3.21.83 \(\int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{17/2}} \, dx\)

Optimal. Leaf size=304 \[ \frac {256 b^4 (a+b x)^{5/2} (-3 a B e+2 A b e+b B d)}{45045 e (d+e x)^{5/2} (b d-a e)^6}+\frac {128 b^3 (a+b x)^{5/2} (-3 a B e+2 A b e+b B d)}{9009 e (d+e x)^{7/2} (b d-a e)^5}+\frac {32 b^2 (a+b x)^{5/2} (-3 a B e+2 A b e+b B d)}{1287 e (d+e x)^{9/2} (b d-a e)^4}+\frac {16 b (a+b x)^{5/2} (-3 a B e+2 A b e+b B d)}{429 e (d+e x)^{11/2} (b d-a e)^3}+\frac {2 (a+b x)^{5/2} (-3 a B e+2 A b e+b B d)}{39 e (d+e x)^{13/2} (b d-a e)^2}-\frac {2 (a+b x)^{5/2} (B d-A e)}{15 e (d+e x)^{15/2} (b d-a e)} \]

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Rubi [A]  time = 0.19, antiderivative size = 304, normalized size of antiderivative = 1.00, number of steps used = 6, 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 {256 b^4 (a+b x)^{5/2} (-3 a B e+2 A b e+b B d)}{45045 e (d+e x)^{5/2} (b d-a e)^6}+\frac {128 b^3 (a+b x)^{5/2} (-3 a B e+2 A b e+b B d)}{9009 e (d+e x)^{7/2} (b d-a e)^5}+\frac {32 b^2 (a+b x)^{5/2} (-3 a B e+2 A b e+b B d)}{1287 e (d+e x)^{9/2} (b d-a e)^4}+\frac {16 b (a+b x)^{5/2} (-3 a B e+2 A b e+b B d)}{429 e (d+e x)^{11/2} (b d-a e)^3}+\frac {2 (a+b x)^{5/2} (-3 a B e+2 A b e+b B d)}{39 e (d+e x)^{13/2} (b d-a e)^2}-\frac {2 (a+b x)^{5/2} (B d-A e)}{15 e (d+e x)^{15/2} (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(3/2)*(A + B*x))/(d + e*x)^(17/2),x]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(5/2))/(15*e*(b*d - a*e)*(d + e*x)^(15/2)) + (2*(b*B*d + 2*A*b*e - 3*a*B*e)*(a + b*x
)^(5/2))/(39*e*(b*d - a*e)^2*(d + e*x)^(13/2)) + (16*b*(b*B*d + 2*A*b*e - 3*a*B*e)*(a + b*x)^(5/2))/(429*e*(b*
d - a*e)^3*(d + e*x)^(11/2)) + (32*b^2*(b*B*d + 2*A*b*e - 3*a*B*e)*(a + b*x)^(5/2))/(1287*e*(b*d - a*e)^4*(d +
e*x)^(9/2)) + (128*b^3*(b*B*d + 2*A*b*e - 3*a*B*e)*(a + b*x)^(5/2))/(9009*e*(b*d - a*e)^5*(d + e*x)^(7/2)) +
(256*b^4*(b*B*d + 2*A*b*e - 3*a*B*e)*(a + b*x)^(5/2))/(45045*e*(b*d - a*e)^6*(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)^{17/2}} \, dx &=-\frac {2 (B d-A e) (a+b x)^{5/2}}{15 e (b d-a e) (d+e x)^{15/2}}+\frac {(b B d+2 A b e-3 a B e) \int \frac {(a+b x)^{3/2}}{(d+e x)^{15/2}} \, dx}{3 e (b d-a e)}\\ &=-\frac {2 (B d-A e) (a+b x)^{5/2}}{15 e (b d-a e) (d+e x)^{15/2}}+\frac {2 (b B d+2 A b e-3 a B e) (a+b x)^{5/2}}{39 e (b d-a e)^2 (d+e x)^{13/2}}+\frac {(8 b (b B d+2 A b e-3 a B e)) \int \frac {(a+b x)^{3/2}}{(d+e x)^{13/2}} \, dx}{39 e (b d-a e)^2}\\ &=-\frac {2 (B d-A e) (a+b x)^{5/2}}{15 e (b d-a e) (d+e x)^{15/2}}+\frac {2 (b B d+2 A b e-3 a B e) (a+b x)^{5/2}}{39 e (b d-a e)^2 (d+e x)^{13/2}}+\frac {16 b (b B d+2 A b e-3 a B e) (a+b x)^{5/2}}{429 e (b d-a e)^3 (d+e x)^{11/2}}+\frac {\left (16 b^2 (b B d+2 A b e-3 a B e)\right ) \int \frac {(a+b x)^{3/2}}{(d+e x)^{11/2}} \, dx}{143 e (b d-a e)^3}\\ &=-\frac {2 (B d-A e) (a+b x)^{5/2}}{15 e (b d-a e) (d+e x)^{15/2}}+\frac {2 (b B d+2 A b e-3 a B e) (a+b x)^{5/2}}{39 e (b d-a e)^2 (d+e x)^{13/2}}+\frac {16 b (b B d+2 A b e-3 a B e) (a+b x)^{5/2}}{429 e (b d-a e)^3 (d+e x)^{11/2}}+\frac {32 b^2 (b B d+2 A b e-3 a B e) (a+b x)^{5/2}}{1287 e (b d-a e)^4 (d+e x)^{9/2}}+\frac {\left (64 b^3 (b B d+2 A b e-3 a B e)\right ) \int \frac {(a+b x)^{3/2}}{(d+e x)^{9/2}} \, dx}{1287 e (b d-a e)^4}\\ &=-\frac {2 (B d-A e) (a+b x)^{5/2}}{15 e (b d-a e) (d+e x)^{15/2}}+\frac {2 (b B d+2 A b e-3 a B e) (a+b x)^{5/2}}{39 e (b d-a e)^2 (d+e x)^{13/2}}+\frac {16 b (b B d+2 A b e-3 a B e) (a+b x)^{5/2}}{429 e (b d-a e)^3 (d+e x)^{11/2}}+\frac {32 b^2 (b B d+2 A b e-3 a B e) (a+b x)^{5/2}}{1287 e (b d-a e)^4 (d+e x)^{9/2}}+\frac {128 b^3 (b B d+2 A b e-3 a B e) (a+b x)^{5/2}}{9009 e (b d-a e)^5 (d+e x)^{7/2}}+\frac {\left (128 b^4 (b B d+2 A b e-3 a B e)\right ) \int \frac {(a+b x)^{3/2}}{(d+e x)^{7/2}} \, dx}{9009 e (b d-a e)^5}\\ &=-\frac {2 (B d-A e) (a+b x)^{5/2}}{15 e (b d-a e) (d+e x)^{15/2}}+\frac {2 (b B d+2 A b e-3 a B e) (a+b x)^{5/2}}{39 e (b d-a e)^2 (d+e x)^{13/2}}+\frac {16 b (b B d+2 A b e-3 a B e) (a+b x)^{5/2}}{429 e (b d-a e)^3 (d+e x)^{11/2}}+\frac {32 b^2 (b B d+2 A b e-3 a B e) (a+b x)^{5/2}}{1287 e (b d-a e)^4 (d+e x)^{9/2}}+\frac {128 b^3 (b B d+2 A b e-3 a B e) (a+b x)^{5/2}}{9009 e (b d-a e)^5 (d+e x)^{7/2}}+\frac {256 b^4 (b B d+2 A b e-3 a B e) (a+b x)^{5/2}}{45045 e (b d-a e)^6 (d+e x)^{5/2}}\\ \end {align*}

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Mathematica [A]  time = 0.41, size = 155, normalized size = 0.51 \begin {gather*} \frac {2 (a+b x)^{5/2} \left (15015 (B d-A e)-\frac {5 (d+e x) \left (8 b (d+e x) \left (2 b (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 )+105 (b d-a e)^3\right )+1155 (b d-a e)^4\right ) (-3 a B e+2 A b e+b B d)}{(b d-a e)^5}\right )}{225225 e (d+e x)^{15/2} (a e-b d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(3/2)*(A + B*x))/(d + e*x)^(17/2),x]

[Out]

(2*(a + b*x)^(5/2)*(15015*(B*d - A*e) - (5*(b*B*d + 2*A*b*e - 3*a*B*e)*(d + e*x)*(1155*(b*d - a*e)^4 + 8*b*(d
+ e*x)*(105*(b*d - a*e)^3 + 2*b*(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)^5))/(225225*e*(-(b*d) + a*e)*(d + e*x)^(15/2))

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IntegrateAlgebraic [A]  time = 0.63, size = 347, normalized size = 1.14 \begin {gather*} -\frac {2 (a+b x)^{15/2} \left (-\frac {9009 A b^5 (d+e x)^5}{(a+b x)^5}+\frac {32175 A b^4 e (d+e x)^4}{(a+b x)^4}-\frac {50050 A b^3 e^2 (d+e x)^3}{(a+b x)^3}+\frac {40950 A b^2 e^3 (d+e x)^2}{(a+b x)^2}-\frac {17325 A b e^4 (d+e x)}{a+b x}+\frac {9009 a b^4 B (d+e x)^5}{(a+b x)^5}-\frac {6435 b^4 B d (d+e x)^4}{(a+b x)^4}-\frac {25740 a b^3 B e (d+e x)^4}{(a+b x)^4}+\frac {20020 b^3 B d e (d+e x)^3}{(a+b x)^3}+\frac {30030 a b^2 B e^2 (d+e x)^3}{(a+b x)^3}-\frac {24570 b^2 B d e^2 (d+e x)^2}{(a+b x)^2}+\frac {3465 a B e^4 (d+e x)}{a+b x}-\frac {16380 a b B e^3 (d+e x)^2}{(a+b x)^2}+\frac {13860 b B d e^3 (d+e x)}{a+b x}+3003 A e^5-3003 B d e^4\right )}{45045 (d+e x)^{15/2} (b d-a e)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((a + b*x)^(3/2)*(A + B*x))/(d + e*x)^(17/2),x]

[Out]

(-2*(a + b*x)^(15/2)*(-3003*B*d*e^4 + 3003*A*e^5 + (13860*b*B*d*e^3*(d + e*x))/(a + b*x) - (17325*A*b*e^4*(d +
e*x))/(a + b*x) + (3465*a*B*e^4*(d + e*x))/(a + b*x) - (24570*b^2*B*d*e^2*(d + e*x)^2)/(a + b*x)^2 + (40950*A
*b^2*e^3*(d + e*x)^2)/(a + b*x)^2 - (16380*a*b*B*e^3*(d + e*x)^2)/(a + b*x)^2 + (20020*b^3*B*d*e*(d + e*x)^3)/
(a + b*x)^3 - (50050*A*b^3*e^2*(d + e*x)^3)/(a + b*x)^3 + (30030*a*b^2*B*e^2*(d + e*x)^3)/(a + b*x)^3 - (6435*
b^4*B*d*(d + e*x)^4)/(a + b*x)^4 + (32175*A*b^4*e*(d + e*x)^4)/(a + b*x)^4 - (25740*a*b^3*B*e*(d + e*x)^4)/(a
+ b*x)^4 - (9009*A*b^5*(d + e*x)^5)/(a + b*x)^5 + (9009*a*b^4*B*(d + e*x)^5)/(a + b*x)^5))/(45045*(b*d - a*e)^
6*(d + e*x)^(15/2))

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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)^(17/2),x, algorithm="fricas")

[Out]

Timed out

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giac [B]  time = 9.52, size = 1483, normalized size = 4.88

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2)*(B*x+A)/(e*x+d)^(17/2),x, algorithm="giac")

[Out]

2/45045*((8*(2*(4*(b*x + a)*(2*(B*b^17*d^2*abs(b)*e^11 - 4*B*a*b^16*d*abs(b)*e^12 + 2*A*b^17*d*abs(b)*e^12 + 3
*B*a^2*b^15*abs(b)*e^13 - 2*A*a*b^16*abs(b)*e^13)*(b*x + a)/(b^9*d^7*e^7 - 7*a*b^8*d^6*e^8 + 21*a^2*b^7*d^5*e^
9 - 35*a^3*b^6*d^4*e^10 + 35*a^4*b^5*d^3*e^11 - 21*a^5*b^4*d^2*e^12 + 7*a^6*b^3*d*e^13 - a^7*b^2*e^14) + 15*(B
*b^18*d^3*abs(b)*e^10 - 5*B*a*b^17*d^2*abs(b)*e^11 + 2*A*b^18*d^2*abs(b)*e^11 + 7*B*a^2*b^16*d*abs(b)*e^12 - 4
*A*a*b^17*d*abs(b)*e^12 - 3*B*a^3*b^15*abs(b)*e^13 + 2*A*a^2*b^16*abs(b)*e^13)/(b^9*d^7*e^7 - 7*a*b^8*d^6*e^8
+ 21*a^2*b^7*d^5*e^9 - 35*a^3*b^6*d^4*e^10 + 35*a^4*b^5*d^3*e^11 - 21*a^5*b^4*d^2*e^12 + 7*a^6*b^3*d*e^13 - a^
7*b^2*e^14)) + 195*(B*b^19*d^4*abs(b)*e^9 - 6*B*a*b^18*d^3*abs(b)*e^10 + 2*A*b^19*d^3*abs(b)*e^10 + 12*B*a^2*b
^17*d^2*abs(b)*e^11 - 6*A*a*b^18*d^2*abs(b)*e^11 - 10*B*a^3*b^16*d*abs(b)*e^12 + 6*A*a^2*b^17*d*abs(b)*e^12 +
3*B*a^4*b^15*abs(b)*e^13 - 2*A*a^3*b^16*abs(b)*e^13)/(b^9*d^7*e^7 - 7*a*b^8*d^6*e^8 + 21*a^2*b^7*d^5*e^9 - 35*
a^3*b^6*d^4*e^10 + 35*a^4*b^5*d^3*e^11 - 21*a^5*b^4*d^2*e^12 + 7*a^6*b^3*d*e^13 - a^7*b^2*e^14))*(b*x + a) + 7
15*(B*b^20*d^5*abs(b)*e^8 - 7*B*a*b^19*d^4*abs(b)*e^9 + 2*A*b^20*d^4*abs(b)*e^9 + 18*B*a^2*b^18*d^3*abs(b)*e^1
0 - 8*A*a*b^19*d^3*abs(b)*e^10 - 22*B*a^3*b^17*d^2*abs(b)*e^11 + 12*A*a^2*b^18*d^2*abs(b)*e^11 + 13*B*a^4*b^16
*d*abs(b)*e^12 - 8*A*a^3*b^17*d*abs(b)*e^12 - 3*B*a^5*b^15*abs(b)*e^13 + 2*A*a^4*b^16*abs(b)*e^13)/(b^9*d^7*e^
7 - 7*a*b^8*d^6*e^8 + 21*a^2*b^7*d^5*e^9 - 35*a^3*b^6*d^4*e^10 + 35*a^4*b^5*d^3*e^11 - 21*a^5*b^4*d^2*e^12 + 7
*a^6*b^3*d*e^13 - a^7*b^2*e^14))*(b*x + a) + 6435*(B*b^21*d^6*abs(b)*e^7 - 8*B*a*b^20*d^5*abs(b)*e^8 + 2*A*b^2
1*d^5*abs(b)*e^8 + 25*B*a^2*b^19*d^4*abs(b)*e^9 - 10*A*a*b^20*d^4*abs(b)*e^9 - 40*B*a^3*b^18*d^3*abs(b)*e^10 +
20*A*a^2*b^19*d^3*abs(b)*e^10 + 35*B*a^4*b^17*d^2*abs(b)*e^11 - 20*A*a^3*b^18*d^2*abs(b)*e^11 - 16*B*a^5*b^16
*d*abs(b)*e^12 + 10*A*a^4*b^17*d*abs(b)*e^12 + 3*B*a^6*b^15*abs(b)*e^13 - 2*A*a^5*b^16*abs(b)*e^13)/(b^9*d^7*e
^7 - 7*a*b^8*d^6*e^8 + 21*a^2*b^7*d^5*e^9 - 35*a^3*b^6*d^4*e^10 + 35*a^4*b^5*d^3*e^11 - 21*a^5*b^4*d^2*e^12 +
7*a^6*b^3*d*e^13 - a^7*b^2*e^14))*(b*x + a) - 9009*(B*a*b^21*d^6*abs(b)*e^7 - A*b^22*d^6*abs(b)*e^7 - 6*B*a^2*
b^20*d^5*abs(b)*e^8 + 6*A*a*b^21*d^5*abs(b)*e^8 + 15*B*a^3*b^19*d^4*abs(b)*e^9 - 15*A*a^2*b^20*d^4*abs(b)*e^9
- 20*B*a^4*b^18*d^3*abs(b)*e^10 + 20*A*a^3*b^19*d^3*abs(b)*e^10 + 15*B*a^5*b^17*d^2*abs(b)*e^11 - 15*A*a^4*b^1
8*d^2*abs(b)*e^11 - 6*B*a^6*b^16*d*abs(b)*e^12 + 6*A*a^5*b^17*d*abs(b)*e^12 + B*a^7*b^15*abs(b)*e^13 - A*a^6*b
^16*abs(b)*e^13)/(b^9*d^7*e^7 - 7*a*b^8*d^6*e^8 + 21*a^2*b^7*d^5*e^9 - 35*a^3*b^6*d^4*e^10 + 35*a^4*b^5*d^3*e^
11 - 21*a^5*b^4*d^2*e^12 + 7*a^6*b^3*d*e^13 - a^7*b^2*e^14))*(b*x + a)^(5/2)/(b^2*d + (b*x + a)*b*e - a*b*e)^(
15/2)

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maple [B]  time = 0.01, size = 722, normalized size = 2.38 \begin {gather*} -\frac {2 \left (b x +a \right )^{\frac {5}{2}} \left (-256 A \,b^{5} e^{5} x^{5}+384 B a \,b^{4} e^{5} x^{5}-128 B \,b^{5} d \,e^{4} x^{5}+640 A a \,b^{4} e^{5} x^{4}-1920 A \,b^{5} d \,e^{4} x^{4}-960 B \,a^{2} b^{3} e^{5} x^{4}+3200 B a \,b^{4} d \,e^{4} x^{4}-960 B \,b^{5} d^{2} e^{3} x^{4}-1120 A \,a^{2} b^{3} e^{5} x^{3}+4800 A a \,b^{4} d \,e^{4} x^{3}-6240 A \,b^{5} d^{2} e^{3} x^{3}+1680 B \,a^{3} b^{2} e^{5} x^{3}-7760 B \,a^{2} b^{3} d \,e^{4} x^{3}+11760 B a \,b^{4} d^{2} e^{3} x^{3}-3120 B \,b^{5} d^{3} e^{2} x^{3}+1680 A \,a^{3} b^{2} e^{5} x^{2}-8400 A \,a^{2} b^{3} d \,e^{4} x^{2}+15600 A a \,b^{4} d^{2} e^{3} x^{2}-11440 A \,b^{5} d^{3} e^{2} x^{2}-2520 B \,a^{4} b \,e^{5} x^{2}+13440 B \,a^{3} b^{2} d \,e^{4} x^{2}-27600 B \,a^{2} b^{3} d^{2} e^{3} x^{2}+24960 B a \,b^{4} d^{3} e^{2} x^{2}-5720 B \,b^{5} d^{4} e \,x^{2}-2310 A \,a^{4} b \,e^{5} x +12600 A \,a^{3} b^{2} d \,e^{4} x -27300 A \,a^{2} b^{3} d^{2} e^{3} x +28600 A a \,b^{4} d^{3} e^{2} x -12870 A \,b^{5} d^{4} e x +3465 B \,a^{5} e^{5} x -20055 B \,a^{4} b d \,e^{4} x +47250 B \,a^{3} b^{2} d^{2} e^{3} x -56550 B \,a^{2} b^{3} d^{3} e^{2} x +33605 B a \,b^{4} d^{4} e x -6435 B \,b^{5} d^{5} x +3003 A \,a^{5} e^{5}-17325 A \,a^{4} b d \,e^{4}+40950 A \,a^{3} b^{2} d^{2} e^{3}-50050 A \,a^{2} b^{3} d^{3} e^{2}+32175 A a \,b^{4} d^{4} e -9009 A \,b^{5} d^{5}+462 B \,a^{5} d \,e^{4}-2520 B \,a^{4} b \,d^{2} e^{3}+5460 B \,a^{3} b^{2} d^{3} e^{2}-5720 B \,a^{2} b^{3} d^{4} e +2574 B a \,b^{4} d^{5}\right )}{45045 \left (e x +d \right )^{\frac {15}{2}} \left (a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}\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)^(17/2),x)

[Out]

-2/45045*(b*x+a)^(5/2)*(-256*A*b^5*e^5*x^5+384*B*a*b^4*e^5*x^5-128*B*b^5*d*e^4*x^5+640*A*a*b^4*e^5*x^4-1920*A*
b^5*d*e^4*x^4-960*B*a^2*b^3*e^5*x^4+3200*B*a*b^4*d*e^4*x^4-960*B*b^5*d^2*e^3*x^4-1120*A*a^2*b^3*e^5*x^3+4800*A
*a*b^4*d*e^4*x^3-6240*A*b^5*d^2*e^3*x^3+1680*B*a^3*b^2*e^5*x^3-7760*B*a^2*b^3*d*e^4*x^3+11760*B*a*b^4*d^2*e^3*
x^3-3120*B*b^5*d^3*e^2*x^3+1680*A*a^3*b^2*e^5*x^2-8400*A*a^2*b^3*d*e^4*x^2+15600*A*a*b^4*d^2*e^3*x^2-11440*A*b
^5*d^3*e^2*x^2-2520*B*a^4*b*e^5*x^2+13440*B*a^3*b^2*d*e^4*x^2-27600*B*a^2*b^3*d^2*e^3*x^2+24960*B*a*b^4*d^3*e^
2*x^2-5720*B*b^5*d^4*e*x^2-2310*A*a^4*b*e^5*x+12600*A*a^3*b^2*d*e^4*x-27300*A*a^2*b^3*d^2*e^3*x+28600*A*a*b^4*
d^3*e^2*x-12870*A*b^5*d^4*e*x+3465*B*a^5*e^5*x-20055*B*a^4*b*d*e^4*x+47250*B*a^3*b^2*d^2*e^3*x-56550*B*a^2*b^3
*d^3*e^2*x+33605*B*a*b^4*d^4*e*x-6435*B*b^5*d^5*x+3003*A*a^5*e^5-17325*A*a^4*b*d*e^4+40950*A*a^3*b^2*d^2*e^3-5
0050*A*a^2*b^3*d^3*e^2+32175*A*a*b^4*d^4*e-9009*A*b^5*d^5+462*B*a^5*d*e^4-2520*B*a^4*b*d^2*e^3+5460*B*a^3*b^2*
d^3*e^2-5720*B*a^2*b^3*d^4*e+2574*B*a*b^4*d^5)/(e*x+d)^(15/2)/(a^6*e^6-6*a^5*b*d*e^5+15*a^4*b^2*d^2*e^4-20*a^3
*b^3*d^3*e^3+15*a^2*b^4*d^4*e^2-6*a*b^5*d^5*e+b^6*d^6)

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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)^(17/2),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 = 3.69, size = 941, normalized size = 3.10 \begin {gather*} -\frac {\sqrt {d+e\,x}\,\left (\frac {\sqrt {a+b\,x}\,\left (924\,B\,a^7\,d\,e^4+6006\,A\,a^7\,e^5-5040\,B\,a^6\,b\,d^2\,e^3-34650\,A\,a^6\,b\,d\,e^4+10920\,B\,a^5\,b^2\,d^3\,e^2+81900\,A\,a^5\,b^2\,d^2\,e^3-11440\,B\,a^4\,b^3\,d^4\,e-100100\,A\,a^4\,b^3\,d^3\,e^2+5148\,B\,a^3\,b^4\,d^5+64350\,A\,a^3\,b^4\,d^4\,e-18018\,A\,a^2\,b^5\,d^5\right )}{45045\,e^8\,{\left (a\,e-b\,d\right )}^6}-\frac {x^2\,\sqrt {a+b\,x}\,\left (-8820\,B\,a^6\,b\,e^5+52416\,B\,a^5\,b^2\,d\,e^4-126\,A\,a^5\,b^2\,e^5-128760\,B\,a^4\,b^3\,d^2\,e^3+1050\,A\,a^4\,b^3\,d\,e^4+165360\,B\,a^3\,b^4\,d^3\,e^2-3900\,A\,a^3\,b^4\,d^2\,e^3-111540\,B\,a^2\,b^5\,d^4\,e+8580\,A\,a^2\,b^5\,d^3\,e^2+20592\,B\,a\,b^6\,d^5-12870\,A\,a\,b^6\,d^4\,e+18018\,A\,b^7\,d^5\right )}{45045\,e^8\,{\left (a\,e-b\,d\right )}^6}+\frac {x\,\sqrt {a+b\,x}\,\left (6930\,B\,a^7\,e^5-38262\,B\,a^6\,b\,d\,e^4+7392\,A\,a^6\,b\,e^5+84420\,B\,a^5\,b^2\,d^2\,e^3-44100\,A\,a^5\,b^2\,d\,e^4-91260\,B\,a^4\,b^3\,d^3\,e^2+109200\,A\,a^4\,b^3\,d^2\,e^3+44330\,B\,a^3\,b^4\,d^4\,e-143000\,A\,a^3\,b^4\,d^3\,e^2-2574\,B\,a^2\,b^5\,d^5+102960\,A\,a^2\,b^5\,d^4\,e-36036\,A\,a\,b^6\,d^5\right )}{45045\,e^8\,{\left (a\,e-b\,d\right )}^6}-\frac {256\,b^6\,x^7\,\sqrt {a+b\,x}\,\left (2\,A\,b\,e-3\,B\,a\,e+B\,b\,d\right )}{45045\,e^4\,{\left (a\,e-b\,d\right )}^6}+\frac {16\,b^3\,x^4\,\sqrt {a+b\,x}\,\left (2\,A\,b\,e-3\,B\,a\,e+B\,b\,d\right )\,\left (a^3\,e^3-9\,a^2\,b\,d\,e^2+39\,a\,b^2\,d^2\,e-143\,b^3\,d^3\right )}{9009\,e^7\,{\left (a\,e-b\,d\right )}^6}+\frac {128\,b^5\,x^6\,\left (a\,e-15\,b\,d\right )\,\sqrt {a+b\,x}\,\left (2\,A\,b\,e-3\,B\,a\,e+B\,b\,d\right )}{45045\,e^5\,{\left (a\,e-b\,d\right )}^6}-\frac {2\,b^2\,x^3\,\sqrt {a+b\,x}\,\left (2\,A\,b\,e-3\,B\,a\,e+B\,b\,d\right )\,\left (7\,a^4\,e^4-60\,a^3\,b\,d\,e^3+234\,a^2\,b^2\,d^2\,e^2-572\,a\,b^3\,d^3\,e+1287\,b^4\,d^4\right )}{9009\,e^8\,{\left (a\,e-b\,d\right )}^6}-\frac {32\,b^4\,x^5\,\sqrt {a+b\,x}\,\left (a^2\,e^2-10\,a\,b\,d\,e+65\,b^2\,d^2\right )\,\left (2\,A\,b\,e-3\,B\,a\,e+B\,b\,d\right )}{15015\,e^6\,{\left (a\,e-b\,d\right )}^6}\right )}{x^8+\frac {d^8}{e^8}+\frac {8\,d\,x^7}{e}+\frac {8\,d^7\,x}{e^7}+\frac {28\,d^2\,x^6}{e^2}+\frac {56\,d^3\,x^5}{e^3}+\frac {70\,d^4\,x^4}{e^4}+\frac {56\,d^5\,x^3}{e^5}+\frac {28\,d^6\,x^2}{e^6}} \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)^(17/2),x)

[Out]

-((d + e*x)^(1/2)*(((a + b*x)^(1/2)*(6006*A*a^7*e^5 + 924*B*a^7*d*e^4 - 18018*A*a^2*b^5*d^5 + 5148*B*a^3*b^4*d
^5 + 64350*A*a^3*b^4*d^4*e - 11440*B*a^4*b^3*d^4*e - 5040*B*a^6*b*d^2*e^3 - 100100*A*a^4*b^3*d^3*e^2 + 81900*A
*a^5*b^2*d^2*e^3 + 10920*B*a^5*b^2*d^3*e^2 - 34650*A*a^6*b*d*e^4))/(45045*e^8*(a*e - b*d)^6) - (x^2*(a + b*x)^
(1/2)*(18018*A*b^7*d^5 + 20592*B*a*b^6*d^5 - 8820*B*a^6*b*e^5 - 126*A*a^5*b^2*e^5 + 1050*A*a^4*b^3*d*e^4 - 111
540*B*a^2*b^5*d^4*e + 52416*B*a^5*b^2*d*e^4 + 8580*A*a^2*b^5*d^3*e^2 - 3900*A*a^3*b^4*d^2*e^3 + 165360*B*a^3*b
^4*d^3*e^2 - 128760*B*a^4*b^3*d^2*e^3 - 12870*A*a*b^6*d^4*e))/(45045*e^8*(a*e - b*d)^6) + (x*(a + b*x)^(1/2)*(
6930*B*a^7*e^5 - 36036*A*a*b^6*d^5 + 7392*A*a^6*b*e^5 - 2574*B*a^2*b^5*d^5 + 102960*A*a^2*b^5*d^4*e - 44100*A*
a^5*b^2*d*e^4 + 44330*B*a^3*b^4*d^4*e - 143000*A*a^3*b^4*d^3*e^2 + 109200*A*a^4*b^3*d^2*e^3 - 91260*B*a^4*b^3*
d^3*e^2 + 84420*B*a^5*b^2*d^2*e^3 - 38262*B*a^6*b*d*e^4))/(45045*e^8*(a*e - b*d)^6) - (256*b^6*x^7*(a + b*x)^(
1/2)*(2*A*b*e - 3*B*a*e + B*b*d))/(45045*e^4*(a*e - b*d)^6) + (16*b^3*x^4*(a + b*x)^(1/2)*(2*A*b*e - 3*B*a*e +
B*b*d)*(a^3*e^3 - 143*b^3*d^3 + 39*a*b^2*d^2*e - 9*a^2*b*d*e^2))/(9009*e^7*(a*e - b*d)^6) + (128*b^5*x^6*(a*e
- 15*b*d)*(a + b*x)^(1/2)*(2*A*b*e - 3*B*a*e + B*b*d))/(45045*e^5*(a*e - b*d)^6) - (2*b^2*x^3*(a + b*x)^(1/2)
*(2*A*b*e - 3*B*a*e + B*b*d)*(7*a^4*e^4 + 1287*b^4*d^4 + 234*a^2*b^2*d^2*e^2 - 572*a*b^3*d^3*e - 60*a^3*b*d*e^
3))/(9009*e^8*(a*e - b*d)^6) - (32*b^4*x^5*(a + b*x)^(1/2)*(a^2*e^2 + 65*b^2*d^2 - 10*a*b*d*e)*(2*A*b*e - 3*B*
a*e + B*b*d))/(15015*e^6*(a*e - b*d)^6)))/(x^8 + d^8/e^8 + (8*d*x^7)/e + (8*d^7*x)/e^7 + (28*d^2*x^6)/e^2 + (5
6*d^3*x^5)/e^3 + (70*d^4*x^4)/e^4 + (56*d^5*x^3)/e^5 + (28*d^6*x^2)/e^6)

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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)**(17/2),x)

[Out]

Timed out

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