3.22.22 \(\int \frac {A+B x}{(a+b x)^{5/2} (d+e x)^{9/2}} \, dx\)
Optimal. Leaf size=298 \[ -\frac {256 b^2 e \sqrt {a+b x} (7 a B e-10 A b e+3 b B d)}{105 \sqrt {d+e x} (b d-a e)^6}-\frac {128 b e \sqrt {a+b x} (7 a B e-10 A b e+3 b B d)}{105 (d+e x)^{3/2} (b d-a e)^5}-\frac {32 e \sqrt {a+b x} (7 a B e-10 A b e+3 b B d)}{35 (d+e x)^{5/2} (b d-a e)^4}-\frac {16 e \sqrt {a+b x} (7 a B e-10 A b e+3 b B d)}{21 b (d+e x)^{7/2} (b d-a e)^3}-\frac {2 (7 a B e-10 A b e+3 b B d)}{3 b \sqrt {a+b x} (d+e x)^{7/2} (b d-a e)^2}-\frac {2 (A b-a B)}{3 b (a+b x)^{3/2} (d+e x)^{7/2} (b d-a e)} \]
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Rubi [A] time = 0.21, antiderivative size = 298, 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^2 e \sqrt {a+b x} (7 a B e-10 A b e+3 b B d)}{105 \sqrt {d+e x} (b d-a e)^6}-\frac {128 b e \sqrt {a+b x} (7 a B e-10 A b e+3 b B d)}{105 (d+e x)^{3/2} (b d-a e)^5}-\frac {32 e \sqrt {a+b x} (7 a B e-10 A b e+3 b B d)}{35 (d+e x)^{5/2} (b d-a e)^4}-\frac {16 e \sqrt {a+b x} (7 a B e-10 A b e+3 b B d)}{21 b (d+e x)^{7/2} (b d-a e)^3}-\frac {2 (7 a B e-10 A b e+3 b B d)}{3 b \sqrt {a+b x} (d+e x)^{7/2} (b d-a e)^2}-\frac {2 (A b-a B)}{3 b (a+b x)^{3/2} (d+e x)^{7/2} (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(A + B*x)/((a + b*x)^(5/2)*(d + e*x)^(9/2)),x]
[Out]
(-2*(A*b - a*B))/(3*b*(b*d - a*e)*(a + b*x)^(3/2)*(d + e*x)^(7/2)) - (2*(3*b*B*d - 10*A*b*e + 7*a*B*e))/(3*b*(
b*d - a*e)^2*Sqrt[a + b*x]*(d + e*x)^(7/2)) - (16*e*(3*b*B*d - 10*A*b*e + 7*a*B*e)*Sqrt[a + b*x])/(21*b*(b*d -
a*e)^3*(d + e*x)^(7/2)) - (32*e*(3*b*B*d - 10*A*b*e + 7*a*B*e)*Sqrt[a + b*x])/(35*(b*d - a*e)^4*(d + e*x)^(5/
2)) - (128*b*e*(3*b*B*d - 10*A*b*e + 7*a*B*e)*Sqrt[a + b*x])/(105*(b*d - a*e)^5*(d + e*x)^(3/2)) - (256*b^2*e*
(3*b*B*d - 10*A*b*e + 7*a*B*e)*Sqrt[a + b*x])/(105*(b*d - a*e)^6*Sqrt[d + e*x])
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}{(a+b x)^{5/2} (d+e x)^{9/2}} \, dx &=-\frac {2 (A b-a B)}{3 b (b d-a e) (a+b x)^{3/2} (d+e x)^{7/2}}+\frac {(3 b B d-10 A b e+7 a B e) \int \frac {1}{(a+b x)^{3/2} (d+e x)^{9/2}} \, dx}{3 b (b d-a e)}\\ &=-\frac {2 (A b-a B)}{3 b (b d-a e) (a+b x)^{3/2} (d+e x)^{7/2}}-\frac {2 (3 b B d-10 A b e+7 a B e)}{3 b (b d-a e)^2 \sqrt {a+b x} (d+e x)^{7/2}}-\frac {(8 e (3 b B d-10 A b e+7 a B e)) \int \frac {1}{\sqrt {a+b x} (d+e x)^{9/2}} \, dx}{3 b (b d-a e)^2}\\ &=-\frac {2 (A b-a B)}{3 b (b d-a e) (a+b x)^{3/2} (d+e x)^{7/2}}-\frac {2 (3 b B d-10 A b e+7 a B e)}{3 b (b d-a e)^2 \sqrt {a+b x} (d+e x)^{7/2}}-\frac {16 e (3 b B d-10 A b e+7 a B e) \sqrt {a+b x}}{21 b (b d-a e)^3 (d+e x)^{7/2}}-\frac {(16 e (3 b B d-10 A b e+7 a B e)) \int \frac {1}{\sqrt {a+b x} (d+e x)^{7/2}} \, dx}{7 (b d-a e)^3}\\ &=-\frac {2 (A b-a B)}{3 b (b d-a e) (a+b x)^{3/2} (d+e x)^{7/2}}-\frac {2 (3 b B d-10 A b e+7 a B e)}{3 b (b d-a e)^2 \sqrt {a+b x} (d+e x)^{7/2}}-\frac {16 e (3 b B d-10 A b e+7 a B e) \sqrt {a+b x}}{21 b (b d-a e)^3 (d+e x)^{7/2}}-\frac {32 e (3 b B d-10 A b e+7 a B e) \sqrt {a+b x}}{35 (b d-a e)^4 (d+e x)^{5/2}}-\frac {(64 b e (3 b B d-10 A b e+7 a B e)) \int \frac {1}{\sqrt {a+b x} (d+e x)^{5/2}} \, dx}{35 (b d-a e)^4}\\ &=-\frac {2 (A b-a B)}{3 b (b d-a e) (a+b x)^{3/2} (d+e x)^{7/2}}-\frac {2 (3 b B d-10 A b e+7 a B e)}{3 b (b d-a e)^2 \sqrt {a+b x} (d+e x)^{7/2}}-\frac {16 e (3 b B d-10 A b e+7 a B e) \sqrt {a+b x}}{21 b (b d-a e)^3 (d+e x)^{7/2}}-\frac {32 e (3 b B d-10 A b e+7 a B e) \sqrt {a+b x}}{35 (b d-a e)^4 (d+e x)^{5/2}}-\frac {128 b e (3 b B d-10 A b e+7 a B e) \sqrt {a+b x}}{105 (b d-a e)^5 (d+e x)^{3/2}}-\frac {\left (128 b^2 e (3 b B d-10 A b e+7 a B e)\right ) \int \frac {1}{\sqrt {a+b x} (d+e x)^{3/2}} \, dx}{105 (b d-a e)^5}\\ &=-\frac {2 (A b-a B)}{3 b (b d-a e) (a+b x)^{3/2} (d+e x)^{7/2}}-\frac {2 (3 b B d-10 A b e+7 a B e)}{3 b (b d-a e)^2 \sqrt {a+b x} (d+e x)^{7/2}}-\frac {16 e (3 b B d-10 A b e+7 a B e) \sqrt {a+b x}}{21 b (b d-a e)^3 (d+e x)^{7/2}}-\frac {32 e (3 b B d-10 A b e+7 a B e) \sqrt {a+b x}}{35 (b d-a e)^4 (d+e x)^{5/2}}-\frac {128 b e (3 b B d-10 A b e+7 a B e) \sqrt {a+b x}}{105 (b d-a e)^5 (d+e x)^{3/2}}-\frac {256 b^2 e (3 b B d-10 A b e+7 a B e) \sqrt {a+b x}}{105 (b d-a e)^6 \sqrt {d+e x}}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 160, normalized size = 0.54 \begin {gather*} \frac {2 \left (2 (a+b x) \left (8 e (a+b x) \left (2 b (d+e x) \left (4 b (d+e x) (-a e+3 b d+2 b e x)+3 (b d-a e)^2\right )+5 (b d-a e)^3\right )+35 (b d-a e)^4\right ) \left (-\frac {7 a B e}{2}+5 A b e-\frac {3}{2} b B d\right )-35 (A b-a B) (b d-a e)^5\right )}{105 b (a+b x)^{3/2} (d+e x)^{7/2} (b d-a e)^6} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x)/((a + b*x)^(5/2)*(d + e*x)^(9/2)),x]
[Out]
(2*(-35*(A*b - a*B)*(b*d - a*e)^5 + 2*((-3*b*B*d)/2 + 5*A*b*e - (7*a*B*e)/2)*(a + b*x)*(35*(b*d - a*e)^4 + 8*e
*(a + b*x)*(5*(b*d - a*e)^3 + 2*b*(d + e*x)*(3*(b*d - a*e)^2 + 4*b*(d + e*x)*(3*b*d - a*e + 2*b*e*x))))))/(105
*b*(b*d - a*e)^6*(a + b*x)^(3/2)*(d + e*x)^(7/2))
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IntegrateAlgebraic [A] time = 0.28, size = 347, normalized size = 1.16 \begin {gather*} \frac {2 (a+b x)^{7/2} \left (-\frac {35 A b^5 (d+e x)^5}{(a+b x)^5}+\frac {525 A b^4 e (d+e x)^4}{(a+b x)^4}+\frac {1050 A b^3 e^2 (d+e x)^3}{(a+b x)^3}-\frac {350 A b^2 e^3 (d+e x)^2}{(a+b x)^2}+\frac {105 A b e^4 (d+e x)}{a+b x}+\frac {35 a b^4 B (d+e x)^5}{(a+b x)^5}-\frac {105 b^4 B d (d+e x)^4}{(a+b x)^4}-\frac {420 a b^3 B e (d+e x)^4}{(a+b x)^4}-\frac {420 b^3 B d e (d+e x)^3}{(a+b x)^3}-\frac {630 a b^2 B e^2 (d+e x)^3}{(a+b x)^3}+\frac {210 b^2 B d e^2 (d+e x)^2}{(a+b x)^2}-\frac {21 a B e^4 (d+e x)}{a+b x}+\frac {140 a b B e^3 (d+e x)^2}{(a+b x)^2}-\frac {84 b B d e^3 (d+e x)}{a+b x}-15 A e^5+15 B d e^4\right )}{105 (d+e x)^{7/2} (b d-a e)^6} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(A + B*x)/((a + b*x)^(5/2)*(d + e*x)^(9/2)),x]
[Out]
(2*(a + b*x)^(7/2)*(15*B*d*e^4 - 15*A*e^5 - (84*b*B*d*e^3*(d + e*x))/(a + b*x) + (105*A*b*e^4*(d + e*x))/(a +
b*x) - (21*a*B*e^4*(d + e*x))/(a + b*x) + (210*b^2*B*d*e^2*(d + e*x)^2)/(a + b*x)^2 - (350*A*b^2*e^3*(d + e*x)
^2)/(a + b*x)^2 + (140*a*b*B*e^3*(d + e*x)^2)/(a + b*x)^2 - (420*b^3*B*d*e*(d + e*x)^3)/(a + b*x)^3 + (1050*A*
b^3*e^2*(d + e*x)^3)/(a + b*x)^3 - (630*a*b^2*B*e^2*(d + e*x)^3)/(a + b*x)^3 - (105*b^4*B*d*(d + e*x)^4)/(a +
b*x)^4 + (525*A*b^4*e*(d + e*x)^4)/(a + b*x)^4 - (420*a*b^3*B*e*(d + e*x)^4)/(a + b*x)^4 - (35*A*b^5*(d + e*x)
^5)/(a + b*x)^5 + (35*a*b^4*B*(d + e*x)^5)/(a + b*x)^5))/(105*(b*d - a*e)^6*(d + e*x)^(7/2))
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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)/(b*x+a)^(5/2)/(e*x+d)^(9/2),x, algorithm="fricas")
[Out]
Timed out
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giac [B] time = 21.23, size = 3473, normalized size = 11.65
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(b*x+a)^(5/2)/(e*x+d)^(9/2),x, algorithm="giac")
[Out]
-2/105*(((b*x + a)*((279*B*b^22*d^13*abs(b)*e^7 - 2837*B*a*b^21*d^12*abs(b)*e^8 - 790*A*b^22*d^12*abs(b)*e^8 +
12282*B*a^2*b^20*d^11*abs(b)*e^9 + 9480*A*a*b^21*d^11*abs(b)*e^9 - 27654*B*a^3*b^19*d^10*abs(b)*e^10 - 52140*
A*a^2*b^20*d^10*abs(b)*e^10 + 25685*B*a^4*b^18*d^9*abs(b)*e^11 + 173800*A*a^3*b^19*d^9*abs(b)*e^11 + 31977*B*a
^5*b^17*d^8*abs(b)*e^12 - 391050*A*a^4*b^18*d^8*abs(b)*e^12 - 146916*B*a^6*b^16*d^7*abs(b)*e^13 + 625680*A*a^5
*b^17*d^7*abs(b)*e^13 + 251196*B*a^7*b^15*d^6*abs(b)*e^14 - 729960*A*a^6*b^16*d^6*abs(b)*e^14 - 266607*B*a^8*b
^14*d^5*abs(b)*e^15 + 625680*A*a^7*b^15*d^5*abs(b)*e^15 + 191565*B*a^9*b^13*d^4*abs(b)*e^16 - 391050*A*a^8*b^1
4*d^4*abs(b)*e^16 - 94006*B*a^10*b^12*d^3*abs(b)*e^17 + 173800*A*a^9*b^13*d^3*abs(b)*e^17 + 30378*B*a^11*b^11*
d^2*abs(b)*e^18 - 52140*A*a^10*b^12*d^2*abs(b)*e^18 - 5853*B*a^12*b^10*d*abs(b)*e^19 + 9480*A*a^11*b^11*d*abs(
b)*e^19 + 511*B*a^13*b^9*abs(b)*e^20 - 790*A*a^12*b^10*abs(b)*e^20)*(b*x + a)/(b^22*d^18*e^3 - 18*a*b^21*d^17*
e^4 + 153*a^2*b^20*d^16*e^5 - 816*a^3*b^19*d^15*e^6 + 3060*a^4*b^18*d^14*e^7 - 8568*a^5*b^17*d^13*e^8 + 18564*
a^6*b^16*d^12*e^9 - 31824*a^7*b^15*d^11*e^10 + 43758*a^8*b^14*d^10*e^11 - 48620*a^9*b^13*d^9*e^12 + 43758*a^10
*b^12*d^8*e^13 - 31824*a^11*b^11*d^7*e^14 + 18564*a^12*b^10*d^6*e^15 - 8568*a^13*b^9*d^5*e^16 + 3060*a^14*b^8*
d^4*e^17 - 816*a^15*b^7*d^3*e^18 + 153*a^16*b^6*d^2*e^19 - 18*a^17*b^5*d*e^20 + a^18*b^4*e^21) + 7*(132*B*b^23
*d^14*abs(b)*e^6 - 1483*B*a*b^22*d^13*abs(b)*e^7 - 365*A*b^23*d^13*abs(b)*e^7 + 7267*B*a^2*b^21*d^12*abs(b)*e^
8 + 4745*A*a*b^22*d^12*abs(b)*e^8 - 19578*B*a^3*b^20*d^11*abs(b)*e^9 - 28470*A*a^2*b^21*d^11*abs(b)*e^9 + 2774
2*B*a^4*b^19*d^10*abs(b)*e^10 + 104390*A*a^3*b^20*d^10*abs(b)*e^10 - 3289*B*a^5*b^18*d^9*abs(b)*e^11 - 260975*
A*a^4*b^19*d^9*abs(b)*e^11 - 73359*B*a^6*b^17*d^8*abs(b)*e^12 + 469755*A*a^5*b^18*d^8*abs(b)*e^12 + 173316*B*a
^7*b^16*d^7*abs(b)*e^13 - 626340*A*a^6*b^17*d^7*abs(b)*e^13 - 229944*B*a^8*b^15*d^6*abs(b)*e^14 + 626340*A*a^7
*b^16*d^6*abs(b)*e^14 + 205491*B*a^9*b^14*d^5*abs(b)*e^15 - 469755*A*a^8*b^15*d^5*abs(b)*e^15 - 128843*B*a^10*
b^13*d^4*abs(b)*e^16 + 260975*A*a^9*b^14*d^4*abs(b)*e^16 + 56342*B*a^11*b^12*d^3*abs(b)*e^17 - 104390*A*a^10*b
^13*d^3*abs(b)*e^17 - 16458*B*a^12*b^11*d^2*abs(b)*e^18 + 28470*A*a^11*b^12*d^2*abs(b)*e^18 + 2897*B*a^13*b^10
*d*abs(b)*e^19 - 4745*A*a^12*b^11*d*abs(b)*e^19 - 233*B*a^14*b^9*abs(b)*e^20 + 365*A*a^13*b^10*abs(b)*e^20)/(b
^22*d^18*e^3 - 18*a*b^21*d^17*e^4 + 153*a^2*b^20*d^16*e^5 - 816*a^3*b^19*d^15*e^6 + 3060*a^4*b^18*d^14*e^7 - 8
568*a^5*b^17*d^13*e^8 + 18564*a^6*b^16*d^12*e^9 - 31824*a^7*b^15*d^11*e^10 + 43758*a^8*b^14*d^10*e^11 - 48620*
a^9*b^13*d^9*e^12 + 43758*a^10*b^12*d^8*e^13 - 31824*a^11*b^11*d^7*e^14 + 18564*a^12*b^10*d^6*e^15 - 8568*a^13
*b^9*d^5*e^16 + 3060*a^14*b^8*d^4*e^17 - 816*a^15*b^7*d^3*e^18 + 153*a^16*b^6*d^2*e^19 - 18*a^17*b^5*d*e^20 +
a^18*b^4*e^21)) + 350*(3*B*b^24*d^15*abs(b)*e^5 - 37*B*a*b^23*d^14*abs(b)*e^6 - 8*A*b^24*d^14*abs(b)*e^6 + 203
*B*a^2*b^22*d^13*abs(b)*e^7 + 112*A*a*b^23*d^13*abs(b)*e^7 - 637*B*a^3*b^21*d^12*abs(b)*e^8 - 728*A*a^2*b^22*d
^12*abs(b)*e^8 + 1183*B*a^4*b^20*d^11*abs(b)*e^9 + 2912*A*a^3*b^21*d^11*abs(b)*e^9 - 1001*B*a^5*b^19*d^10*abs(
b)*e^10 - 8008*A*a^4*b^20*d^10*abs(b)*e^10 - 1001*B*a^6*b^18*d^9*abs(b)*e^11 + 16016*A*a^5*b^19*d^9*abs(b)*e^1
1 + 4719*B*a^7*b^17*d^8*abs(b)*e^12 - 24024*A*a^6*b^18*d^8*abs(b)*e^12 - 8151*B*a^8*b^16*d^7*abs(b)*e^13 + 274
56*A*a^7*b^17*d^7*abs(b)*e^13 + 9009*B*a^9*b^15*d^6*abs(b)*e^14 - 24024*A*a^8*b^16*d^6*abs(b)*e^14 - 7007*B*a^
10*b^14*d^5*abs(b)*e^15 + 16016*A*a^9*b^15*d^5*abs(b)*e^15 + 3913*B*a^11*b^13*d^4*abs(b)*e^16 - 8008*A*a^10*b^
14*d^4*abs(b)*e^16 - 1547*B*a^12*b^12*d^3*abs(b)*e^17 + 2912*A*a^11*b^13*d^3*abs(b)*e^17 + 413*B*a^13*b^11*d^2
*abs(b)*e^18 - 728*A*a^12*b^12*d^2*abs(b)*e^18 - 67*B*a^14*b^10*d*abs(b)*e^19 + 112*A*a^13*b^11*d*abs(b)*e^19
+ 5*B*a^15*b^9*abs(b)*e^20 - 8*A*a^14*b^10*abs(b)*e^20)/(b^22*d^18*e^3 - 18*a*b^21*d^17*e^4 + 153*a^2*b^20*d^1
6*e^5 - 816*a^3*b^19*d^15*e^6 + 3060*a^4*b^18*d^14*e^7 - 8568*a^5*b^17*d^13*e^8 + 18564*a^6*b^16*d^12*e^9 - 31
824*a^7*b^15*d^11*e^10 + 43758*a^8*b^14*d^10*e^11 - 48620*a^9*b^13*d^9*e^12 + 43758*a^10*b^12*d^8*e^13 - 31824
*a^11*b^11*d^7*e^14 + 18564*a^12*b^10*d^6*e^15 - 8568*a^13*b^9*d^5*e^16 + 3060*a^14*b^8*d^4*e^17 - 816*a^15*b^
7*d^3*e^18 + 153*a^16*b^6*d^2*e^19 - 18*a^17*b^5*d*e^20 + a^18*b^4*e^21))*(b*x + a) + 210*(2*B*b^25*d^16*abs(b
)*e^4 - 27*B*a*b^24*d^15*abs(b)*e^5 - 5*A*b^25*d^15*abs(b)*e^5 + 165*B*a^2*b^23*d^14*abs(b)*e^6 + 75*A*a*b^24*
d^14*abs(b)*e^6 - 595*B*a^3*b^22*d^13*abs(b)*e^7 - 525*A*a^2*b^23*d^13*abs(b)*e^7 + 1365*B*a^4*b^21*d^12*abs(b
)*e^8 + 2275*A*a^3*b^22*d^12*abs(b)*e^8 - 1911*B*a^5*b^20*d^11*abs(b)*e^9 - 6825*A*a^4*b^21*d^11*abs(b)*e^9 +
1001*B*a^6*b^19*d^10*abs(b)*e^10 + 15015*A*a^5*b^20*d^10*abs(b)*e^10 + 2145*B*a^7*b^18*d^9*abs(b)*e^11 - 25025
*A*a^6*b^19*d^9*abs(b)*e^11 - 6435*B*a^8*b^17*d^8*abs(b)*e^12 + 32175*A*a^7*b^18*d^8*abs(b)*e^12 + 9295*B*a^9*
b^16*d^7*abs(b)*e^13 - 32175*A*a^8*b^17*d^7*abs(b)*e^13 - 9009*B*a^10*b^15*d^6*abs(b)*e^14 + 25025*A*a^9*b^16*
d^6*abs(b)*e^14 + 6279*B*a^11*b^14*d^5*abs(b)*e^15 - 15015*A*a^10*b^15*d^5*abs(b)*e^15 - 3185*B*a^12*b^13*d^4*
abs(b)*e^16 + 6825*A*a^11*b^14*d^4*abs(b)*e^16 + 1155*B*a^13*b^12*d^3*abs(b)*e^17 - 2275*A*a^12*b^13*d^3*abs(b
)*e^17 - 285*B*a^14*b^11*d^2*abs(b)*e^18 + 525*A*a^13*b^12*d^2*abs(b)*e^18 + 43*B*a^15*b^10*d*abs(b)*e^19 - 75
*A*a^14*b^11*d*abs(b)*e^19 - 3*B*a^16*b^9*abs(b)*e^20 + 5*A*a^15*b^10*abs(b)*e^20)/(b^22*d^18*e^3 - 18*a*b^21*
d^17*e^4 + 153*a^2*b^20*d^16*e^5 - 816*a^3*b^19*d^15*e^6 + 3060*a^4*b^18*d^14*e^7 - 8568*a^5*b^17*d^13*e^8 + 1
8564*a^6*b^16*d^12*e^9 - 31824*a^7*b^15*d^11*e^10 + 43758*a^8*b^14*d^10*e^11 - 48620*a^9*b^13*d^9*e^12 + 43758
*a^10*b^12*d^8*e^13 - 31824*a^11*b^11*d^7*e^14 + 18564*a^12*b^10*d^6*e^15 - 8568*a^13*b^9*d^5*e^16 + 3060*a^14
*b^8*d^4*e^17 - 816*a^15*b^7*d^3*e^18 + 153*a^16*b^6*d^2*e^19 - 18*a^17*b^5*d*e^20 + a^18*b^4*e^21))*sqrt(b*x
+ a)/(b^2*d + (b*x + a)*b*e - a*b*e)^(7/2) - 4/3*(3*B*b^(19/2)*d^3*e^(1/2) + 5*B*a*b^(17/2)*d^2*e^(3/2) - 14*A
*b^(19/2)*d^2*e^(3/2) - 6*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*B*b^(15/2)*d
^2*e^(1/2) - 19*B*a^2*b^(15/2)*d*e^(5/2) + 28*A*a*b^(17/2)*d*e^(5/2) - 18*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqr
t(b^2*d + (b*x + a)*b*e - a*b*e))^2*B*a*b^(13/2)*d*e^(3/2) + 30*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d +
(b*x + a)*b*e - a*b*e))^2*A*b^(15/2)*d*e^(3/2) + 3*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e
- a*b*e))^4*B*b^(11/2)*d*e^(1/2) + 11*B*a^3*b^(13/2)*e^(7/2) - 14*A*a^2*b^(15/2)*e^(7/2) + 24*(sqrt(b*x + a)*
sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*B*a^2*b^(11/2)*e^(5/2) - 30*(sqrt(b*x + a)*sqrt(b)*e^
(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*A*a*b^(13/2)*e^(5/2) + 9*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(
b^2*d + (b*x + a)*b*e - a*b*e))^4*B*a*b^(9/2)*e^(3/2) - 12*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x
+ a)*b*e - a*b*e))^4*A*b^(11/2)*e^(3/2))/((b^5*d^5*abs(b) - 5*a*b^4*d^4*abs(b)*e + 10*a^2*b^3*d^3*abs(b)*e^2 -
10*a^3*b^2*d^2*abs(b)*e^3 + 5*a^4*b*d*abs(b)*e^4 - a^5*abs(b)*e^5)*(b^2*d - a*b*e - (sqrt(b*x + a)*sqrt(b)*e^
(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2)^3)
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maple [B] time = 0.02, size = 722, normalized size = 2.42 \begin {gather*} -\frac {2 \left (-1280 A \,b^{5} e^{5} x^{5}+896 B a \,b^{4} e^{5} x^{5}+384 B \,b^{5} d \,e^{4} x^{5}-1920 A a \,b^{4} e^{5} x^{4}-4480 A \,b^{5} d \,e^{4} x^{4}+1344 B \,a^{2} b^{3} e^{5} x^{4}+3712 B a \,b^{4} d \,e^{4} x^{4}+1344 B \,b^{5} d^{2} e^{3} x^{4}-480 A \,a^{2} b^{3} e^{5} x^{3}-6720 A a \,b^{4} d \,e^{4} x^{3}-5600 A \,b^{5} d^{2} e^{3} x^{3}+336 B \,a^{3} b^{2} e^{5} x^{3}+4848 B \,a^{2} b^{3} d \,e^{4} x^{3}+5936 B a \,b^{4} d^{2} e^{3} x^{3}+1680 B \,b^{5} d^{3} e^{2} x^{3}+80 A \,a^{3} b^{2} e^{5} x^{2}-1680 A \,a^{2} b^{3} d \,e^{4} x^{2}-8400 A a \,b^{4} d^{2} e^{3} x^{2}-2800 A \,b^{5} d^{3} e^{2} x^{2}-56 B \,a^{4} b \,e^{5} x^{2}+1152 B \,a^{3} b^{2} d \,e^{4} x^{2}+6384 B \,a^{2} b^{3} d^{2} e^{3} x^{2}+4480 B a \,b^{4} d^{3} e^{2} x^{2}+840 B \,b^{5} d^{4} e \,x^{2}-30 A \,a^{4} b \,e^{5} x +280 A \,a^{3} b^{2} d \,e^{4} x -2100 A \,a^{2} b^{3} d^{2} e^{3} x -4200 A a \,b^{4} d^{3} e^{2} x -350 A \,b^{5} d^{4} e x +21 B \,a^{5} e^{5} x -187 B \,a^{4} b d \,e^{4} x +1386 B \,a^{3} b^{2} d^{2} e^{3} x +3570 B \,a^{2} b^{3} d^{3} e^{2} x +1505 B a \,b^{4} d^{4} e x +105 B \,b^{5} d^{5} x +15 A \,a^{5} e^{5}-105 A \,a^{4} b d \,e^{4}+350 A \,a^{3} b^{2} d^{2} e^{3}-1050 A \,a^{2} b^{3} d^{3} e^{2}-525 A a \,b^{4} d^{4} e +35 A \,b^{5} d^{5}+6 B \,a^{5} d \,e^{4}-56 B \,a^{4} b \,d^{2} e^{3}+420 B \,a^{3} b^{2} d^{3} e^{2}+840 B \,a^{2} b^{3} d^{4} e +70 B a \,b^{4} d^{5}\right )}{105 \left (b x +a \right )^{\frac {3}{2}} \left (e x +d \right )^{\frac {7}{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)/(b*x+a)^(5/2)/(e*x+d)^(9/2),x)
[Out]
-2/105*(-1280*A*b^5*e^5*x^5+896*B*a*b^4*e^5*x^5+384*B*b^5*d*e^4*x^5-1920*A*a*b^4*e^5*x^4-4480*A*b^5*d*e^4*x^4+
1344*B*a^2*b^3*e^5*x^4+3712*B*a*b^4*d*e^4*x^4+1344*B*b^5*d^2*e^3*x^4-480*A*a^2*b^3*e^5*x^3-6720*A*a*b^4*d*e^4*
x^3-5600*A*b^5*d^2*e^3*x^3+336*B*a^3*b^2*e^5*x^3+4848*B*a^2*b^3*d*e^4*x^3+5936*B*a*b^4*d^2*e^3*x^3+1680*B*b^5*
d^3*e^2*x^3+80*A*a^3*b^2*e^5*x^2-1680*A*a^2*b^3*d*e^4*x^2-8400*A*a*b^4*d^2*e^3*x^2-2800*A*b^5*d^3*e^2*x^2-56*B
*a^4*b*e^5*x^2+1152*B*a^3*b^2*d*e^4*x^2+6384*B*a^2*b^3*d^2*e^3*x^2+4480*B*a*b^4*d^3*e^2*x^2+840*B*b^5*d^4*e*x^
2-30*A*a^4*b*e^5*x+280*A*a^3*b^2*d*e^4*x-2100*A*a^2*b^3*d^2*e^3*x-4200*A*a*b^4*d^3*e^2*x-350*A*b^5*d^4*e*x+21*
B*a^5*e^5*x-187*B*a^4*b*d*e^4*x+1386*B*a^3*b^2*d^2*e^3*x+3570*B*a^2*b^3*d^3*e^2*x+1505*B*a*b^4*d^4*e*x+105*B*b
^5*d^5*x+15*A*a^5*e^5-105*A*a^4*b*d*e^4+350*A*a^3*b^2*d^2*e^3-1050*A*a^2*b^3*d^3*e^2-525*A*a*b^4*d^4*e+35*A*b^
5*d^5+6*B*a^5*d*e^4-56*B*a^4*b*d^2*e^3+420*B*a^3*b^2*d^3*e^2+840*B*a^2*b^3*d^4*e+70*B*a*b^4*d^5)/(b*x+a)^(3/2)
/(e*x+d)^(7/2)/(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)/(b*x+a)^(5/2)/(e*x+d)^(9/2),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 = 3.23, size = 596, normalized size = 2.00 \begin {gather*} -\frac {\sqrt {d+e\,x}\,\left (\frac {12\,B\,a^5\,d\,e^4+30\,A\,a^5\,e^5-112\,B\,a^4\,b\,d^2\,e^3-210\,A\,a^4\,b\,d\,e^4+840\,B\,a^3\,b^2\,d^3\,e^2+700\,A\,a^3\,b^2\,d^2\,e^3+1680\,B\,a^2\,b^3\,d^4\,e-2100\,A\,a^2\,b^3\,d^3\,e^2+140\,B\,a\,b^4\,d^5-1050\,A\,a\,b^4\,d^4\,e+70\,A\,b^5\,d^5}{105\,b\,e^4\,{\left (a\,e-b\,d\right )}^6}+\frac {256\,b^3\,x^5\,\left (7\,B\,a\,e-10\,A\,b\,e+3\,B\,b\,d\right )}{105\,{\left (a\,e-b\,d\right )}^6}+\frac {16\,x^2\,\left (7\,B\,a\,e-10\,A\,b\,e+3\,B\,b\,d\right )\,\left (-a^3\,e^3+21\,a^2\,b\,d\,e^2+105\,a\,b^2\,d^2\,e+35\,b^3\,d^3\right )}{105\,e^3\,{\left (a\,e-b\,d\right )}^6}+\frac {128\,b^2\,x^4\,\left (3\,a\,e+7\,b\,d\right )\,\left (7\,B\,a\,e-10\,A\,b\,e+3\,B\,b\,d\right )}{105\,e\,{\left (a\,e-b\,d\right )}^6}+\frac {32\,b\,x^3\,\left (3\,a^2\,e^2+42\,a\,b\,d\,e+35\,b^2\,d^2\right )\,\left (7\,B\,a\,e-10\,A\,b\,e+3\,B\,b\,d\right )}{105\,e^2\,{\left (a\,e-b\,d\right )}^6}+\frac {2\,x\,\left (7\,B\,a\,e-10\,A\,b\,e+3\,B\,b\,d\right )\,\left (3\,a^4\,e^4-28\,a^3\,b\,d\,e^3+210\,a^2\,b^2\,d^2\,e^2+420\,a\,b^3\,d^3\,e+35\,b^4\,d^4\right )}{105\,b\,e^4\,{\left (a\,e-b\,d\right )}^6}\right )}{x^5\,\sqrt {a+b\,x}+\frac {a\,d^4\,\sqrt {a+b\,x}}{b\,e^4}+\frac {x^4\,\left (a\,e+4\,b\,d\right )\,\sqrt {a+b\,x}}{b\,e}+\frac {2\,d\,x^3\,\left (2\,a\,e+3\,b\,d\right )\,\sqrt {a+b\,x}}{b\,e^2}+\frac {d^3\,x\,\left (4\,a\,e+b\,d\right )\,\sqrt {a+b\,x}}{b\,e^4}+\frac {2\,d^2\,x^2\,\left (3\,a\,e+2\,b\,d\right )\,\sqrt {a+b\,x}}{b\,e^3}} \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)^(9/2)),x)
[Out]
-((d + e*x)^(1/2)*((30*A*a^5*e^5 + 70*A*b^5*d^5 + 140*B*a*b^4*d^5 + 12*B*a^5*d*e^4 + 1680*B*a^2*b^3*d^4*e - 11
2*B*a^4*b*d^2*e^3 - 2100*A*a^2*b^3*d^3*e^2 + 700*A*a^3*b^2*d^2*e^3 + 840*B*a^3*b^2*d^3*e^2 - 1050*A*a*b^4*d^4*
e - 210*A*a^4*b*d*e^4)/(105*b*e^4*(a*e - b*d)^6) + (256*b^3*x^5*(7*B*a*e - 10*A*b*e + 3*B*b*d))/(105*(a*e - b*
d)^6) + (16*x^2*(7*B*a*e - 10*A*b*e + 3*B*b*d)*(35*b^3*d^3 - a^3*e^3 + 105*a*b^2*d^2*e + 21*a^2*b*d*e^2))/(105
*e^3*(a*e - b*d)^6) + (128*b^2*x^4*(3*a*e + 7*b*d)*(7*B*a*e - 10*A*b*e + 3*B*b*d))/(105*e*(a*e - b*d)^6) + (32
*b*x^3*(3*a^2*e^2 + 35*b^2*d^2 + 42*a*b*d*e)*(7*B*a*e - 10*A*b*e + 3*B*b*d))/(105*e^2*(a*e - b*d)^6) + (2*x*(7
*B*a*e - 10*A*b*e + 3*B*b*d)*(3*a^4*e^4 + 35*b^4*d^4 + 210*a^2*b^2*d^2*e^2 + 420*a*b^3*d^3*e - 28*a^3*b*d*e^3)
)/(105*b*e^4*(a*e - b*d)^6)))/(x^5*(a + b*x)^(1/2) + (a*d^4*(a + b*x)^(1/2))/(b*e^4) + (x^4*(a*e + 4*b*d)*(a +
b*x)^(1/2))/(b*e) + (2*d*x^3*(2*a*e + 3*b*d)*(a + b*x)^(1/2))/(b*e^2) + (d^3*x*(4*a*e + b*d)*(a + b*x)^(1/2))
/(b*e^4) + (2*d^2*x^2*(3*a*e + 2*b*d)*(a + b*x)^(1/2))/(b*e^3))
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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)/(b*x+a)**(5/2)/(e*x+d)**(9/2),x)
[Out]
Timed out
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