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

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

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Rubi [A]  time = 0.16, 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 = {78, 45, 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 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)^{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 = 0.48, size = 135, normalized size = 0.53 \begin {gather*} \frac {2 (a+b x)^{5/2} \left (1155 (B d-A e)-\frac {(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 ) (-13 a B e+8 A b e+5 b B d)}{(b d-a e)^4}\right )}{15015 e (d+e x)^{13/2} (a e-b d)} \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 - A*e) - ((5*b*B*d + 8*A*b*e - 13*a*B*e)*(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)^4))/(15015*e*(-(b*d) + a*e)*(
d + e*x)^(13/2))

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IntegrateAlgebraic [A]  time = 0.42, size = 276, normalized size = 1.08 \begin {gather*} -\frac {2 (a+b x)^{13/2} \left (-\frac {3003 A b^4 (d+e x)^4}{(a+b x)^4}+\frac {8580 A b^3 e (d+e x)^3}{(a+b x)^3}-\frac {10010 A b^2 e^2 (d+e x)^2}{(a+b x)^2}+\frac {5460 A b e^3 (d+e x)}{a+b x}+\frac {3003 a b^3 B (d+e x)^4}{(a+b x)^4}-\frac {2145 b^3 B d (d+e x)^3}{(a+b x)^3}-\frac {6435 a b^2 B e (d+e x)^3}{(a+b x)^3}+\frac {5005 b^2 B d e (d+e x)^2}{(a+b x)^2}-\frac {1365 a B e^3 (d+e x)}{a+b x}+\frac {5005 a b B e^2 (d+e x)^2}{(a+b x)^2}-\frac {4095 b B d e^2 (d+e x)}{a+b x}-1155 A e^4+1155 B d e^3\right )}{15015 (d+e x)^{13/2} (b d-a e)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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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)^(15/2),x, algorithm="fricas")

[Out]

Timed out

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giac [B]  time = 6.95, size = 1091, normalized size = 4.28

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)^(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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maple [B]  time = 0.01, size = 505, normalized size = 1.98 \begin {gather*} -\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 )} \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)^(15/2),x)

[Out]

-2/15015*(b*x+a)^(5/2)*(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^4-546
0*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)/(e*x+d)^(13/2)/(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)

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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(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 = 3.11, 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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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)**(15/2),x)

[Out]

Timed out

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