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

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

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Rubi [A]  time = 0.12, 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 = {78, 45, 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 45

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 78

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] || LtQ
[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*}

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Mathematica [A]  time = 0.39, size = 114, normalized size = 0.57 \begin {gather*} \frac {2 (a+b x)^{5/2} \left (315 (B d-A e)-\frac {(d+e x) \left (4 b (d+e x) (-5 a e+7 b d+2 b e x)+35 (b d-a e)^2\right ) (-11 a B e+6 A b e+5 b B d)}{(b d-a e)^3}\right )}{3465 e (d+e x)^{11/2} (a e-b d)} \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 - A*e) - ((5*b*B*d + 6*A*b*e - 11*a*B*e)*(d + e*x)*(35*(b*d - a*e)^2 + 4*b*(d + e
*x)*(7*b*d - 5*a*e + 2*b*e*x)))/(b*d - a*e)^3))/(3465*e*(-(b*d) + a*e)*(d + e*x)^(11/2))

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IntegrateAlgebraic [A]  time = 0.31, size = 205, normalized size = 1.02 \begin {gather*} -\frac {2 (a+b x)^{11/2} \left (-\frac {693 A b^3 (d+e x)^3}{(a+b x)^3}+\frac {1485 A b^2 e (d+e x)^2}{(a+b x)^2}-\frac {1155 A b e^2 (d+e x)}{a+b x}+\frac {693 a b^2 B (d+e x)^3}{(a+b x)^3}-\frac {495 b^2 B d (d+e x)^2}{(a+b x)^2}+\frac {385 a B e^2 (d+e x)}{a+b x}-\frac {990 a b B e (d+e x)^2}{(a+b x)^2}+\frac {770 b B d e (d+e x)}{a+b x}+315 A e^3-315 B d e^2\right )}{3465 (d+e x)^{11/2} (b d-a e)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F(-1)]  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, algorithm="fricas")

[Out]

Timed out

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giac [B]  time = 5.20, 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)

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maple [A]  time = 0.01, size = 322, normalized size = 1.60 \begin {gather*} -\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 \,a^{3} 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 )} \end {gather*}

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)

[Out]

-2/3465*(b*x+a)^(5/2)*(-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-5
94*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*a^3*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)/(e*x+d)^(11/2
)/(a^4*e^4-4*a^3*b*d*e^3+6*a^2*b^2*d^2*e^2-4*a*b^3*d^3*e+b^4*d^4)

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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)^(3/2)*(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(a*e-b*d>0)', see `assume?` for
more details)Is a*e-b*d zero or nonzero?

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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)

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sympy [F(-1)]  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

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