*a*b^12*abs(b)*e^9)*(b*x + a)/(b^7*d^5*e^5 - 5*a*b^6*d^4*e^6 + 10*a^2*b^5*d^3*e^7 - 10*a
^3*b^4*d^2*e^8 + 5*a^4*b^3*d*e^9 - a^5*b^2*e^10) + 11*(5*B*b^14*d^3*abs(b)*e^6 - 21*B*a*b^13*d^2*abs(b)*e^7 +
6*A*b^14*d^2*abs(b)*e^7 + 27*B*a^2*b^12*d*abs(b)*e^8 - 12*A*a*b^13*d*abs(b)*e^8 - 11*B*a^3*b^11*abs(b)*e^9 + 6
*A*a^2*b^12*abs(b)*e^9)/(b^7*d^5*e^5 - 5*a*b^6*d^4*e^6 + 10*a^2*b^5*d^3*e^7 - 10*a^3*b^4*d^2*e^8 + 5*a^4*b^3*d
*e^9 - a^5*b^2*e^10)) + 99*(5*B*b^15*d^4*abs(b)*e^5 - 26*B*a*b^14*d^3*abs(b)*e^6 + 6*A*b^15*d^3*abs(b)*e^6 + 4
8*B*a^2*b^13*d^2*abs(b)*e^7 - 18*A*a*b^14*d^2*abs(b)*e^7 - 38*B*a^3*b^12*d*abs(b)*e^8 + 18*A*a^2*b^13*d*abs(b)
*e^8 + 11*B*a^4*b^11*abs(b)*e^9 - 6*A*a^3*b^12*abs(b)*e^9)/(b^7*d^5*e^5 - 5*a*b^6*d^4*e^6 + 10*a^2*b^5*d^3*e^7
- 10*a^3*b^4*d^2*e^8 + 5*a^4*b^3*d*e^9 - a^5*b^2*e^10))*(b*x + a) - 693*(B*a*b^15*d^4*abs(b)*e^5 - A*b^16*d^4
*abs(b)*e^5 - 4*B*a^2*b^14*d^3*abs(b)*e^6 + 4*A*a*b^15*d^3*abs(b)*e^6 + 6*B*a^3*b^13*d^2*abs(b)*e^7 - 6*A*a^2*
b^14*d^2*abs(b)*e^7 - 4*B*a^4*b^12*d*abs(b)*e^8 + 4*A*a^3*b^13*d*abs(b)*e^8 + B*a^5*b^11*abs(b)*e^9 - A*a^4*b^
12*abs(b)*e^9)/(b^7*d^5*e^5 - 5*a*b^6*d^4*e^6 + 10*a^2*b^5*d^3*e^7 - 10*a^3*b^4*d^2*e^8 + 5*a^4*b^3*d*e^9 - a^
5*b^2*e^10))*(b*x + a)^(5/2)/(b^2*d + (b*x + a)*b*e - a*b*e)^(11/2)

________________________________________________________________________________________

Mupad [B]
time = 2.68, size = 570, normalized size = 2.84 \begin {gather*} -\frac {\sqrt {d+e\,x}\,\left (\frac {\sqrt {a+b\,x}\,\left (140\,B\,a^5\,d\,e^2+630\,A\,a^5\,e^3-440\,B\,a^4\,b\,d^2\,e-2310\,A\,a^4\,b\,d\,e^2+396\,B\,a^3\,b^2\,d^3+2970\,A\,a^3\,b^2\,d^2\,e-1386\,A\,a^2\,b^3\,d^3\right )}{3465\,e^6\,{\left (a\,e-b\,d\right )}^4}+\frac {x\,\sqrt {a+b\,x}\,\left (770\,B\,a^5\,e^3-2490\,B\,a^4\,b\,d\,e^2+840\,A\,a^4\,b\,e^3+2398\,B\,a^3\,b^2\,d^2\,e-3300\,A\,a^3\,b^2\,d\,e^2-198\,B\,a^2\,b^3\,d^3+4752\,A\,a^2\,b^3\,d^2\,e-2772\,A\,a\,b^4\,d^3\right )}{3465\,e^6\,{\left (a\,e-b\,d\right )}^4}-\frac {x^2\,\sqrt {a+b\,x}\,\left (-1100\,B\,a^4\,b\,e^3+4232\,B\,a^3\,b^2\,d\,e^2-30\,A\,a^3\,b^2\,e^3-5676\,B\,a^2\,b^3\,d^2\,e+198\,A\,a^2\,b^3\,d\,e^2+1584\,B\,a\,b^4\,d^3-594\,A\,a\,b^4\,d^2\,e+1386\,A\,b^5\,d^3\right )}{3465\,e^6\,{\left (a\,e-b\,d\right )}^4}-\frac {16\,b^4\,x^5\,\sqrt {a+b\,x}\,\left (6\,A\,b\,e-11\,B\,a\,e+5\,B\,b\,d\right )}{3465\,e^4\,{\left (a\,e-b\,d\right )}^4}+\frac {8\,b^3\,x^4\,\left (a\,e-11\,b\,d\right )\,\sqrt {a+b\,x}\,\left (6\,A\,b\,e-11\,B\,a\,e+5\,B\,b\,d\right )}{3465\,e^5\,{\left (a\,e-b\,d\right )}^4}-\frac {2\,b^2\,x^3\,\sqrt {a+b\,x}\,\left (3\,a^2\,e^2-22\,a\,b\,d\,e+99\,b^2\,d^2\right )\,\left (6\,A\,b\,e-11\,B\,a\,e+5\,B\,b\,d\right )}{3465\,e^6\,{\left (a\,e-b\,d\right )}^4}\right )}{x^6+\frac {d^6}{e^6}+\frac {6\,d\,x^5}{e}+\frac {6\,d^5\,x}{e^5}+\frac {15\,d^2\,x^4}{e^2}+\frac {20\,d^3\,x^3}{e^3}+\frac {15\,d^4\,x^2}{e^4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^(3/2))/(d + e*x)^(13/2),x)

[Out]

-((d + e*x)^(1/2)*(((a + b*x)^(1/2)*(630*A*a^5*e^3 + 140*B*a^5*d*e^2 - 1386*A*a^2*b^3*d^3 + 396*B*a^3*b^2*d^3
+ 2970*A*a^3*b^2*d^2*e - 2310*A*a^4*b*d*e^2 - 440*B*a^4*b*d^2*e))/(3465*e^6*(a*e - b*d)^4) + (x*(a + b*x)^(1/2
)*(770*B*a^5*e^3 - 2772*A*a*b^4*d^3 + 840*A*a^4*b*e^3 - 198*B*a^2*b^3*d^3 + 4752*A*a^2*b^3*d^2*e - 3300*A*a^3*
b^2*d*e^2 + 2398*B*a^3*b^2*d^2*e - 2490*B*a^4*b*d*e^2))/(3465*e^6*(a*e - b*d)^4) - (x^2*(a + b*x)^(1/2)*(1386*
A*b^5*d^3 + 1584*B*a*b^4*d^3 - 1100*B*a^4*b*e^3 - 30*A*a^3*b^2*e^3 + 198*A*a^2*b^3*d*e^2 - 5676*B*a^2*b^3*d^2*
e + 4232*B*a^3*b^2*d*e^2 - 594*A*a*b^4*d^2*e))/(3465*e^6*(a*e - b*d)^4) - (16*b^4*x^5*(a + b*x)^(1/2)*(6*A*b*e
- 11*B*a*e + 5*B*b*d))/(3465*e^4*(a*e - b*d)^4) + (8*b^3*x^4*(a*e - 11*b*d)*(a + b*x)^(1/2)*(6*A*b*e - 11*B*a
*e + 5*B*b*d))/(3465*e^5*(a*e - b*d)^4) - (2*b^2*x^3*(a + b*x)^(1/2)*(3*a^2*e^2 + 99*b^2*d^2 - 22*a*b*d*e)*(6*
A*b*e - 11*B*a*e + 5*B*b*d))/(3465*e^6*(a*e - b*d)^4)))/(x^6 + d^6/e^6 + (6*d*x^5)/e + (6*d^5*x)/e^5 + (15*d^2
*x^4)/e^2 + (20*d^3*x^3)/e^3 + (15*d^4*x^2)/e^4)

________________________________________________________________________________________