3.23.21 \(\int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{15/2}} \, dx\) [2221]

Optimal. Leaf size=255 \[ -\frac {2 (B d-A e) (a+b x)^{5/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac {2 (5 b B d+8 A b e-13 a B e) (a+b x)^{5/2}}{143 e (b d-a e)^2 (d+e x)^{11/2}}+\frac {4 b (5 b B d+8 A b e-13 a B e) (a+b x)^{5/2}}{429 e (b d-a e)^3 (d+e x)^{9/2}}+\frac {16 b^2 (5 b B d+8 A b e-13 a B e) (a+b x)^{5/2}}{3003 e (b d-a e)^4 (d+e x)^{7/2}}+\frac {32 b^3 (5 b B d+8 A b e-13 a B e) (a+b x)^{5/2}}{15015 e (b d-a e)^5 (d+e x)^{5/2}} \]

[Out]

-2/13*(-A*e+B*d)*(b*x+a)^(5/2)/e/(-a*e+b*d)/(e*x+d)^(13/2)+2/143*(8*A*b*e-13*B*a*e+5*B*b*d)*(b*x+a)^(5/2)/e/(-
a*e+b*d)^2/(e*x+d)^(11/2)+4/429*b*(8*A*b*e-13*B*a*e+5*B*b*d)*(b*x+a)^(5/2)/e/(-a*e+b*d)^3/(e*x+d)^(9/2)+16/300
3*b^2*(8*A*b*e-13*B*a*e+5*B*b*d)*(b*x+a)^(5/2)/e/(-a*e+b*d)^4/(e*x+d)^(7/2)+32/15015*b^3*(8*A*b*e-13*B*a*e+5*B
*b*d)*(b*x+a)^(5/2)/e/(-a*e+b*d)^5/(e*x+d)^(5/2)

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Rubi [A]
time = 0.11, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 5, 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 {32 b^3 (a+b x)^{5/2} (-13 a B e+8 A b e+5 b B d)}{15015 e (d+e x)^{5/2} (b d-a e)^5}+\frac {16 b^2 (a+b x)^{5/2} (-13 a B e+8 A b e+5 b B d)}{3003 e (d+e x)^{7/2} (b d-a e)^4}+\frac {4 b (a+b x)^{5/2} (-13 a B e+8 A b e+5 b B d)}{429 e (d+e x)^{9/2} (b d-a e)^3}+\frac {2 (a+b x)^{5/2} (-13 a B e+8 A b e+5 b B d)}{143 e (d+e x)^{11/2} (b d-a e)^2}-\frac {2 (a+b x)^{5/2} (B d-A e)}{13 e (d+e x)^{13/2} (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(3/2)*(A + B*x))/(d + e*x)^(15/2),x]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(5/2))/(13*e*(b*d - a*e)*(d + e*x)^(13/2)) + (2*(5*b*B*d + 8*A*b*e - 13*a*B*e)*(a +
b*x)^(5/2))/(143*e*(b*d - a*e)^2*(d + e*x)^(11/2)) + (4*b*(5*b*B*d + 8*A*b*e - 13*a*B*e)*(a + b*x)^(5/2))/(429
*e*(b*d - a*e)^3*(d + e*x)^(9/2)) + (16*b^2*(5*b*B*d + 8*A*b*e - 13*a*B*e)*(a + b*x)^(5/2))/(3003*e*(b*d - a*e
)^4*(d + e*x)^(7/2)) + (32*b^3*(5*b*B*d + 8*A*b*e - 13*a*B*e)*(a + b*x)^(5/2))/(15015*e*(b*d - a*e)^5*(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)^{15/2}} \, dx &=-\frac {2 (B d-A e) (a+b x)^{5/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac {(5 b B d+8 A b e-13 a B e) \int \frac {(a+b x)^{3/2}}{(d+e x)^{13/2}} \, dx}{13 e (b d-a e)}\\ &=-\frac {2 (B d-A e) (a+b x)^{5/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac {2 (5 b B d+8 A b e-13 a B e) (a+b x)^{5/2}}{143 e (b d-a e)^2 (d+e x)^{11/2}}+\frac {(6 b (5 b B d+8 A b e-13 a B e)) \int \frac {(a+b x)^{3/2}}{(d+e x)^{11/2}} \, dx}{143 e (b d-a e)^2}\\ &=-\frac {2 (B d-A e) (a+b x)^{5/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac {2 (5 b B d+8 A b e-13 a B e) (a+b x)^{5/2}}{143 e (b d-a e)^2 (d+e x)^{11/2}}+\frac {4 b (5 b B d+8 A b e-13 a B e) (a+b x)^{5/2}}{429 e (b d-a e)^3 (d+e x)^{9/2}}+\frac {\left (8 b^2 (5 b B d+8 A b e-13 a B e)\right ) \int \frac {(a+b x)^{3/2}}{(d+e x)^{9/2}} \, dx}{429 e (b d-a e)^3}\\ &=-\frac {2 (B d-A e) (a+b x)^{5/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac {2 (5 b B d+8 A b e-13 a B e) (a+b x)^{5/2}}{143 e (b d-a e)^2 (d+e x)^{11/2}}+\frac {4 b (5 b B d+8 A b e-13 a B e) (a+b x)^{5/2}}{429 e (b d-a e)^3 (d+e x)^{9/2}}+\frac {16 b^2 (5 b B d+8 A b e-13 a B e) (a+b x)^{5/2}}{3003 e (b d-a e)^4 (d+e x)^{7/2}}+\frac {\left (16 b^3 (5 b B d+8 A b e-13 a B e)\right ) \int \frac {(a+b x)^{3/2}}{(d+e x)^{7/2}} \, dx}{3003 e (b d-a e)^4}\\ &=-\frac {2 (B d-A e) (a+b x)^{5/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac {2 (5 b B d+8 A b e-13 a B e) (a+b x)^{5/2}}{143 e (b d-a e)^2 (d+e x)^{11/2}}+\frac {4 b (5 b B d+8 A b e-13 a B e) (a+b x)^{5/2}}{429 e (b d-a e)^3 (d+e x)^{9/2}}+\frac {16 b^2 (5 b B d+8 A b e-13 a B e) (a+b x)^{5/2}}{3003 e (b d-a e)^4 (d+e x)^{7/2}}+\frac {32 b^3 (5 b B d+8 A b e-13 a B e) (a+b x)^{5/2}}{15015 e (b d-a e)^5 (d+e x)^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.45, size = 270, normalized size = 1.06 \begin {gather*} \frac {2 (a+b x)^{5/2} \left (-1155 B d e^3 (a+b x)^4+1155 A e^4 (a+b x)^4+4095 b B d e^2 (a+b x)^3 (d+e x)-5460 A b e^3 (a+b x)^3 (d+e x)+1365 a B e^3 (a+b x)^3 (d+e x)-5005 b^2 B d e (a+b x)^2 (d+e x)^2+10010 A b^2 e^2 (a+b x)^2 (d+e x)^2-5005 a b B e^2 (a+b x)^2 (d+e x)^2+2145 b^3 B d (a+b x) (d+e x)^3-8580 A b^3 e (a+b x) (d+e x)^3+6435 a b^2 B e (a+b x) (d+e x)^3+3003 A b^4 (d+e x)^4-3003 a b^3 B (d+e x)^4\right )}{15015 (b d-a e)^5 (d+e x)^{13/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(3/2)*(A + B*x))/(d + e*x)^(15/2),x]

[Out]

(2*(a + b*x)^(5/2)*(-1155*B*d*e^3*(a + b*x)^4 + 1155*A*e^4*(a + b*x)^4 + 4095*b*B*d*e^2*(a + b*x)^3*(d + e*x)
- 5460*A*b*e^3*(a + b*x)^3*(d + e*x) + 1365*a*B*e^3*(a + b*x)^3*(d + e*x) - 5005*b^2*B*d*e*(a + b*x)^2*(d + e*
x)^2 + 10010*A*b^2*e^2*(a + b*x)^2*(d + e*x)^2 - 5005*a*b*B*e^2*(a + b*x)^2*(d + e*x)^2 + 2145*b^3*B*d*(a + b*
x)*(d + e*x)^3 - 8580*A*b^3*e*(a + b*x)*(d + e*x)^3 + 6435*a*b^2*B*e*(a + b*x)*(d + e*x)^3 + 3003*A*b^4*(d + e
*x)^4 - 3003*a*b^3*B*(d + e*x)^4))/(15015*(b*d - a*e)^5*(d + e*x)^(13/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(604\) vs. \(2(225)=450\).
time = 0.09, size = 605, normalized size = 2.37

method	result	size
gosper	\(-\frac {2 \left (b x +a \right )^{\frac {5}{2}} \left (128 A \,b^{4} e^{4} x^{4}-208 B a \,b^{3} e^{4} x^{4}+80 B \,b^{4} d \,e^{3} x^{4}-320 A a \,b^{3} e^{4} x^{3}+832 A \,b^{4} d \,e^{3} x^{3}+520 B \,a^{2} b^{2} e^{4} x^{3}-1552 B a \,b^{3} d \,e^{3} x^{3}+520 B \,b^{4} d^{2} e^{2} x^{3}+560 A \,a^{2} b^{2} e^{4} x^{2}-2080 A a \,b^{3} d \,e^{3} x^{2}+2288 A \,b^{4} d^{2} e^{2} x^{2}-910 B \,a^{3} b \,e^{4} x^{2}+3730 B \,a^{2} b^{2} d \,e^{3} x^{2}-5018 B a \,b^{3} d^{2} e^{2} x^{2}+1430 B \,b^{4} d^{3} e \,x^{2}-840 A \,a^{3} b \,e^{4} x +3640 A \,a^{2} b^{2} d \,e^{3} x -5720 A a \,b^{3} d^{2} e^{2} x +3432 A \,b^{4} d^{3} e x +1365 B \,a^{4} e^{4} x -6440 B \,a^{3} b d \,e^{3} x +11570 B \,a^{2} b^{2} d^{2} e^{2} x -9152 B a \,b^{3} d^{3} e x +2145 B \,b^{4} d^{4} x +1155 A \,a^{4} e^{4}-5460 A \,a^{3} b d \,e^{3}+10010 A \,a^{2} b^{2} d^{2} e^{2}-8580 A a \,b^{3} d^{3} e +3003 A \,b^{4} d^{4}+210 B \,a^{4} d \,e^{3}-910 B \,a^{3} b \,d^{2} e^{2}+1430 B \,a^{2} b^{2} d^{3} e -858 B a \,b^{3} d^{4}\right )}{15015 \left (e x +d \right )^{\frac {13}{2}} \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}\)	\(505\)
default	\(-\frac {2 \left (128 A \,b^{5} e^{4} x^{5}-208 B a \,b^{4} e^{4} x^{5}+80 B \,b^{5} d \,e^{3} x^{5}-192 A a \,b^{4} e^{4} x^{4}+832 A \,b^{5} d \,e^{3} x^{4}+312 B \,a^{2} b^{3} e^{4} x^{4}-1472 B a \,b^{4} d \,e^{3} x^{4}+520 B \,b^{5} d^{2} e^{2} x^{4}+240 A \,a^{2} b^{3} e^{4} x^{3}-1248 A a \,b^{4} d \,e^{3} x^{3}+2288 A \,b^{5} d^{2} e^{2} x^{3}-390 B \,a^{3} b^{2} e^{4} x^{3}+2178 B \,a^{2} b^{3} d \,e^{3} x^{3}-4498 B a \,b^{4} d^{2} e^{2} x^{3}+1430 B \,b^{5} d^{3} e \,x^{3}-280 A \,a^{3} b^{2} e^{4} x^{2}+1560 A \,a^{2} b^{3} d \,e^{3} x^{2}-3432 A a \,b^{4} d^{2} e^{2} x^{2}+3432 A \,b^{5} d^{3} e \,x^{2}+455 B \,a^{4} b \,e^{4} x^{2}-2710 B \,a^{3} b^{2} d \,e^{3} x^{2}+6552 B \,a^{2} b^{3} d^{2} e^{2} x^{2}-7722 B a \,b^{4} d^{3} e \,x^{2}+2145 B \,b^{5} d^{4} x^{2}+315 A \,a^{4} b \,e^{4} x -1820 A \,a^{3} b^{2} d \,e^{3} x +4290 A \,a^{2} b^{3} d^{2} e^{2} x -5148 A a \,b^{4} d^{3} e x +3003 A \,b^{5} d^{4} x +1365 B \,a^{5} e^{4} x -6230 B \,a^{4} b d \,e^{3} x +10660 B \,a^{3} b^{2} d^{2} e^{2} x -7722 B \,a^{2} b^{3} d^{3} e x +1287 B a \,b^{4} d^{4} x +1155 A \,a^{5} e^{4}-5460 A \,a^{4} b d \,e^{3}+10010 A \,a^{3} b^{2} d^{2} e^{2}-8580 A \,a^{2} b^{3} d^{3} e +3003 A a \,b^{4} d^{4}+210 B \,a^{5} d \,e^{3}-910 B \,a^{4} b \,d^{2} e^{2}+1430 B \,a^{3} b^{2} d^{3} e -858 B \,a^{2} b^{3} d^{4}\right ) \left (b x +a \right )^{\frac {3}{2}}}{15015 \left (e x +d \right )^{\frac {13}{2}} \left (a e -b d \right )^{5}}\)	\(605\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(3/2)*(B*x+A)/(e*x+d)^(15/2),x,method=_RETURNVERBOSE)

[Out]

-2/15015*(128*A*b^5*e^4*x^5-208*B*a*b^4*e^4*x^5+80*B*b^5*d*e^3*x^5-192*A*a*b^4*e^4*x^4+832*A*b^5*d*e^3*x^4+312
*B*a^2*b^3*e^4*x^4-1472*B*a*b^4*d*e^3*x^4+520*B*b^5*d^2*e^2*x^4+240*A*a^2*b^3*e^4*x^3-1248*A*a*b^4*d*e^3*x^3+2
288*A*b^5*d^2*e^2*x^3-390*B*a^3*b^2*e^4*x^3+2178*B*a^2*b^3*d*e^3*x^3-4498*B*a*b^4*d^2*e^2*x^3+1430*B*b^5*d^3*e
*x^3-280*A*a^3*b^2*e^4*x^2+1560*A*a^2*b^3*d*e^3*x^2-3432*A*a*b^4*d^2*e^2*x^2+3432*A*b^5*d^3*e*x^2+455*B*a^4*b*
e^4*x^2-2710*B*a^3*b^2*d*e^3*x^2+6552*B*a^2*b^3*d^2*e^2*x^2-7722*B*a*b^4*d^3*e*x^2+2145*B*b^5*d^4*x^2+315*A*a^
4*b*e^4*x-1820*A*a^3*b^2*d*e^3*x+4290*A*a^2*b^3*d^2*e^2*x-5148*A*a*b^4*d^3*e*x+3003*A*b^5*d^4*x+1365*B*a^5*e^4
*x-6230*B*a^4*b*d*e^3*x+10660*B*a^3*b^2*d^2*e^2*x-7722*B*a^2*b^3*d^3*e*x+1287*B*a*b^4*d^4*x+1155*A*a^5*e^4-546
0*A*a^4*b*d*e^3+10010*A*a^3*b^2*d^2*e^2-8580*A*a^2*b^3*d^3*e+3003*A*a*b^4*d^4+210*B*a^5*d*e^3-910*B*a^4*b*d^2*
e^2+1430*B*a^3*b^2*d^3*e-858*B*a^2*b^3*d^4)*(b*x+a)^(3/2)/(e*x+d)^(13/2)/(a*e-b*d)^5

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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)^(15/2),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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1230 vs. \(2 (239) = 478\).
time = 167.84, size = 1230, normalized size = 4.82 \begin {gather*} \frac {2 \, {\left (2145 \, B b^{6} d^{4} x^{3} + 429 \, {\left (8 \, B a b^{5} + 7 \, A b^{6}\right )} d^{4} x^{2} + 429 \, {\left (B a^{2} b^{4} + 14 \, A a b^{5}\right )} d^{4} x - 429 \, {\left (2 \, B a^{3} b^{3} - 7 \, A a^{2} b^{4}\right )} d^{4} + {\left (1155 \, A a^{6} - 16 \, {\left (13 \, B a b^{5} - 8 \, A b^{6}\right )} x^{6} + 8 \, {\left (13 \, B a^{2} b^{4} - 8 \, A a b^{5}\right )} x^{5} - 6 \, {\left (13 \, B a^{3} b^{3} - 8 \, A a^{2} b^{4}\right )} x^{4} + 5 \, {\left (13 \, B a^{4} b^{2} - 8 \, A a^{3} b^{3}\right )} x^{3} + 35 \, {\left (52 \, B a^{5} b + A a^{4} b^{2}\right )} x^{2} + 105 \, {\left (13 \, B a^{6} + 14 \, A a^{5} b\right )} x\right )} e^{4} + 2 \, {\left (40 \, B b^{6} d x^{6} - 8 \, {\left (87 \, B a b^{5} - 52 \, A b^{6}\right )} d x^{5} + {\left (353 \, B a^{2} b^{4} - 208 \, A a b^{5}\right )} d x^{4} - 2 \, {\left (133 \, B a^{3} b^{3} - 78 \, A a^{2} b^{4}\right )} d x^{3} - 10 \, {\left (447 \, B a^{4} b^{2} + 13 \, A a^{3} b^{3}\right )} d x^{2} - 70 \, {\left (43 \, B a^{5} b + 52 \, A a^{4} b^{2}\right )} d x + 105 \, {\left (B a^{6} - 26 \, A a^{5} b\right )} d\right )} e^{3} + 26 \, {\left (20 \, B b^{6} d^{2} x^{5} - {\left (153 \, B a b^{5} - 88 \, A b^{6}\right )} d^{2} x^{4} + {\left (79 \, B a^{2} b^{4} - 44 \, A a b^{5}\right )} d^{2} x^{3} + {\left (662 \, B a^{3} b^{3} + 33 \, A a^{2} b^{4}\right )} d^{2} x^{2} + 25 \, {\left (15 \, B a^{4} b^{2} + 22 \, A a^{3} b^{3}\right )} d^{2} x - 35 \, {\left (B a^{5} b - 11 \, A a^{4} b^{2}\right )} d^{2}\right )} e^{2} + 286 \, {\left (5 \, B b^{6} d^{3} x^{4} - 2 \, {\left (11 \, B a b^{5} - 6 \, A b^{6}\right )} d^{3} x^{3} - 6 \, {\left (9 \, B a^{2} b^{4} + A a b^{5}\right )} d^{3} x^{2} - 2 \, {\left (11 \, B a^{3} b^{3} + 24 \, A a^{2} b^{4}\right )} d^{3} x + 5 \, {\left (B a^{4} b^{2} - 6 \, A a^{3} b^{3}\right )} d^{3}\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d}}{15015 \, {\left (b^{5} d^{12} - a^{5} x^{7} e^{12} + {\left (5 \, a^{4} b d x^{7} - 7 \, a^{5} d x^{6}\right )} e^{11} - {\left (10 \, a^{3} b^{2} d^{2} x^{7} - 35 \, a^{4} b d^{2} x^{6} + 21 \, a^{5} d^{2} x^{5}\right )} e^{10} + 5 \, {\left (2 \, a^{2} b^{3} d^{3} x^{7} - 14 \, a^{3} b^{2} d^{3} x^{6} + 21 \, a^{4} b d^{3} x^{5} - 7 \, a^{5} d^{3} x^{4}\right )} e^{9} - 5 \, {\left (a b^{4} d^{4} x^{7} - 14 \, a^{2} b^{3} d^{4} x^{6} + 42 \, a^{3} b^{2} d^{4} x^{5} - 35 \, a^{4} b d^{4} x^{4} + 7 \, a^{5} d^{4} x^{3}\right )} e^{8} + {\left (b^{5} d^{5} x^{7} - 35 \, a b^{4} d^{5} x^{6} + 210 \, a^{2} b^{3} d^{5} x^{5} - 350 \, a^{3} b^{2} d^{5} x^{4} + 175 \, a^{4} b d^{5} x^{3} - 21 \, a^{5} d^{5} x^{2}\right )} e^{7} + 7 \, {\left (b^{5} d^{6} x^{6} - 15 \, a b^{4} d^{6} x^{5} + 50 \, a^{2} b^{3} d^{6} x^{4} - 50 \, a^{3} b^{2} d^{6} x^{3} + 15 \, a^{4} b d^{6} x^{2} - a^{5} d^{6} x\right )} e^{6} + {\left (21 \, b^{5} d^{7} x^{5} - 175 \, a b^{4} d^{7} x^{4} + 350 \, a^{2} b^{3} d^{7} x^{3} - 210 \, a^{3} b^{2} d^{7} x^{2} + 35 \, a^{4} b d^{7} x - a^{5} d^{7}\right )} e^{5} + 5 \, {\left (7 \, b^{5} d^{8} x^{4} - 35 \, a b^{4} d^{8} x^{3} + 42 \, a^{2} b^{3} d^{8} x^{2} - 14 \, a^{3} b^{2} d^{8} x + a^{4} b d^{8}\right )} e^{4} + 5 \, {\left (7 \, b^{5} d^{9} x^{3} - 21 \, a b^{4} d^{9} x^{2} + 14 \, a^{2} b^{3} d^{9} x - 2 \, a^{3} b^{2} d^{9}\right )} e^{3} + {\left (21 \, b^{5} d^{10} x^{2} - 35 \, a b^{4} d^{10} x + 10 \, a^{2} b^{3} d^{10}\right )} e^{2} + {\left (7 \, b^{5} d^{11} x - 5 \, a b^{4} d^{11}\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)^(15/2),x, algorithm="fricas")

[Out]

2/15015*(2145*B*b^6*d^4*x^3 + 429*(8*B*a*b^5 + 7*A*b^6)*d^4*x^2 + 429*(B*a^2*b^4 + 14*A*a*b^5)*d^4*x - 429*(2*
B*a^3*b^3 - 7*A*a^2*b^4)*d^4 + (1155*A*a^6 - 16*(13*B*a*b^5 - 8*A*b^6)*x^6 + 8*(13*B*a^2*b^4 - 8*A*a*b^5)*x^5
- 6*(13*B*a^3*b^3 - 8*A*a^2*b^4)*x^4 + 5*(13*B*a^4*b^2 - 8*A*a^3*b^3)*x^3 + 35*(52*B*a^5*b + A*a^4*b^2)*x^2 +
105*(13*B*a^6 + 14*A*a^5*b)*x)*e^4 + 2*(40*B*b^6*d*x^6 - 8*(87*B*a*b^5 - 52*A*b^6)*d*x^5 + (353*B*a^2*b^4 - 20
8*A*a*b^5)*d*x^4 - 2*(133*B*a^3*b^3 - 78*A*a^2*b^4)*d*x^3 - 10*(447*B*a^4*b^2 + 13*A*a^3*b^3)*d*x^2 - 70*(43*B
*a^5*b + 52*A*a^4*b^2)*d*x + 105*(B*a^6 - 26*A*a^5*b)*d)*e^3 + 26*(20*B*b^6*d^2*x^5 - (153*B*a*b^5 - 88*A*b^6)
*d^2*x^4 + (79*B*a^2*b^4 - 44*A*a*b^5)*d^2*x^3 + (662*B*a^3*b^3 + 33*A*a^2*b^4)*d^2*x^2 + 25*(15*B*a^4*b^2 + 2
2*A*a^3*b^3)*d^2*x - 35*(B*a^5*b - 11*A*a^4*b^2)*d^2)*e^2 + 286*(5*B*b^6*d^3*x^4 - 2*(11*B*a*b^5 - 6*A*b^6)*d^
3*x^3 - 6*(9*B*a^2*b^4 + A*a*b^5)*d^3*x^2 - 2*(11*B*a^3*b^3 + 24*A*a^2*b^4)*d^3*x + 5*(B*a^4*b^2 - 6*A*a^3*b^3
)*d^3)*e)*sqrt(b*x + a)*sqrt(x*e + d)/(b^5*d^12 - a^5*x^7*e^12 + (5*a^4*b*d*x^7 - 7*a^5*d*x^6)*e^11 - (10*a^3*
b^2*d^2*x^7 - 35*a^4*b*d^2*x^6 + 21*a^5*d^2*x^5)*e^10 + 5*(2*a^2*b^3*d^3*x^7 - 14*a^3*b^2*d^3*x^6 + 21*a^4*b*d
^3*x^5 - 7*a^5*d^3*x^4)*e^9 - 5*(a*b^4*d^4*x^7 - 14*a^2*b^3*d^4*x^6 + 42*a^3*b^2*d^4*x^5 - 35*a^4*b*d^4*x^4 +
7*a^5*d^4*x^3)*e^8 + (b^5*d^5*x^7 - 35*a*b^4*d^5*x^6 + 210*a^2*b^3*d^5*x^5 - 350*a^3*b^2*d^5*x^4 + 175*a^4*b*d
^5*x^3 - 21*a^5*d^5*x^2)*e^7 + 7*(b^5*d^6*x^6 - 15*a*b^4*d^6*x^5 + 50*a^2*b^3*d^6*x^4 - 50*a^3*b^2*d^6*x^3 + 1
5*a^4*b*d^6*x^2 - a^5*d^6*x)*e^6 + (21*b^5*d^7*x^5 - 175*a*b^4*d^7*x^4 + 350*a^2*b^3*d^7*x^3 - 210*a^3*b^2*d^7
*x^2 + 35*a^4*b*d^7*x - a^5*d^7)*e^5 + 5*(7*b^5*d^8*x^4 - 35*a*b^4*d^8*x^3 + 42*a^2*b^3*d^8*x^2 - 14*a^3*b^2*d
^8*x + a^4*b*d^8)*e^4 + 5*(7*b^5*d^9*x^3 - 21*a*b^4*d^9*x^2 + 14*a^2*b^3*d^9*x - 2*a^3*b^2*d^9)*e^3 + (21*b^5*
d^10*x^2 - 35*a*b^4*d^10*x + 10*a^2*b^3*d^10)*e^2 + (7*b^5*d^11*x - 5*a*b^4*d^11)*e)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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)**(15/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4061 deep

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1091 vs. \(2 (239) = 478\).
time = 2.29, size = 1091, normalized size = 4.28 \begin {gather*} \frac {2 \, {\left ({\left (2 \, {\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (5 \, B b^{15} d^{2} {\left | b \right |} e^{9} - 18 \, B a b^{14} d {\left | b \right |} e^{10} + 8 \, A b^{15} d {\left | b \right |} e^{10} + 13 \, B a^{2} b^{13} {\left | b \right |} e^{11} - 8 \, A a b^{14} {\left | b \right |} e^{11}\right )} {\left (b x + a\right )}}{b^{8} d^{6} e^{6} - 6 \, a b^{7} d^{5} e^{7} + 15 \, a^{2} b^{6} d^{4} e^{8} - 20 \, a^{3} b^{5} d^{3} e^{9} + 15 \, a^{4} b^{4} d^{2} e^{10} - 6 \, a^{5} b^{3} d e^{11} + a^{6} b^{2} e^{12}} + \frac {13 \, {\left (5 \, B b^{16} d^{3} {\left | b \right |} e^{8} - 23 \, B a b^{15} d^{2} {\left | b \right |} e^{9} + 8 \, A b^{16} d^{2} {\left | b \right |} e^{9} + 31 \, B a^{2} b^{14} d {\left | b \right |} e^{10} - 16 \, A a b^{15} d {\left | b \right |} e^{10} - 13 \, B a^{3} b^{13} {\left | b \right |} e^{11} + 8 \, A a^{2} b^{14} {\left | b \right |} e^{11}\right )}}{b^{8} d^{6} e^{6} - 6 \, a b^{7} d^{5} e^{7} + 15 \, a^{2} b^{6} d^{4} e^{8} - 20 \, a^{3} b^{5} d^{3} e^{9} + 15 \, a^{4} b^{4} d^{2} e^{10} - 6 \, a^{5} b^{3} d e^{11} + a^{6} b^{2} e^{12}}\right )} + \frac {143 \, {\left (5 \, B b^{17} d^{4} {\left | b \right |} e^{7} - 28 \, B a b^{16} d^{3} {\left | b \right |} e^{8} + 8 \, A b^{17} d^{3} {\left | b \right |} e^{8} + 54 \, B a^{2} b^{15} d^{2} {\left | b \right |} e^{9} - 24 \, A a b^{16} d^{2} {\left | b \right |} e^{9} - 44 \, B a^{3} b^{14} d {\left | b \right |} e^{10} + 24 \, A a^{2} b^{15} d {\left | b \right |} e^{10} + 13 \, B a^{4} b^{13} {\left | b \right |} e^{11} - 8 \, A a^{3} b^{14} {\left | b \right |} e^{11}\right )}}{b^{8} d^{6} e^{6} - 6 \, a b^{7} d^{5} e^{7} + 15 \, a^{2} b^{6} d^{4} e^{8} - 20 \, a^{3} b^{5} d^{3} e^{9} + 15 \, a^{4} b^{4} d^{2} e^{10} - 6 \, a^{5} b^{3} d e^{11} + a^{6} b^{2} e^{12}}\right )} {\left (b x + a\right )} + \frac {429 \, {\left (5 \, B b^{18} d^{5} {\left | b \right |} e^{6} - 33 \, B a b^{17} d^{4} {\left | b \right |} e^{7} + 8 \, A b^{18} d^{4} {\left | b \right |} e^{7} + 82 \, B a^{2} b^{16} d^{3} {\left | b \right |} e^{8} - 32 \, A a b^{17} d^{3} {\left | b \right |} e^{8} - 98 \, B a^{3} b^{15} d^{2} {\left | b \right |} e^{9} + 48 \, A a^{2} b^{16} d^{2} {\left | b \right |} e^{9} + 57 \, B a^{4} b^{14} d {\left | b \right |} e^{10} - 32 \, A a^{3} b^{15} d {\left | b \right |} e^{10} - 13 \, B a^{5} b^{13} {\left | b \right |} e^{11} + 8 \, A a^{4} b^{14} {\left | b \right |} e^{11}\right )}}{b^{8} d^{6} e^{6} - 6 \, a b^{7} d^{5} e^{7} + 15 \, a^{2} b^{6} d^{4} e^{8} - 20 \, a^{3} b^{5} d^{3} e^{9} + 15 \, a^{4} b^{4} d^{2} e^{10} - 6 \, a^{5} b^{3} d e^{11} + a^{6} b^{2} e^{12}}\right )} {\left (b x + a\right )} - \frac {3003 \, {\left (B a b^{18} d^{5} {\left | b \right |} e^{6} - A b^{19} d^{5} {\left | b \right |} e^{6} - 5 \, B a^{2} b^{17} d^{4} {\left | b \right |} e^{7} + 5 \, A a b^{18} d^{4} {\left | b \right |} e^{7} + 10 \, B a^{3} b^{16} d^{3} {\left | b \right |} e^{8} - 10 \, A a^{2} b^{17} d^{3} {\left | b \right |} e^{8} - 10 \, B a^{4} b^{15} d^{2} {\left | b \right |} e^{9} + 10 \, A a^{3} b^{16} d^{2} {\left | b \right |} e^{9} + 5 \, B a^{5} b^{14} d {\left | b \right |} e^{10} - 5 \, A a^{4} b^{15} d {\left | b \right |} e^{10} - B a^{6} b^{13} {\left | b \right |} e^{11} + A a^{5} b^{14} {\left | b \right |} e^{11}\right )}}{b^{8} d^{6} e^{6} - 6 \, a b^{7} d^{5} e^{7} + 15 \, a^{2} b^{6} d^{4} e^{8} - 20 \, a^{3} b^{5} d^{3} e^{9} + 15 \, a^{4} b^{4} d^{2} e^{10} - 6 \, a^{5} b^{3} d e^{11} + a^{6} b^{2} e^{12}}\right )} {\left (b x + a\right )}^{\frac {5}{2}}}{15015 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {13}{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)^(15/2),x, algorithm="giac")

[Out]

2/15015*((2*(4*(b*x + a)*(2*(5*B*b^15*d^2*abs(b)*e^9 - 18*B*a*b^14*d*abs(b)*e^10 + 8*A*b^15*d*abs(b)*e^10 + 13
*B*a^2*b^13*abs(b)*e^11 - 8*A*a*b^14*abs(b)*e^11)*(b*x + a)/(b^8*d^6*e^6 - 6*a*b^7*d^5*e^7 + 15*a^2*b^6*d^4*e^
8 - 20*a^3*b^5*d^3*e^9 + 15*a^4*b^4*d^2*e^10 - 6*a^5*b^3*d*e^11 + a^6*b^2*e^12) + 13*(5*B*b^16*d^3*abs(b)*e^8
- 23*B*a*b^15*d^2*abs(b)*e^9 + 8*A*b^16*d^2*abs(b)*e^9 + 31*B*a^2*b^14*d*abs(b)*e^10 - 16*A*a*b^15*d*abs(b)*e^
10 - 13*B*a^3*b^13*abs(b)*e^11 + 8*A*a^2*b^14*abs(b)*e^11)/(b^8*d^6*e^6 - 6*a*b^7*d^5*e^7 + 15*a^2*b^6*d^4*e^8
- 20*a^3*b^5*d^3*e^9 + 15*a^4*b^4*d^2*e^10 - 6*a^5*b^3*d*e^11 + a^6*b^2*e^12)) + 143*(5*B*b^17*d^4*abs(b)*e^7
- 28*B*a*b^16*d^3*abs(b)*e^8 + 8*A*b^17*d^3*abs(b)*e^8 + 54*B*a^2*b^15*d^2*abs(b)*e^9 - 24*A*a*b^16*d^2*abs(b
)*e^9 - 44*B*a^3*b^14*d*abs(b)*e^10 + 24*A*a^2*b^15*d*abs(b)*e^10 + 13*B*a^4*b^13*abs(b)*e^11 - 8*A*a^3*b^14*a
bs(b)*e^11)/(b^8*d^6*e^6 - 6*a*b^7*d^5*e^7 + 15*a^2*b^6*d^4*e^8 - 20*a^3*b^5*d^3*e^9 + 15*a^4*b^4*d^2*e^10 - 6
*a^5*b^3*d*e^11 + a^6*b^2*e^12))*(b*x + a) + 429*(5*B*b^18*d^5*abs(b)*e^6 - 33*B*a*b^17*d^4*abs(b)*e^7 + 8*A*b
^18*d^4*abs(b)*e^7 + 82*B*a^2*b^16*d^3*abs(b)*e^8 - 32*A*a*b^17*d^3*abs(b)*e^8 - 98*B*a^3*b^15*d^2*abs(b)*e^9
+ 48*A*a^2*b^16*d^2*abs(b)*e^9 + 57*B*a^4*b^14*d*abs(b)*e^10 - 32*A*a^3*b^15*d*abs(b)*e^10 - 13*B*a^5*b^13*abs
(b)*e^11 + 8*A*a^4*b^14*abs(b)*e^11)/(b^8*d^6*e^6 - 6*a*b^7*d^5*e^7 + 15*a^2*b^6*d^4*e^8 - 20*a^3*b^5*d^3*e^9
+ 15*a^4*b^4*d^2*e^10 - 6*a^5*b^3*d*e^11 + a^6*b^2*e^12))*(b*x + a) - 3003*(B*a*b^18*d^5*abs(b)*e^6 - A*b^19*d
^5*abs(b)*e^6 - 5*B*a^2*b^17*d^4*abs(b)*e^7 + 5*A*a*b^18*d^4*abs(b)*e^7 + 10*B*a^3*b^16*d^3*abs(b)*e^8 - 10*A*
a^2*b^17*d^3*abs(b)*e^8 - 10*B*a^4*b^15*d^2*abs(b)*e^9 + 10*A*a^3*b^16*d^2*abs(b)*e^9 + 5*B*a^5*b^14*d*abs(b)*
e^10 - 5*A*a^4*b^15*d*abs(b)*e^10 - B*a^6*b^13*abs(b)*e^11 + A*a^5*b^14*abs(b)*e^11)/(b^8*d^6*e^6 - 6*a*b^7*d^
5*e^7 + 15*a^2*b^6*d^4*e^8 - 20*a^3*b^5*d^3*e^9 + 15*a^4*b^4*d^2*e^10 - 6*a^5*b^3*d*e^11 + a^6*b^2*e^12))*(b*x
+ a)^(5/2)/(b^2*d + (b*x + a)*b*e - a*b*e)^(13/2)

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Mupad [B]
time = 3.11, size = 752, normalized size = 2.95 \begin {gather*} -\frac {\sqrt {d+e\,x}\,\left (\frac {\sqrt {a+b\,x}\,\left (420\,B\,a^6\,d\,e^3+2310\,A\,a^6\,e^4-1820\,B\,a^5\,b\,d^2\,e^2-10920\,A\,a^5\,b\,d\,e^3+2860\,B\,a^4\,b^2\,d^3\,e+20020\,A\,a^4\,b^2\,d^2\,e^2-1716\,B\,a^3\,b^3\,d^4-17160\,A\,a^3\,b^3\,d^3\,e+6006\,A\,a^2\,b^4\,d^4\right )}{15015\,e^7\,{\left (a\,e-b\,d\right )}^5}+\frac {x\,\sqrt {a+b\,x}\,\left (2730\,B\,a^6\,e^4-12040\,B\,a^5\,b\,d\,e^3+2940\,A\,a^5\,b\,e^4+19500\,B\,a^4\,b^2\,d^2\,e^2-14560\,A\,a^4\,b^2\,d\,e^3-12584\,B\,a^3\,b^3\,d^3\,e+28600\,A\,a^3\,b^3\,d^2\,e^2+858\,B\,a^2\,b^4\,d^4-27456\,A\,a^2\,b^4\,d^3\,e+12012\,A\,a\,b^5\,d^4\right )}{15015\,e^7\,{\left (a\,e-b\,d\right )}^5}+\frac {x^2\,\sqrt {a+b\,x}\,\left (3640\,B\,a^5\,b\,e^4-17880\,B\,a^4\,b^2\,d\,e^3+70\,A\,a^4\,b^2\,e^4+34424\,B\,a^3\,b^3\,d^2\,e^2-520\,A\,a^3\,b^3\,d\,e^3-30888\,B\,a^2\,b^4\,d^3\,e+1716\,A\,a^2\,b^4\,d^2\,e^2+6864\,B\,a\,b^5\,d^4-3432\,A\,a\,b^5\,d^3\,e+6006\,A\,b^6\,d^4\right )}{15015\,e^7\,{\left (a\,e-b\,d\right )}^5}+\frac {32\,b^5\,x^6\,\sqrt {a+b\,x}\,\left (8\,A\,b\,e-13\,B\,a\,e+5\,B\,b\,d\right )}{15015\,e^4\,{\left (a\,e-b\,d\right )}^5}-\frac {2\,b^2\,x^3\,\sqrt {a+b\,x}\,\left (8\,A\,b\,e-13\,B\,a\,e+5\,B\,b\,d\right )\,\left (5\,a^3\,e^3-39\,a^2\,b\,d\,e^2+143\,a\,b^2\,d^2\,e-429\,b^3\,d^3\right )}{15015\,e^7\,{\left (a\,e-b\,d\right )}^5}-\frac {16\,b^4\,x^5\,\left (a\,e-13\,b\,d\right )\,\sqrt {a+b\,x}\,\left (8\,A\,b\,e-13\,B\,a\,e+5\,B\,b\,d\right )}{15015\,e^5\,{\left (a\,e-b\,d\right )}^5}+\frac {4\,b^3\,x^4\,\sqrt {a+b\,x}\,\left (3\,a^2\,e^2-26\,a\,b\,d\,e+143\,b^2\,d^2\right )\,\left (8\,A\,b\,e-13\,B\,a\,e+5\,B\,b\,d\right )}{15015\,e^6\,{\left (a\,e-b\,d\right )}^5}\right )}{x^7+\frac {d^7}{e^7}+\frac {7\,d\,x^6}{e}+\frac {7\,d^6\,x}{e^6}+\frac {21\,d^2\,x^5}{e^2}+\frac {35\,d^3\,x^4}{e^3}+\frac {35\,d^4\,x^3}{e^4}+\frac {21\,d^5\,x^2}{e^5}} \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)^(15/2),x)

[Out]

-((d + e*x)^(1/2)*(((a + b*x)^(1/2)*(2310*A*a^6*e^4 + 420*B*a^6*d*e^3 + 6006*A*a^2*b^4*d^4 - 1716*B*a^3*b^3*d^
4 - 17160*A*a^3*b^3*d^3*e + 2860*B*a^4*b^2*d^3*e - 1820*B*a^5*b*d^2*e^2 + 20020*A*a^4*b^2*d^2*e^2 - 10920*A*a^
5*b*d*e^3))/(15015*e^7*(a*e - b*d)^5) + (x*(a + b*x)^(1/2)*(2730*B*a^6*e^4 + 12012*A*a*b^5*d^4 + 2940*A*a^5*b*
e^4 + 858*B*a^2*b^4*d^4 - 27456*A*a^2*b^4*d^3*e - 14560*A*a^4*b^2*d*e^3 - 12584*B*a^3*b^3*d^3*e + 28600*A*a^3*
b^3*d^2*e^2 + 19500*B*a^4*b^2*d^2*e^2 - 12040*B*a^5*b*d*e^3))/(15015*e^7*(a*e - b*d)^5) + (x^2*(a + b*x)^(1/2)
*(6006*A*b^6*d^4 + 6864*B*a*b^5*d^4 + 3640*B*a^5*b*e^4 + 70*A*a^4*b^2*e^4 - 520*A*a^3*b^3*d*e^3 - 30888*B*a^2*
b^4*d^3*e - 17880*B*a^4*b^2*d*e^3 + 1716*A*a^2*b^4*d^2*e^2 + 34424*B*a^3*b^3*d^2*e^2 - 3432*A*a*b^5*d^3*e))/(1
5015*e^7*(a*e - b*d)^5) + (32*b^5*x^6*(a + b*x)^(1/2)*(8*A*b*e - 13*B*a*e + 5*B*b*d))/(15015*e^4*(a*e - b*d)^5
) - (2*b^2*x^3*(a + b*x)^(1/2)*(8*A*b*e - 13*B*a*e + 5*B*b*d)*(5*a^3*e^3 - 429*b^3*d^3 + 143*a*b^2*d^2*e - 39*
a^2*b*d*e^2))/(15015*e^7*(a*e - b*d)^5) - (16*b^4*x^5*(a*e - 13*b*d)*(a + b*x)^(1/2)*(8*A*b*e - 13*B*a*e + 5*B
*b*d))/(15015*e^5*(a*e - b*d)^5) + (4*b^3*x^4*(a + b*x)^(1/2)*(3*a^2*e^2 + 143*b^2*d^2 - 26*a*b*d*e)*(8*A*b*e
- 13*B*a*e + 5*B*b*d))/(15015*e^6*(a*e - b*d)^5)))/(x^7 + d^7/e^7 + (7*d*x^6)/e + (7*d^6*x)/e^6 + (21*d^2*x^5)
/e^2 + (35*d^3*x^4)/e^3 + (35*d^4*x^3)/e^4 + (21*d^5*x^2)/e^5)

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