3.23.35 \(\int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{19/2}} \, dx\) [2235]

Optimal. Leaf size=309 \[ -\frac {2 (B d-A e) (a+b x)^{7/2}}{17 e (b d-a e) (d+e x)^{17/2}}+\frac {2 (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{255 e (b d-a e)^2 (d+e x)^{15/2}}+\frac {16 b (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{3315 e (b d-a e)^3 (d+e x)^{13/2}}+\frac {32 b^2 (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{12155 e (b d-a e)^4 (d+e x)^{11/2}}+\frac {128 b^3 (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{109395 e (b d-a e)^5 (d+e x)^{9/2}}+\frac {256 b^4 (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{765765 e (b d-a e)^6 (d+e x)^{7/2}} \]

[Out]

-2/17*(-A*e+B*d)*(b*x+a)^(7/2)/e/(-a*e+b*d)/(e*x+d)^(17/2)+2/255*(10*A*b*e-17*B*a*e+7*B*b*d)*(b*x+a)^(7/2)/e/(
-a*e+b*d)^2/(e*x+d)^(15/2)+16/3315*b*(10*A*b*e-17*B*a*e+7*B*b*d)*(b*x+a)^(7/2)/e/(-a*e+b*d)^3/(e*x+d)^(13/2)+3
2/12155*b^2*(10*A*b*e-17*B*a*e+7*B*b*d)*(b*x+a)^(7/2)/e/(-a*e+b*d)^4/(e*x+d)^(11/2)+128/109395*b^3*(10*A*b*e-1
7*B*a*e+7*B*b*d)*(b*x+a)^(7/2)/e/(-a*e+b*d)^5/(e*x+d)^(9/2)+256/765765*b^4*(10*A*b*e-17*B*a*e+7*B*b*d)*(b*x+a)
^(7/2)/e/(-a*e+b*d)^6/(e*x+d)^(7/2)

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Rubi [A]
time = 0.13, antiderivative size = 309, 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 = {79, 47, 37} \begin {gather*} \frac {256 b^4 (a+b x)^{7/2} (-17 a B e+10 A b e+7 b B d)}{765765 e (d+e x)^{7/2} (b d-a e)^6}+\frac {128 b^3 (a+b x)^{7/2} (-17 a B e+10 A b e+7 b B d)}{109395 e (d+e x)^{9/2} (b d-a e)^5}+\frac {32 b^2 (a+b x)^{7/2} (-17 a B e+10 A b e+7 b B d)}{12155 e (d+e x)^{11/2} (b d-a e)^4}+\frac {16 b (a+b x)^{7/2} (-17 a B e+10 A b e+7 b B d)}{3315 e (d+e x)^{13/2} (b d-a e)^3}+\frac {2 (a+b x)^{7/2} (-17 a B e+10 A b e+7 b B d)}{255 e (d+e x)^{15/2} (b d-a e)^2}-\frac {2 (a+b x)^{7/2} (B d-A e)}{17 e (d+e x)^{17/2} (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(19/2),x]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(7/2))/(17*e*(b*d - a*e)*(d + e*x)^(17/2)) + (2*(7*b*B*d + 10*A*b*e - 17*a*B*e)*(a +
b*x)^(7/2))/(255*e*(b*d - a*e)^2*(d + e*x)^(15/2)) + (16*b*(7*b*B*d + 10*A*b*e - 17*a*B*e)*(a + b*x)^(7/2))/(
3315*e*(b*d - a*e)^3*(d + e*x)^(13/2)) + (32*b^2*(7*b*B*d + 10*A*b*e - 17*a*B*e)*(a + b*x)^(7/2))/(12155*e*(b*
d - a*e)^4*(d + e*x)^(11/2)) + (128*b^3*(7*b*B*d + 10*A*b*e - 17*a*B*e)*(a + b*x)^(7/2))/(109395*e*(b*d - a*e)
^5*(d + e*x)^(9/2)) + (256*b^4*(7*b*B*d + 10*A*b*e - 17*a*B*e)*(a + b*x)^(7/2))/(765765*e*(b*d - a*e)^6*(d + e
*x)^(7/2))

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)^{5/2} (A+B x)}{(d+e x)^{19/2}} \, dx &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{17 e (b d-a e) (d+e x)^{17/2}}+\frac {(7 b B d+10 A b e-17 a B e) \int \frac {(a+b x)^{5/2}}{(d+e x)^{17/2}} \, dx}{17 e (b d-a e)}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{17 e (b d-a e) (d+e x)^{17/2}}+\frac {2 (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{255 e (b d-a e)^2 (d+e x)^{15/2}}+\frac {(8 b (7 b B d+10 A b e-17 a B e)) \int \frac {(a+b x)^{5/2}}{(d+e x)^{15/2}} \, dx}{255 e (b d-a e)^2}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{17 e (b d-a e) (d+e x)^{17/2}}+\frac {2 (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{255 e (b d-a e)^2 (d+e x)^{15/2}}+\frac {16 b (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{3315 e (b d-a e)^3 (d+e x)^{13/2}}+\frac {\left (16 b^2 (7 b B d+10 A b e-17 a B e)\right ) \int \frac {(a+b x)^{5/2}}{(d+e x)^{13/2}} \, dx}{1105 e (b d-a e)^3}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{17 e (b d-a e) (d+e x)^{17/2}}+\frac {2 (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{255 e (b d-a e)^2 (d+e x)^{15/2}}+\frac {16 b (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{3315 e (b d-a e)^3 (d+e x)^{13/2}}+\frac {32 b^2 (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{12155 e (b d-a e)^4 (d+e x)^{11/2}}+\frac {\left (64 b^3 (7 b B d+10 A b e-17 a B e)\right ) \int \frac {(a+b x)^{5/2}}{(d+e x)^{11/2}} \, dx}{12155 e (b d-a e)^4}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{17 e (b d-a e) (d+e x)^{17/2}}+\frac {2 (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{255 e (b d-a e)^2 (d+e x)^{15/2}}+\frac {16 b (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{3315 e (b d-a e)^3 (d+e x)^{13/2}}+\frac {32 b^2 (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{12155 e (b d-a e)^4 (d+e x)^{11/2}}+\frac {128 b^3 (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{109395 e (b d-a e)^5 (d+e x)^{9/2}}+\frac {\left (128 b^4 (7 b B d+10 A b e-17 a B e)\right ) \int \frac {(a+b x)^{5/2}}{(d+e x)^{9/2}} \, dx}{109395 e (b d-a e)^5}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{17 e (b d-a e) (d+e x)^{17/2}}+\frac {2 (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{255 e (b d-a e)^2 (d+e x)^{15/2}}+\frac {16 b (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{3315 e (b d-a e)^3 (d+e x)^{13/2}}+\frac {32 b^2 (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{12155 e (b d-a e)^4 (d+e x)^{11/2}}+\frac {128 b^3 (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{109395 e (b d-a e)^5 (d+e x)^{9/2}}+\frac {256 b^4 (7 b B d+10 A b e-17 a B e) (a+b x)^{7/2}}{765765 e (b d-a e)^6 (d+e x)^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 0.69, size = 341, normalized size = 1.10 \begin {gather*} \frac {2 (a+b x)^{7/2} \left (45045 B d e^4 (a+b x)^5-45045 A e^5 (a+b x)^5-204204 b B d e^3 (a+b x)^4 (d+e x)+255255 A b e^4 (a+b x)^4 (d+e x)-51051 a B e^4 (a+b x)^4 (d+e x)+353430 b^2 B d e^2 (a+b x)^3 (d+e x)^2-589050 A b^2 e^3 (a+b x)^3 (d+e x)^2+235620 a b B e^3 (a+b x)^3 (d+e x)^2-278460 b^3 B d e (a+b x)^2 (d+e x)^3+696150 A b^3 e^2 (a+b x)^2 (d+e x)^3-417690 a b^2 B e^2 (a+b x)^2 (d+e x)^3+85085 b^4 B d (a+b x) (d+e x)^4-425425 A b^4 e (a+b x) (d+e x)^4+340340 a b^3 B e (a+b x) (d+e x)^4+109395 A b^5 (d+e x)^5-109395 a b^4 B (d+e x)^5\right )}{765765 (b d-a e)^6 (d+e x)^{17/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(19/2),x]

[Out]

(2*(a + b*x)^(7/2)*(45045*B*d*e^4*(a + b*x)^5 - 45045*A*e^5*(a + b*x)^5 - 204204*b*B*d*e^3*(a + b*x)^4*(d + e*
x) + 255255*A*b*e^4*(a + b*x)^4*(d + e*x) - 51051*a*B*e^4*(a + b*x)^4*(d + e*x) + 353430*b^2*B*d*e^2*(a + b*x)
^3*(d + e*x)^2 - 589050*A*b^2*e^3*(a + b*x)^3*(d + e*x)^2 + 235620*a*b*B*e^3*(a + b*x)^3*(d + e*x)^2 - 278460*
b^3*B*d*e*(a + b*x)^2*(d + e*x)^3 + 696150*A*b^3*e^2*(a + b*x)^2*(d + e*x)^3 - 417690*a*b^2*B*e^2*(a + b*x)^2*
(d + e*x)^3 + 85085*b^4*B*d*(a + b*x)*(d + e*x)^4 - 425425*A*b^4*e*(a + b*x)*(d + e*x)^4 + 340340*a*b^3*B*e*(a
+ b*x)*(d + e*x)^4 + 109395*A*b^5*(d + e*x)^5 - 109395*a*b^4*B*(d + e*x)^5))/(765765*(b*d - a*e)^6*(d + e*x)^
(17/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1040\) vs. \(2(273)=546\).
time = 0.09, size = 1041, normalized size = 3.37

method	result	size
gosper	\(-\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (-1280 A \,b^{5} e^{5} x^{5}+2176 B a \,b^{4} e^{5} x^{5}-896 B \,b^{5} d \,e^{4} x^{5}+4480 A a \,b^{4} e^{5} x^{4}-10880 A \,b^{5} d \,e^{4} x^{4}-7616 B \,a^{2} b^{3} e^{5} x^{4}+21632 B a \,b^{4} d \,e^{4} x^{4}-7616 B \,b^{5} d^{2} e^{3} x^{4}-10080 A \,a^{2} b^{3} e^{5} x^{3}+38080 A a \,b^{4} d \,e^{4} x^{3}-40800 A \,b^{5} d^{2} e^{3} x^{3}+17136 B \,a^{3} b^{2} e^{5} x^{3}-71792 B \,a^{2} b^{3} d \,e^{4} x^{3}+96016 B a \,b^{4} d^{2} e^{3} x^{3}-28560 B \,b^{5} d^{3} e^{2} x^{3}+18480 A \,a^{3} b^{2} e^{5} x^{2}-85680 A \,a^{2} b^{3} d \,e^{4} x^{2}+142800 A a \,b^{4} d^{2} e^{3} x^{2}-88400 A \,b^{5} d^{3} e^{2} x^{2}-31416 B \,a^{4} b \,e^{5} x^{2}+158592 B \,a^{3} b^{2} d \,e^{4} x^{2}-302736 B \,a^{2} b^{3} d^{2} e^{3} x^{2}+250240 B a \,b^{4} d^{3} e^{2} x^{2}-61880 B \,b^{5} d^{4} e \,x^{2}-30030 A \,a^{4} b \,e^{5} x +157080 A \,a^{3} b^{2} d \,e^{4} x -321300 A \,a^{2} b^{3} d^{2} e^{3} x +309400 A a \,b^{4} d^{3} e^{2} x -121550 A \,b^{5} d^{4} e x +51051 B \,a^{5} e^{5} x -288057 B \,a^{4} b d \,e^{4} x +656166 B \,a^{3} b^{2} d^{2} e^{3} x -750890 B \,a^{2} b^{3} d^{3} e^{2} x +423215 B a \,b^{4} d^{4} e x -85085 B \,b^{5} d^{5} x +45045 A \,a^{5} e^{5}-255255 A \,a^{4} b d \,e^{4}+589050 A \,a^{3} b^{2} d^{2} e^{3}-696150 A \,a^{2} b^{3} d^{3} e^{2}+425425 A a \,b^{4} d^{4} e -109395 A \,b^{5} d^{5}+6006 B \,a^{5} d \,e^{4}-31416 B \,a^{4} b \,d^{2} e^{3}+64260 B \,a^{3} b^{2} d^{3} e^{2}-61880 B \,a^{2} b^{3} d^{4} e +24310 B a \,b^{4} d^{5}\right )}{765765 \left (e x +d \right )^{\frac {17}{2}} \left (a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}\right )}\)	\(722\)
default	\(-\frac {2 \left (-1280 A \,b^{7} e^{5} x^{7}+2176 B a \,b^{6} e^{5} x^{7}-896 B \,b^{7} d \,e^{4} x^{7}+1920 A a \,b^{6} e^{5} x^{6}-10880 A \,b^{7} d \,e^{4} x^{6}-3264 B \,a^{2} b^{5} e^{5} x^{6}+19840 B a \,b^{6} d \,e^{4} x^{6}-7616 B \,b^{7} d^{2} e^{3} x^{6}-2400 A \,a^{2} b^{5} e^{5} x^{5}+16320 A a \,b^{6} d \,e^{4} x^{5}-40800 A \,b^{7} d^{2} e^{3} x^{5}+4080 B \,a^{3} b^{4} e^{5} x^{5}-29424 B \,a^{2} b^{5} d \,e^{4} x^{5}+80784 B a \,b^{6} d^{2} e^{3} x^{5}-28560 B \,b^{7} d^{3} e^{2} x^{5}+2800 A \,a^{3} b^{4} e^{5} x^{4}-20400 A \,a^{2} b^{5} d \,e^{4} x^{4}+61200 A a \,b^{6} d^{2} e^{3} x^{4}-88400 A \,b^{7} d^{3} e^{2} x^{4}-4760 B \,a^{4} b^{3} e^{5} x^{4}+36640 B \,a^{3} b^{4} d \,e^{4} x^{4}-118320 B \,a^{2} b^{5} d^{2} e^{3} x^{4}+193120 B a \,b^{6} d^{3} e^{2} x^{4}-61880 B \,b^{7} d^{4} e \,x^{4}-3150 A \,a^{4} b^{3} e^{5} x^{3}+23800 A \,a^{3} b^{4} d \,e^{4} x^{3}-76500 A \,a^{2} b^{5} d^{2} e^{3} x^{3}+132600 A a \,b^{6} d^{3} e^{2} x^{3}-121550 A \,b^{7} d^{4} e \,x^{3}+5355 B \,a^{5} b^{2} e^{5} x^{3}-42665 B \,a^{4} b^{3} d \,e^{4} x^{3}+146710 B \,a^{3} b^{4} d^{2} e^{3} x^{3}-278970 B \,a^{2} b^{5} d^{3} e^{2} x^{3}+299455 B a \,b^{6} d^{4} e \,x^{3}-85085 B \,b^{7} d^{5} x^{3}+3465 A \,a^{5} b^{2} e^{5} x^{2}-26775 A \,a^{4} b^{3} d \,e^{4} x^{2}+89250 A \,a^{3} b^{4} d^{2} e^{3} x^{2}-165750 A \,a^{2} b^{5} d^{3} e^{2} x^{2}+182325 A a \,b^{6} d^{4} e \,x^{2}-109395 A \,b^{7} d^{5} x^{2}+70686 B \,a^{6} b \,e^{5} x^{2}-411516 B \,a^{5} b^{2} d \,e^{4} x^{2}+978180 B \,a^{4} b^{3} d^{2} e^{3} x^{2}-1187280 B \,a^{3} b^{4} d^{3} e^{2} x^{2}+722670 B \,a^{2} b^{5} d^{4} e \,x^{2}-145860 B a \,b^{6} d^{5} x^{2}+60060 A \,a^{6} b \,e^{5} x -353430 A \,a^{5} b^{2} d \,e^{4} x +856800 A \,a^{4} b^{3} d^{2} e^{3} x -1082900 A \,a^{3} b^{4} d^{3} e^{2} x +729300 A \,a^{2} b^{5} d^{4} e x -218790 A a \,b^{6} d^{5} x +51051 B \,a^{7} e^{5} x -276045 B \,a^{6} b d \,e^{4} x +593334 B \,a^{5} b^{2} d^{2} e^{3} x -622370 B \,a^{4} b^{3} d^{3} e^{2} x +299455 B \,a^{3} b^{4} d^{4} e x -36465 B \,a^{2} b^{5} d^{5} x +45045 A \,a^{7} e^{5}-255255 A \,a^{6} b d \,e^{4}+589050 A \,a^{5} b^{2} d^{2} e^{3}-696150 A \,a^{4} b^{3} d^{3} e^{2}+425425 A \,a^{3} b^{4} d^{4} e -109395 A \,a^{2} b^{5} d^{5}+6006 B \,a^{7} d \,e^{4}-31416 B \,a^{6} b \,d^{2} e^{3}+64260 B \,a^{5} b^{2} d^{3} e^{2}-61880 B \,a^{4} b^{3} d^{4} e +24310 B \,a^{3} b^{4} d^{5}\right ) \left (b x +a \right )^{\frac {3}{2}}}{765765 \left (e x +d \right )^{\frac {17}{2}} \left (a e -b d \right )^{6}}\)	\(1041\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(19/2),x,method=_RETURNVERBOSE)

[Out]

-2/765765*(-1280*A*b^7*e^5*x^7+2176*B*a*b^6*e^5*x^7-896*B*b^7*d*e^4*x^7+1920*A*a*b^6*e^5*x^6-10880*A*b^7*d*e^4
*x^6-3264*B*a^2*b^5*e^5*x^6+19840*B*a*b^6*d*e^4*x^6-7616*B*b^7*d^2*e^3*x^6-2400*A*a^2*b^5*e^5*x^5+16320*A*a*b^
6*d*e^4*x^5-40800*A*b^7*d^2*e^3*x^5+4080*B*a^3*b^4*e^5*x^5-29424*B*a^2*b^5*d*e^4*x^5+80784*B*a*b^6*d^2*e^3*x^5
-28560*B*b^7*d^3*e^2*x^5+2800*A*a^3*b^4*e^5*x^4-20400*A*a^2*b^5*d*e^4*x^4+61200*A*a*b^6*d^2*e^3*x^4-88400*A*b^
7*d^3*e^2*x^4-4760*B*a^4*b^3*e^5*x^4+36640*B*a^3*b^4*d*e^4*x^4-118320*B*a^2*b^5*d^2*e^3*x^4+193120*B*a*b^6*d^3
*e^2*x^4-61880*B*b^7*d^4*e*x^4-3150*A*a^4*b^3*e^5*x^3+23800*A*a^3*b^4*d*e^4*x^3-76500*A*a^2*b^5*d^2*e^3*x^3+13
2600*A*a*b^6*d^3*e^2*x^3-121550*A*b^7*d^4*e*x^3+5355*B*a^5*b^2*e^5*x^3-42665*B*a^4*b^3*d*e^4*x^3+146710*B*a^3*
b^4*d^2*e^3*x^3-278970*B*a^2*b^5*d^3*e^2*x^3+299455*B*a*b^6*d^4*e*x^3-85085*B*b^7*d^5*x^3+3465*A*a^5*b^2*e^5*x
^2-26775*A*a^4*b^3*d*e^4*x^2+89250*A*a^3*b^4*d^2*e^3*x^2-165750*A*a^2*b^5*d^3*e^2*x^2+182325*A*a*b^6*d^4*e*x^2
-109395*A*b^7*d^5*x^2+70686*B*a^6*b*e^5*x^2-411516*B*a^5*b^2*d*e^4*x^2+978180*B*a^4*b^3*d^2*e^3*x^2-1187280*B*
a^3*b^4*d^3*e^2*x^2+722670*B*a^2*b^5*d^4*e*x^2-145860*B*a*b^6*d^5*x^2+60060*A*a^6*b*e^5*x-353430*A*a^5*b^2*d*e
^4*x+856800*A*a^4*b^3*d^2*e^3*x-1082900*A*a^3*b^4*d^3*e^2*x+729300*A*a^2*b^5*d^4*e*x-218790*A*a*b^6*d^5*x+5105
1*B*a^7*e^5*x-276045*B*a^6*b*d*e^4*x+593334*B*a^5*b^2*d^2*e^3*x-622370*B*a^4*b^3*d^3*e^2*x+299455*B*a^3*b^4*d^
4*e*x-36465*B*a^2*b^5*d^5*x+45045*A*a^7*e^5-255255*A*a^6*b*d*e^4+589050*A*a^5*b^2*d^2*e^3-696150*A*a^4*b^3*d^3
*e^2+425425*A*a^3*b^4*d^4*e-109395*A*a^2*b^5*d^5+6006*B*a^7*d*e^4-31416*B*a^6*b*d^2*e^3+64260*B*a^5*b^2*d^3*e^
2-61880*B*a^4*b^3*d^4*e+24310*B*a^3*b^4*d^5)*(b*x+a)^(3/2)/(e*x+d)^(17/2)/(a*e-b*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)^(5/2)*(B*x+A)/(e*x+d)^(19/2),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 [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)^(5/2)*(B*x+A)/(e*x+d)^(19/2),x, algorithm="fricas")

[Out]

Timed out

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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)**(5/2)*(B*x+A)/(e*x+d)**(19/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 8437 deep

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1752 vs. \(2 (290) = 580\).
time = 2.17, size = 1752, normalized size = 5.67 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(19/2),x, algorithm="giac")

[Out]

2/765765*((8*(2*(4*(b*x + a)*(2*(7*B*b^20*d^3*abs(b)*e^12 - 31*B*a*b^19*d^2*abs(b)*e^13 + 10*A*b^20*d^2*abs(b)
*e^13 + 41*B*a^2*b^18*d*abs(b)*e^14 - 20*A*a*b^19*d*abs(b)*e^14 - 17*B*a^3*b^17*abs(b)*e^15 + 10*A*a^2*b^18*ab
s(b)*e^15)*(b*x + a)/(b^10*d^8*e^8 - 8*a*b^9*d^7*e^9 + 28*a^2*b^8*d^6*e^10 - 56*a^3*b^7*d^5*e^11 + 70*a^4*b^6*
d^4*e^12 - 56*a^5*b^5*d^3*e^13 + 28*a^6*b^4*d^2*e^14 - 8*a^7*b^3*d*e^15 + a^8*b^2*e^16) + 17*(7*B*b^21*d^4*abs
(b)*e^11 - 38*B*a*b^20*d^3*abs(b)*e^12 + 10*A*b^21*d^3*abs(b)*e^12 + 72*B*a^2*b^19*d^2*abs(b)*e^13 - 30*A*a*b^
20*d^2*abs(b)*e^13 - 58*B*a^3*b^18*d*abs(b)*e^14 + 30*A*a^2*b^19*d*abs(b)*e^14 + 17*B*a^4*b^17*abs(b)*e^15 - 1
0*A*a^3*b^18*abs(b)*e^15)/(b^10*d^8*e^8 - 8*a*b^9*d^7*e^9 + 28*a^2*b^8*d^6*e^10 - 56*a^3*b^7*d^5*e^11 + 70*a^4
*b^6*d^4*e^12 - 56*a^5*b^5*d^3*e^13 + 28*a^6*b^4*d^2*e^14 - 8*a^7*b^3*d*e^15 + a^8*b^2*e^16)) + 255*(7*B*b^22*
d^5*abs(b)*e^10 - 45*B*a*b^21*d^4*abs(b)*e^11 + 10*A*b^22*d^4*abs(b)*e^11 + 110*B*a^2*b^20*d^3*abs(b)*e^12 - 4
0*A*a*b^21*d^3*abs(b)*e^12 - 130*B*a^3*b^19*d^2*abs(b)*e^13 + 60*A*a^2*b^20*d^2*abs(b)*e^13 + 75*B*a^4*b^18*d*
abs(b)*e^14 - 40*A*a^3*b^19*d*abs(b)*e^14 - 17*B*a^5*b^17*abs(b)*e^15 + 10*A*a^4*b^18*abs(b)*e^15)/(b^10*d^8*e
^8 - 8*a*b^9*d^7*e^9 + 28*a^2*b^8*d^6*e^10 - 56*a^3*b^7*d^5*e^11 + 70*a^4*b^6*d^4*e^12 - 56*a^5*b^5*d^3*e^13 +
28*a^6*b^4*d^2*e^14 - 8*a^7*b^3*d*e^15 + a^8*b^2*e^16))*(b*x + a) + 1105*(7*B*b^23*d^6*abs(b)*e^9 - 52*B*a*b^
22*d^5*abs(b)*e^10 + 10*A*b^23*d^5*abs(b)*e^10 + 155*B*a^2*b^21*d^4*abs(b)*e^11 - 50*A*a*b^22*d^4*abs(b)*e^11
- 240*B*a^3*b^20*d^3*abs(b)*e^12 + 100*A*a^2*b^21*d^3*abs(b)*e^12 + 205*B*a^4*b^19*d^2*abs(b)*e^13 - 100*A*a^3
*b^20*d^2*abs(b)*e^13 - 92*B*a^5*b^18*d*abs(b)*e^14 + 50*A*a^4*b^19*d*abs(b)*e^14 + 17*B*a^6*b^17*abs(b)*e^15
- 10*A*a^5*b^18*abs(b)*e^15)/(b^10*d^8*e^8 - 8*a*b^9*d^7*e^9 + 28*a^2*b^8*d^6*e^10 - 56*a^3*b^7*d^5*e^11 + 70*
a^4*b^6*d^4*e^12 - 56*a^5*b^5*d^3*e^13 + 28*a^6*b^4*d^2*e^14 - 8*a^7*b^3*d*e^15 + a^8*b^2*e^16))*(b*x + a) + 1
2155*(7*B*b^24*d^7*abs(b)*e^8 - 59*B*a*b^23*d^6*abs(b)*e^9 + 10*A*b^24*d^6*abs(b)*e^9 + 207*B*a^2*b^22*d^5*abs
(b)*e^10 - 60*A*a*b^23*d^5*abs(b)*e^10 - 395*B*a^3*b^21*d^4*abs(b)*e^11 + 150*A*a^2*b^22*d^4*abs(b)*e^11 + 445
*B*a^4*b^20*d^3*abs(b)*e^12 - 200*A*a^3*b^21*d^3*abs(b)*e^12 - 297*B*a^5*b^19*d^2*abs(b)*e^13 + 150*A*a^4*b^20
*d^2*abs(b)*e^13 + 109*B*a^6*b^18*d*abs(b)*e^14 - 60*A*a^5*b^19*d*abs(b)*e^14 - 17*B*a^7*b^17*abs(b)*e^15 + 10
*A*a^6*b^18*abs(b)*e^15)/(b^10*d^8*e^8 - 8*a*b^9*d^7*e^9 + 28*a^2*b^8*d^6*e^10 - 56*a^3*b^7*d^5*e^11 + 70*a^4*
b^6*d^4*e^12 - 56*a^5*b^5*d^3*e^13 + 28*a^6*b^4*d^2*e^14 - 8*a^7*b^3*d*e^15 + a^8*b^2*e^16))*(b*x + a) - 10939
5*(B*a*b^24*d^7*abs(b)*e^8 - A*b^25*d^7*abs(b)*e^8 - 7*B*a^2*b^23*d^6*abs(b)*e^9 + 7*A*a*b^24*d^6*abs(b)*e^9 +
21*B*a^3*b^22*d^5*abs(b)*e^10 - 21*A*a^2*b^23*d^5*abs(b)*e^10 - 35*B*a^4*b^21*d^4*abs(b)*e^11 + 35*A*a^3*b^22
*d^4*abs(b)*e^11 + 35*B*a^5*b^20*d^3*abs(b)*e^12 - 35*A*a^4*b^21*d^3*abs(b)*e^12 - 21*B*a^6*b^19*d^2*abs(b)*e^
13 + 21*A*a^5*b^20*d^2*abs(b)*e^13 + 7*B*a^7*b^18*d*abs(b)*e^14 - 7*A*a^6*b^19*d*abs(b)*e^14 - B*a^8*b^17*abs(
b)*e^15 + A*a^7*b^18*abs(b)*e^15)/(b^10*d^8*e^8 - 8*a*b^9*d^7*e^9 + 28*a^2*b^8*d^6*e^10 - 56*a^3*b^7*d^5*e^11
+ 70*a^4*b^6*d^4*e^12 - 56*a^5*b^5*d^3*e^13 + 28*a^6*b^4*d^2*e^14 - 8*a^7*b^3*d*e^15 + a^8*b^2*e^16))*(b*x + a
)^(7/2)/(b^2*d + (b*x + a)*b*e - a*b*e)^(17/2)

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Mupad [B]
time = 4.46, size = 1144, normalized size = 3.70 \begin {gather*} \frac {\sqrt {d+e\,x}\,\left (\frac {x^2\,\sqrt {a+b\,x}\,\left (-243474\,B\,a^7\,b\,e^5+1375122\,B\,a^6\,b^2\,d\,e^4-127050\,A\,a^6\,b^2\,e^5-3143028\,B\,a^5\,b^3\,d^2\,e^3+760410\,A\,a^5\,b^3\,d\,e^4+3619300\,B\,a^4\,b^4\,d^3\,e^2-1892100\,A\,a^4\,b^4\,d^2\,e^3-2044250\,B\,a^3\,b^5\,d^4\,e+2497300\,A\,a^3\,b^5\,d^3\,e^2+364650\,B\,a^2\,b^6\,d^5-1823250\,A\,a^2\,b^6\,d^4\,e+656370\,A\,a\,b^7\,d^5\right )}{765765\,e^9\,{\left (a\,e-b\,d\right )}^6}-\frac {x\,\sqrt {a+b\,x}\,\left (102102\,B\,a^8\,e^5-540078\,B\,a^7\,b\,d\,e^4+210210\,A\,a^7\,b\,e^5+1123836\,B\,a^6\,b^2\,d^2\,e^3-1217370\,A\,a^6\,b^2\,d\,e^4-1116220\,B\,a^5\,b^3\,d^3\,e^2+2891700\,A\,a^5\,b^3\,d^2\,e^3+475150\,B\,a^4\,b^4\,d^4\,e-3558100\,A\,a^4\,b^4\,d^3\,e^2-24310\,B\,a^3\,b^5\,d^5+2309450\,A\,a^3\,b^5\,d^4\,e-656370\,A\,a^2\,b^6\,d^5\right )}{765765\,e^9\,{\left (a\,e-b\,d\right )}^6}-\frac {\sqrt {a+b\,x}\,\left (12012\,B\,a^8\,d\,e^4+90090\,A\,a^8\,e^5-62832\,B\,a^7\,b\,d^2\,e^3-510510\,A\,a^7\,b\,d\,e^4+128520\,B\,a^6\,b^2\,d^3\,e^2+1178100\,A\,a^6\,b^2\,d^2\,e^3-123760\,B\,a^5\,b^3\,d^4\,e-1392300\,A\,a^5\,b^3\,d^3\,e^2+48620\,B\,a^4\,b^4\,d^5+850850\,A\,a^4\,b^4\,d^4\,e-218790\,A\,a^3\,b^5\,d^5\right )}{765765\,e^9\,{\left (a\,e-b\,d\right )}^6}+\frac {x^3\,\sqrt {a+b\,x}\,\left (-152082\,B\,a^6\,b^2\,e^5+908362\,B\,a^5\,b^3\,d\,e^4-630\,A\,a^5\,b^3\,e^5-2249780\,B\,a^4\,b^4\,d^2\,e^3+5950\,A\,a^4\,b^4\,d\,e^4+2932500\,B\,a^3\,b^5\,d^3\,e^2-25500\,A\,a^3\,b^5\,d^2\,e^3-2044250\,B\,a^2\,b^6\,d^4\,e+66300\,A\,a^2\,b^6\,d^3\,e^2+461890\,B\,a\,b^7\,d^5-121550\,A\,a\,b^7\,d^4\,e+218790\,A\,b^8\,d^5\right )}{765765\,e^9\,{\left (a\,e-b\,d\right )}^6}+\frac {256\,b^7\,x^8\,\sqrt {a+b\,x}\,\left (10\,A\,b\,e-17\,B\,a\,e+7\,B\,b\,d\right )}{765765\,e^5\,{\left (a\,e-b\,d\right )}^6}-\frac {16\,b^4\,x^5\,\sqrt {a+b\,x}\,\left (10\,A\,b\,e-17\,B\,a\,e+7\,B\,b\,d\right )\,\left (5\,a^3\,e^3-51\,a^2\,b\,d\,e^2+255\,a\,b^2\,d^2\,e-1105\,b^3\,d^3\right )}{765765\,e^8\,{\left (a\,e-b\,d\right )}^6}-\frac {128\,b^6\,x^7\,\left (a\,e-17\,b\,d\right )\,\sqrt {a+b\,x}\,\left (10\,A\,b\,e-17\,B\,a\,e+7\,B\,b\,d\right )}{765765\,e^6\,{\left (a\,e-b\,d\right )}^6}+\frac {2\,b^3\,x^4\,\sqrt {a+b\,x}\,\left (10\,A\,b\,e-17\,B\,a\,e+7\,B\,b\,d\right )\,\left (7\,a^4\,e^4-68\,a^3\,b\,d\,e^3+306\,a^2\,b^2\,d^2\,e^2-884\,a\,b^3\,d^3\,e+2431\,b^4\,d^4\right )}{153153\,e^9\,{\left (a\,e-b\,d\right )}^6}+\frac {32\,b^5\,x^6\,\sqrt {a+b\,x}\,\left (3\,a^2\,e^2-34\,a\,b\,d\,e+255\,b^2\,d^2\right )\,\left (10\,A\,b\,e-17\,B\,a\,e+7\,B\,b\,d\right )}{765765\,e^7\,{\left (a\,e-b\,d\right )}^6}\right )}{x^9+\frac {d^9}{e^9}+\frac {9\,d\,x^8}{e}+\frac {9\,d^8\,x}{e^8}+\frac {36\,d^2\,x^7}{e^2}+\frac {84\,d^3\,x^6}{e^3}+\frac {126\,d^4\,x^5}{e^4}+\frac {126\,d^5\,x^4}{e^5}+\frac {84\,d^6\,x^3}{e^6}+\frac {36\,d^7\,x^2}{e^7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^(5/2))/(d + e*x)^(19/2),x)

[Out]

((d + e*x)^(1/2)*((x^2*(a + b*x)^(1/2)*(656370*A*a*b^7*d^5 - 243474*B*a^7*b*e^5 - 127050*A*a^6*b^2*e^5 + 36465
0*B*a^2*b^6*d^5 - 1823250*A*a^2*b^6*d^4*e + 760410*A*a^5*b^3*d*e^4 - 2044250*B*a^3*b^5*d^4*e + 1375122*B*a^6*b
^2*d*e^4 + 2497300*A*a^3*b^5*d^3*e^2 - 1892100*A*a^4*b^4*d^2*e^3 + 3619300*B*a^4*b^4*d^3*e^2 - 3143028*B*a^5*b
^3*d^2*e^3))/(765765*e^9*(a*e - b*d)^6) - (x*(a + b*x)^(1/2)*(102102*B*a^8*e^5 + 210210*A*a^7*b*e^5 - 656370*A
*a^2*b^6*d^5 - 24310*B*a^3*b^5*d^5 + 2309450*A*a^3*b^5*d^4*e - 1217370*A*a^6*b^2*d*e^4 + 475150*B*a^4*b^4*d^4*
e - 3558100*A*a^4*b^4*d^3*e^2 + 2891700*A*a^5*b^3*d^2*e^3 - 1116220*B*a^5*b^3*d^3*e^2 + 1123836*B*a^6*b^2*d^2*
e^3 - 540078*B*a^7*b*d*e^4))/(765765*e^9*(a*e - b*d)^6) - ((a + b*x)^(1/2)*(90090*A*a^8*e^5 + 12012*B*a^8*d*e^
4 - 218790*A*a^3*b^5*d^5 + 48620*B*a^4*b^4*d^5 + 850850*A*a^4*b^4*d^4*e - 123760*B*a^5*b^3*d^4*e - 62832*B*a^7
*b*d^2*e^3 - 1392300*A*a^5*b^3*d^3*e^2 + 1178100*A*a^6*b^2*d^2*e^3 + 128520*B*a^6*b^2*d^3*e^2 - 510510*A*a^7*b
*d*e^4))/(765765*e^9*(a*e - b*d)^6) + (x^3*(a + b*x)^(1/2)*(218790*A*b^8*d^5 + 461890*B*a*b^7*d^5 - 630*A*a^5*
b^3*e^5 - 152082*B*a^6*b^2*e^5 + 5950*A*a^4*b^4*d*e^4 - 2044250*B*a^2*b^6*d^4*e + 908362*B*a^5*b^3*d*e^4 + 663
00*A*a^2*b^6*d^3*e^2 - 25500*A*a^3*b^5*d^2*e^3 + 2932500*B*a^3*b^5*d^3*e^2 - 2249780*B*a^4*b^4*d^2*e^3 - 12155
0*A*a*b^7*d^4*e))/(765765*e^9*(a*e - b*d)^6) + (256*b^7*x^8*(a + b*x)^(1/2)*(10*A*b*e - 17*B*a*e + 7*B*b*d))/(
765765*e^5*(a*e - b*d)^6) - (16*b^4*x^5*(a + b*x)^(1/2)*(10*A*b*e - 17*B*a*e + 7*B*b*d)*(5*a^3*e^3 - 1105*b^3*
d^3 + 255*a*b^2*d^2*e - 51*a^2*b*d*e^2))/(765765*e^8*(a*e - b*d)^6) - (128*b^6*x^7*(a*e - 17*b*d)*(a + b*x)^(1
/2)*(10*A*b*e - 17*B*a*e + 7*B*b*d))/(765765*e^6*(a*e - b*d)^6) + (2*b^3*x^4*(a + b*x)^(1/2)*(10*A*b*e - 17*B*
a*e + 7*B*b*d)*(7*a^4*e^4 + 2431*b^4*d^4 + 306*a^2*b^2*d^2*e^2 - 884*a*b^3*d^3*e - 68*a^3*b*d*e^3))/(153153*e^
9*(a*e - b*d)^6) + (32*b^5*x^6*(a + b*x)^(1/2)*(3*a^2*e^2 + 255*b^2*d^2 - 34*a*b*d*e)*(10*A*b*e - 17*B*a*e + 7
*B*b*d))/(765765*e^7*(a*e - b*d)^6)))/(x^9 + d^9/e^9 + (9*d*x^8)/e + (9*d^8*x)/e^8 + (36*d^2*x^7)/e^2 + (84*d^
3*x^6)/e^3 + (126*d^4*x^5)/e^4 + (126*d^5*x^4)/e^5 + (84*d^6*x^3)/e^6 + (36*d^7*x^2)/e^7)

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