3.22.13 \(\int \frac {A+B x}{(a+b x)^{3/2} (d+e x)^{9/2}} \, dx\)
Optimal. Leaf size=237 \[ \frac {32 b^2 \sqrt {a+b x} (7 a B e-8 A b e+b B d)}{35 \sqrt {d+e x} (b d-a e)^5}+\frac {16 b \sqrt {a+b x} (7 a B e-8 A b e+b B d)}{35 (d+e x)^{3/2} (b d-a e)^4}+\frac {12 \sqrt {a+b x} (7 a B e-8 A b e+b B d)}{35 (d+e x)^{5/2} (b d-a e)^3}+\frac {2 \sqrt {a+b x} (7 a B e-8 A b e+b B d)}{7 b (d+e x)^{7/2} (b d-a e)^2}-\frac {2 (A b-a B)}{b \sqrt {a+b x} (d+e x)^{7/2} (b d-a e)} \]
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Rubi [A] time = 0.15, antiderivative size = 237, 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^2 \sqrt {a+b x} (7 a B e-8 A b e+b B d)}{35 \sqrt {d+e x} (b d-a e)^5}+\frac {16 b \sqrt {a+b x} (7 a B e-8 A b e+b B d)}{35 (d+e x)^{3/2} (b d-a e)^4}+\frac {12 \sqrt {a+b x} (7 a B e-8 A b e+b B d)}{35 (d+e x)^{5/2} (b d-a e)^3}+\frac {2 \sqrt {a+b x} (7 a B e-8 A b e+b B d)}{7 b (d+e x)^{7/2} (b d-a e)^2}-\frac {2 (A b-a B)}{b \sqrt {a+b x} (d+e x)^{7/2} (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(A + B*x)/((a + b*x)^(3/2)*(d + e*x)^(9/2)),x]
[Out]
(-2*(A*b - a*B))/(b*(b*d - a*e)*Sqrt[a + b*x]*(d + e*x)^(7/2)) + (2*(b*B*d - 8*A*b*e + 7*a*B*e)*Sqrt[a + b*x])
/(7*b*(b*d - a*e)^2*(d + e*x)^(7/2)) + (12*(b*B*d - 8*A*b*e + 7*a*B*e)*Sqrt[a + b*x])/(35*(b*d - a*e)^3*(d + e
*x)^(5/2)) + (16*b*(b*B*d - 8*A*b*e + 7*a*B*e)*Sqrt[a + b*x])/(35*(b*d - a*e)^4*(d + e*x)^(3/2)) + (32*b^2*(b*
B*d - 8*A*b*e + 7*a*B*e)*Sqrt[a + b*x])/(35*(b*d - a*e)^5*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)^{3/2} (d+e x)^{9/2}} \, dx &=-\frac {2 (A b-a B)}{b (b d-a e) \sqrt {a+b x} (d+e x)^{7/2}}+\frac {(b B d-8 A b e+7 a B e) \int \frac {1}{\sqrt {a+b x} (d+e x)^{9/2}} \, dx}{b (b d-a e)}\\ &=-\frac {2 (A b-a B)}{b (b d-a e) \sqrt {a+b x} (d+e x)^{7/2}}+\frac {2 (b B d-8 A b e+7 a B e) \sqrt {a+b x}}{7 b (b d-a e)^2 (d+e x)^{7/2}}+\frac {(6 (b B d-8 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)^2}\\ &=-\frac {2 (A b-a B)}{b (b d-a e) \sqrt {a+b x} (d+e x)^{7/2}}+\frac {2 (b B d-8 A b e+7 a B e) \sqrt {a+b x}}{7 b (b d-a e)^2 (d+e x)^{7/2}}+\frac {12 (b B d-8 A b e+7 a B e) \sqrt {a+b x}}{35 (b d-a e)^3 (d+e x)^{5/2}}+\frac {(24 b (b B d-8 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)^3}\\ &=-\frac {2 (A b-a B)}{b (b d-a e) \sqrt {a+b x} (d+e x)^{7/2}}+\frac {2 (b B d-8 A b e+7 a B e) \sqrt {a+b x}}{7 b (b d-a e)^2 (d+e x)^{7/2}}+\frac {12 (b B d-8 A b e+7 a B e) \sqrt {a+b x}}{35 (b d-a e)^3 (d+e x)^{5/2}}+\frac {16 b (b B d-8 A b e+7 a B e) \sqrt {a+b x}}{35 (b d-a e)^4 (d+e x)^{3/2}}+\frac {\left (16 b^2 (b B d-8 A b e+7 a B e)\right ) \int \frac {1}{\sqrt {a+b x} (d+e x)^{3/2}} \, dx}{35 (b d-a e)^4}\\ &=-\frac {2 (A b-a B)}{b (b d-a e) \sqrt {a+b x} (d+e x)^{7/2}}+\frac {2 (b B d-8 A b e+7 a B e) \sqrt {a+b x}}{7 b (b d-a e)^2 (d+e x)^{7/2}}+\frac {12 (b B d-8 A b e+7 a B e) \sqrt {a+b x}}{35 (b d-a e)^3 (d+e x)^{5/2}}+\frac {16 b (b B d-8 A b e+7 a B e) \sqrt {a+b x}}{35 (b d-a e)^4 (d+e x)^{3/2}}+\frac {32 b^2 (b B d-8 A b e+7 a B e) \sqrt {a+b x}}{35 (b d-a e)^5 \sqrt {d+e x}}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 135, normalized size = 0.57 \begin {gather*} \frac {2 \left (-(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 ) (-7 a B e+8 A b e-b B d)-35 (A b-a B) (b d-a e)^4\right )}{35 b \sqrt {a+b x} (d+e x)^{7/2} (b d-a e)^5} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x)/((a + b*x)^(3/2)*(d + e*x)^(9/2)),x]
[Out]
(2*(-35*(A*b - a*B)*(b*d - a*e)^4 - (-(b*B*d) + 8*A*b*e - 7*a*B*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)))))/(35*b*(b*d - a*e)^5*Sqrt[a + b*x]*(d + e*x)^(7/2)
)
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IntegrateAlgebraic [A] time = 0.23, size = 276, normalized size = 1.16 \begin {gather*} \frac {2 (a+b x)^{7/2} \left (-\frac {35 A b^4 (d+e x)^4}{(a+b x)^4}-\frac {140 A b^3 e (d+e x)^3}{(a+b x)^3}+\frac {70 A b^2 e^2 (d+e x)^2}{(a+b x)^2}-\frac {28 A b e^3 (d+e x)}{a+b x}+\frac {35 a b^3 B (d+e x)^4}{(a+b x)^4}+\frac {35 b^3 B d (d+e x)^3}{(a+b x)^3}+\frac {105 a b^2 B e (d+e x)^3}{(a+b x)^3}-\frac {35 b^2 B d e (d+e x)^2}{(a+b x)^2}+\frac {7 a B e^3 (d+e x)}{a+b x}-\frac {35 a b B e^2 (d+e x)^2}{(a+b x)^2}+\frac {21 b B d e^2 (d+e x)}{a+b x}+5 A e^4-5 B d e^3\right )}{35 (d+e x)^{7/2} (b d-a e)^5} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(A + B*x)/((a + b*x)^(3/2)*(d + e*x)^(9/2)),x]
[Out]
(2*(a + b*x)^(7/2)*(-5*B*d*e^3 + 5*A*e^4 + (21*b*B*d*e^2*(d + e*x))/(a + b*x) - (28*A*b*e^3*(d + e*x))/(a + b*
x) + (7*a*B*e^3*(d + e*x))/(a + b*x) - (35*b^2*B*d*e*(d + e*x)^2)/(a + b*x)^2 + (70*A*b^2*e^2*(d + e*x)^2)/(a
+ b*x)^2 - (35*a*b*B*e^2*(d + e*x)^2)/(a + b*x)^2 + (35*b^3*B*d*(d + e*x)^3)/(a + b*x)^3 - (140*A*b^3*e*(d + e
*x)^3)/(a + b*x)^3 + (105*a*b^2*B*e*(d + e*x)^3)/(a + b*x)^3 - (35*A*b^4*(d + e*x)^4)/(a + b*x)^4 + (35*a*b^3*
B*(d + e*x)^4)/(a + b*x)^4))/(35*(b*d - a*e)^5*(d + e*x)^(7/2))
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fricas [B] time = 97.94, size = 887, normalized size = 3.74 \begin {gather*} \frac {2 \, {\left (5 \, A a^{4} e^{4} + 35 \, {\left (2 \, B a b^{3} - A b^{4}\right )} d^{4} + 70 \, {\left (B a^{2} b^{2} - 2 \, A a b^{3}\right )} d^{3} e - 14 \, {\left (B a^{3} b - 5 \, A a^{2} b^{2}\right )} d^{2} e^{2} + 2 \, {\left (B a^{4} - 14 \, A a^{3} b\right )} d e^{3} + 16 \, {\left (B b^{4} d e^{3} + {\left (7 \, B a b^{3} - 8 \, A b^{4}\right )} e^{4}\right )} x^{4} + 8 \, {\left (7 \, B b^{4} d^{2} e^{2} + 2 \, {\left (25 \, B a b^{3} - 28 \, A b^{4}\right )} d e^{3} + {\left (7 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} e^{4}\right )} x^{3} + 2 \, {\left (35 \, B b^{4} d^{3} e + 7 \, {\left (37 \, B a b^{3} - 40 \, A b^{4}\right )} d^{2} e^{2} + {\left (97 \, B a^{2} b^{2} - 112 \, A a b^{3}\right )} d e^{3} - {\left (7 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} e^{4}\right )} x^{2} + {\left (35 \, B b^{4} d^{4} + 280 \, {\left (B a b^{3} - A b^{4}\right )} d^{3} e + 14 \, {\left (17 \, B a^{2} b^{2} - 20 \, A a b^{3}\right )} d^{2} e^{2} - 8 \, {\left (6 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} d e^{3} + {\left (7 \, B a^{4} - 8 \, A a^{3} b\right )} e^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{35 \, {\left (a b^{5} d^{9} - 5 \, a^{2} b^{4} d^{8} e + 10 \, a^{3} b^{3} d^{7} e^{2} - 10 \, a^{4} b^{2} d^{6} e^{3} + 5 \, a^{5} b d^{5} e^{4} - a^{6} d^{4} e^{5} + {\left (b^{6} d^{5} e^{4} - 5 \, a b^{5} d^{4} e^{5} + 10 \, a^{2} b^{4} d^{3} e^{6} - 10 \, a^{3} b^{3} d^{2} e^{7} + 5 \, a^{4} b^{2} d e^{8} - a^{5} b e^{9}\right )} x^{5} + {\left (4 \, b^{6} d^{6} e^{3} - 19 \, a b^{5} d^{5} e^{4} + 35 \, a^{2} b^{4} d^{4} e^{5} - 30 \, a^{3} b^{3} d^{3} e^{6} + 10 \, a^{4} b^{2} d^{2} e^{7} + a^{5} b d e^{8} - a^{6} e^{9}\right )} x^{4} + 2 \, {\left (3 \, b^{6} d^{7} e^{2} - 13 \, a b^{5} d^{6} e^{3} + 20 \, a^{2} b^{4} d^{5} e^{4} - 10 \, a^{3} b^{3} d^{4} e^{5} - 5 \, a^{4} b^{2} d^{3} e^{6} + 7 \, a^{5} b d^{2} e^{7} - 2 \, a^{6} d e^{8}\right )} x^{3} + 2 \, {\left (2 \, b^{6} d^{8} e - 7 \, a b^{5} d^{7} e^{2} + 5 \, a^{2} b^{4} d^{6} e^{3} + 10 \, a^{3} b^{3} d^{5} e^{4} - 20 \, a^{4} b^{2} d^{4} e^{5} + 13 \, a^{5} b d^{3} e^{6} - 3 \, a^{6} d^{2} e^{7}\right )} x^{2} + {\left (b^{6} d^{9} - a b^{5} d^{8} e - 10 \, a^{2} b^{4} d^{7} e^{2} + 30 \, a^{3} b^{3} d^{6} e^{3} - 35 \, a^{4} b^{2} d^{5} e^{4} + 19 \, a^{5} b d^{4} e^{5} - 4 \, a^{6} d^{3} e^{6}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(b*x+a)^(3/2)/(e*x+d)^(9/2),x, algorithm="fricas")
[Out]
2/35*(5*A*a^4*e^4 + 35*(2*B*a*b^3 - A*b^4)*d^4 + 70*(B*a^2*b^2 - 2*A*a*b^3)*d^3*e - 14*(B*a^3*b - 5*A*a^2*b^2)
*d^2*e^2 + 2*(B*a^4 - 14*A*a^3*b)*d*e^3 + 16*(B*b^4*d*e^3 + (7*B*a*b^3 - 8*A*b^4)*e^4)*x^4 + 8*(7*B*b^4*d^2*e^
2 + 2*(25*B*a*b^3 - 28*A*b^4)*d*e^3 + (7*B*a^2*b^2 - 8*A*a*b^3)*e^4)*x^3 + 2*(35*B*b^4*d^3*e + 7*(37*B*a*b^3 -
40*A*b^4)*d^2*e^2 + (97*B*a^2*b^2 - 112*A*a*b^3)*d*e^3 - (7*B*a^3*b - 8*A*a^2*b^2)*e^4)*x^2 + (35*B*b^4*d^4 +
280*(B*a*b^3 - A*b^4)*d^3*e + 14*(17*B*a^2*b^2 - 20*A*a*b^3)*d^2*e^2 - 8*(6*B*a^3*b - 7*A*a^2*b^2)*d*e^3 + (7
*B*a^4 - 8*A*a^3*b)*e^4)*x)*sqrt(b*x + a)*sqrt(e*x + d)/(a*b^5*d^9 - 5*a^2*b^4*d^8*e + 10*a^3*b^3*d^7*e^2 - 10
*a^4*b^2*d^6*e^3 + 5*a^5*b*d^5*e^4 - a^6*d^4*e^5 + (b^6*d^5*e^4 - 5*a*b^5*d^4*e^5 + 10*a^2*b^4*d^3*e^6 - 10*a^
3*b^3*d^2*e^7 + 5*a^4*b^2*d*e^8 - a^5*b*e^9)*x^5 + (4*b^6*d^6*e^3 - 19*a*b^5*d^5*e^4 + 35*a^2*b^4*d^4*e^5 - 30
*a^3*b^3*d^3*e^6 + 10*a^4*b^2*d^2*e^7 + a^5*b*d*e^8 - a^6*e^9)*x^4 + 2*(3*b^6*d^7*e^2 - 13*a*b^5*d^6*e^3 + 20*
a^2*b^4*d^5*e^4 - 10*a^3*b^3*d^4*e^5 - 5*a^4*b^2*d^3*e^6 + 7*a^5*b*d^2*e^7 - 2*a^6*d*e^8)*x^3 + 2*(2*b^6*d^8*e
- 7*a*b^5*d^7*e^2 + 5*a^2*b^4*d^6*e^3 + 10*a^3*b^3*d^5*e^4 - 20*a^4*b^2*d^4*e^5 + 13*a^5*b*d^3*e^6 - 3*a^6*d^
2*e^7)*x^2 + (b^6*d^9 - a*b^5*d^8*e - 10*a^2*b^4*d^7*e^2 + 30*a^3*b^3*d^6*e^3 - 35*a^4*b^2*d^5*e^4 + 19*a^5*b*
d^4*e^5 - 4*a^6*d^3*e^6)*x)
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giac [B] time = 8.78, size = 2477, normalized size = 10.45
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(b*x+a)^(3/2)/(e*x+d)^(9/2),x, algorithm="giac")
[Out]
4*(B^2*a^2*b^9*e - 2*A*B*a*b^10*e + A^2*b^11*e)/((B*a*b^(13/2)*d*e^(1/2) - A*b^(15/2)*d*e^(1/2) - B*a^2*b^(11/
2)*e^(3/2) + A*a*b^(13/2)*e^(3/2) - (sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*B*
a*b^(9/2)*e^(1/2) + (sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*A*b^(11/2)*e^(1/2)
)*(b^4*d^4*abs(b) - 4*a*b^3*d^3*abs(b)*e + 6*a^2*b^2*d^2*abs(b)*e^2 - 4*a^3*b*d*abs(b)*e^3 + a^4*abs(b)*e^4))
+ 2/35*(((b*x + a)*((16*B*b^19*d^10*abs(b)*e^6 - 67*B*a*b^18*d^9*abs(b)*e^7 - 93*A*b^19*d^9*abs(b)*e^7 - 117*B
*a^2*b^17*d^8*abs(b)*e^8 + 837*A*a*b^18*d^8*abs(b)*e^8 + 1428*B*a^3*b^16*d^7*abs(b)*e^9 - 3348*A*a^2*b^17*d^7*
abs(b)*e^9 - 4452*B*a^4*b^15*d^6*abs(b)*e^10 + 7812*A*a^3*b^16*d^6*abs(b)*e^10 + 7686*B*a^5*b^14*d^5*abs(b)*e^
11 - 11718*A*a^4*b^15*d^5*abs(b)*e^11 - 8358*B*a^6*b^13*d^4*abs(b)*e^12 + 11718*A*a^5*b^14*d^4*abs(b)*e^12 + 5
892*B*a^7*b^12*d^3*abs(b)*e^13 - 7812*A*a^6*b^13*d^3*abs(b)*e^13 - 2628*B*a^8*b^11*d^2*abs(b)*e^14 + 3348*A*a^
7*b^12*d^2*abs(b)*e^14 + 677*B*a^9*b^10*d*abs(b)*e^15 - 837*A*a^8*b^11*d*abs(b)*e^15 - 77*B*a^10*b^9*abs(b)*e^
16 + 93*A*a^9*b^10*abs(b)*e^16)*(b*x + a)/(b^18*d^14*e^3 - 14*a*b^17*d^13*e^4 + 91*a^2*b^16*d^12*e^5 - 364*a^3
*b^15*d^11*e^6 + 1001*a^4*b^14*d^10*e^7 - 2002*a^5*b^13*d^9*e^8 + 3003*a^6*b^12*d^8*e^9 - 3432*a^7*b^11*d^7*e^
10 + 3003*a^8*b^10*d^6*e^11 - 2002*a^9*b^9*d^5*e^12 + 1001*a^10*b^8*d^4*e^13 - 364*a^11*b^7*d^3*e^14 + 91*a^12
*b^6*d^2*e^15 - 14*a^13*b^5*d*e^16 + a^14*b^4*e^17) + 28*(2*B*b^20*d^11*abs(b)*e^5 - 11*B*a*b^19*d^10*abs(b)*e
^6 - 11*A*b^20*d^10*abs(b)*e^6 + 110*A*a*b^19*d^9*abs(b)*e^7 + 165*B*a^3*b^17*d^8*abs(b)*e^8 - 495*A*a^2*b^18*
d^8*abs(b)*e^8 - 660*B*a^4*b^16*d^7*abs(b)*e^9 + 1320*A*a^3*b^17*d^7*abs(b)*e^9 + 1386*B*a^5*b^15*d^6*abs(b)*e
^10 - 2310*A*a^4*b^16*d^6*abs(b)*e^10 - 1848*B*a^6*b^14*d^5*abs(b)*e^11 + 2772*A*a^5*b^15*d^5*abs(b)*e^11 + 16
50*B*a^7*b^13*d^4*abs(b)*e^12 - 2310*A*a^6*b^14*d^4*abs(b)*e^12 - 990*B*a^8*b^12*d^3*abs(b)*e^13 + 1320*A*a^7*
b^13*d^3*abs(b)*e^13 + 385*B*a^9*b^11*d^2*abs(b)*e^14 - 495*A*a^8*b^12*d^2*abs(b)*e^14 - 88*B*a^10*b^10*d*abs(
b)*e^15 + 110*A*a^9*b^11*d*abs(b)*e^15 + 9*B*a^11*b^9*abs(b)*e^16 - 11*A*a^10*b^10*abs(b)*e^16)/(b^18*d^14*e^3
- 14*a*b^17*d^13*e^4 + 91*a^2*b^16*d^12*e^5 - 364*a^3*b^15*d^11*e^6 + 1001*a^4*b^14*d^10*e^7 - 2002*a^5*b^13*
d^9*e^8 + 3003*a^6*b^12*d^8*e^9 - 3432*a^7*b^11*d^7*e^10 + 3003*a^8*b^10*d^6*e^11 - 2002*a^9*b^9*d^5*e^12 + 10
01*a^10*b^8*d^4*e^13 - 364*a^11*b^7*d^3*e^14 + 91*a^12*b^6*d^2*e^15 - 14*a^13*b^5*d*e^16 + a^14*b^4*e^17)) + 7
0*(B*b^21*d^12*abs(b)*e^4 - 7*B*a*b^20*d^11*abs(b)*e^5 - 5*A*b^21*d^11*abs(b)*e^5 + 11*B*a^2*b^19*d^10*abs(b)*
e^6 + 55*A*a*b^20*d^10*abs(b)*e^6 + 55*B*a^3*b^18*d^9*abs(b)*e^7 - 275*A*a^2*b^19*d^9*abs(b)*e^7 - 330*B*a^4*b
^17*d^8*abs(b)*e^8 + 825*A*a^3*b^18*d^8*abs(b)*e^8 + 858*B*a^5*b^16*d^7*abs(b)*e^9 - 1650*A*a^4*b^17*d^7*abs(b
)*e^9 - 1386*B*a^6*b^15*d^6*abs(b)*e^10 + 2310*A*a^5*b^16*d^6*abs(b)*e^10 + 1518*B*a^7*b^14*d^5*abs(b)*e^11 -
2310*A*a^6*b^15*d^5*abs(b)*e^11 - 1155*B*a^8*b^13*d^4*abs(b)*e^12 + 1650*A*a^7*b^14*d^4*abs(b)*e^12 + 605*B*a^
9*b^12*d^3*abs(b)*e^13 - 825*A*a^8*b^13*d^3*abs(b)*e^13 - 209*B*a^10*b^11*d^2*abs(b)*e^14 + 275*A*a^9*b^12*d^2
*abs(b)*e^14 + 43*B*a^11*b^10*d*abs(b)*e^15 - 55*A*a^10*b^11*d*abs(b)*e^15 - 4*B*a^12*b^9*abs(b)*e^16 + 5*A*a^
11*b^10*abs(b)*e^16)/(b^18*d^14*e^3 - 14*a*b^17*d^13*e^4 + 91*a^2*b^16*d^12*e^5 - 364*a^3*b^15*d^11*e^6 + 1001
*a^4*b^14*d^10*e^7 - 2002*a^5*b^13*d^9*e^8 + 3003*a^6*b^12*d^8*e^9 - 3432*a^7*b^11*d^7*e^10 + 3003*a^8*b^10*d^
6*e^11 - 2002*a^9*b^9*d^5*e^12 + 1001*a^10*b^8*d^4*e^13 - 364*a^11*b^7*d^3*e^14 + 91*a^12*b^6*d^2*e^15 - 14*a^
13*b^5*d*e^16 + a^14*b^4*e^17))*(b*x + a) + 35*(B*b^22*d^13*abs(b)*e^3 - 9*B*a*b^21*d^12*abs(b)*e^4 - 4*A*b^22
*d^12*abs(b)*e^4 + 30*B*a^2*b^20*d^11*abs(b)*e^5 + 48*A*a*b^21*d^11*abs(b)*e^5 - 22*B*a^3*b^19*d^10*abs(b)*e^6
- 264*A*a^2*b^20*d^10*abs(b)*e^6 - 165*B*a^4*b^18*d^9*abs(b)*e^7 + 880*A*a^3*b^19*d^9*abs(b)*e^7 + 693*B*a^5*
b^17*d^8*abs(b)*e^8 - 1980*A*a^4*b^18*d^8*abs(b)*e^8 - 1452*B*a^6*b^16*d^7*abs(b)*e^9 + 3168*A*a^5*b^17*d^7*ab
s(b)*e^9 + 1980*B*a^7*b^15*d^6*abs(b)*e^10 - 3696*A*a^6*b^16*d^6*abs(b)*e^10 - 1881*B*a^8*b^14*d^5*abs(b)*e^11
+ 3168*A*a^7*b^15*d^5*abs(b)*e^11 + 1265*B*a^9*b^13*d^4*abs(b)*e^12 - 1980*A*a^8*b^14*d^4*abs(b)*e^12 - 594*B
*a^10*b^12*d^3*abs(b)*e^13 + 880*A*a^9*b^13*d^3*abs(b)*e^13 + 186*B*a^11*b^11*d^2*abs(b)*e^14 - 264*A*a^10*b^1
2*d^2*abs(b)*e^14 - 35*B*a^12*b^10*d*abs(b)*e^15 + 48*A*a^11*b^11*d*abs(b)*e^15 + 3*B*a^13*b^9*abs(b)*e^16 - 4
*A*a^12*b^10*abs(b)*e^16)/(b^18*d^14*e^3 - 14*a*b^17*d^13*e^4 + 91*a^2*b^16*d^12*e^5 - 364*a^3*b^15*d^11*e^6 +
1001*a^4*b^14*d^10*e^7 - 2002*a^5*b^13*d^9*e^8 + 3003*a^6*b^12*d^8*e^9 - 3432*a^7*b^11*d^7*e^10 + 3003*a^8*b^
10*d^6*e^11 - 2002*a^9*b^9*d^5*e^12 + 1001*a^10*b^8*d^4*e^13 - 364*a^11*b^7*d^3*e^14 + 91*a^12*b^6*d^2*e^15 -
14*a^13*b^5*d*e^16 + a^14*b^4*e^17))*sqrt(b*x + a)/(b^2*d + (b*x + a)*b*e - a*b*e)^(7/2)
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maple [B] time = 0.01, size = 505, normalized size = 2.13 \begin {gather*} -\frac {2 \left (-128 A \,b^{4} e^{4} x^{4}+112 B a \,b^{3} e^{4} x^{4}+16 B \,b^{4} d \,e^{3} x^{4}-64 A a \,b^{3} e^{4} x^{3}-448 A \,b^{4} d \,e^{3} x^{3}+56 B \,a^{2} b^{2} e^{4} x^{3}+400 B a \,b^{3} d \,e^{3} x^{3}+56 B \,b^{4} d^{2} e^{2} x^{3}+16 A \,a^{2} b^{2} e^{4} x^{2}-224 A a \,b^{3} d \,e^{3} x^{2}-560 A \,b^{4} d^{2} e^{2} x^{2}-14 B \,a^{3} b \,e^{4} x^{2}+194 B \,a^{2} b^{2} d \,e^{3} x^{2}+518 B a \,b^{3} d^{2} e^{2} x^{2}+70 B \,b^{4} d^{3} e \,x^{2}-8 A \,a^{3} b \,e^{4} x +56 A \,a^{2} b^{2} d \,e^{3} x -280 A a \,b^{3} d^{2} e^{2} x -280 A \,b^{4} d^{3} e x +7 B \,a^{4} e^{4} x -48 B \,a^{3} b d \,e^{3} x +238 B \,a^{2} b^{2} d^{2} e^{2} x +280 B a \,b^{3} d^{3} e x +35 B \,b^{4} d^{4} x +5 A \,a^{4} e^{4}-28 A \,a^{3} b d \,e^{3}+70 A \,a^{2} b^{2} d^{2} e^{2}-140 A a \,b^{3} d^{3} e -35 A \,b^{4} d^{4}+2 B \,a^{4} d \,e^{3}-14 B \,a^{3} b \,d^{2} e^{2}+70 B \,a^{2} b^{2} d^{3} e +70 B a \,b^{3} d^{4}\right )}{35 \sqrt {b x +a}\, \left (e x +d \right )^{\frac {7}{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)/(b*x+a)^(3/2)/(e*x+d)^(9/2),x)
[Out]
-2/35*(-128*A*b^4*e^4*x^4+112*B*a*b^3*e^4*x^4+16*B*b^4*d*e^3*x^4-64*A*a*b^3*e^4*x^3-448*A*b^4*d*e^3*x^3+56*B*a
^2*b^2*e^4*x^3+400*B*a*b^3*d*e^3*x^3+56*B*b^4*d^2*e^2*x^3+16*A*a^2*b^2*e^4*x^2-224*A*a*b^3*d*e^3*x^2-560*A*b^4
*d^2*e^2*x^2-14*B*a^3*b*e^4*x^2+194*B*a^2*b^2*d*e^3*x^2+518*B*a*b^3*d^2*e^2*x^2+70*B*b^4*d^3*e*x^2-8*A*a^3*b*e
^4*x+56*A*a^2*b^2*d*e^3*x-280*A*a*b^3*d^2*e^2*x-280*A*b^4*d^3*e*x+7*B*a^4*e^4*x-48*B*a^3*b*d*e^3*x+238*B*a^2*b
^2*d^2*e^2*x+280*B*a*b^3*d^3*e*x+35*B*b^4*d^4*x+5*A*a^4*e^4-28*A*a^3*b*d*e^3+70*A*a^2*b^2*d^2*e^2-140*A*a*b^3*
d^3*e-35*A*b^4*d^4+2*B*a^4*d*e^3-14*B*a^3*b*d^2*e^2+70*B*a^2*b^2*d^3*e+70*B*a*b^3*d^4)/(b*x+a)^(1/2)/(e*x+d)^(
7/2)/(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)/(b*x+a)^(3/2)/(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 = 2.68, size = 409, normalized size = 1.73 \begin {gather*} -\frac {\sqrt {d+e\,x}\,\left (\frac {4\,B\,a^4\,d\,e^3+10\,A\,a^4\,e^4-28\,B\,a^3\,b\,d^2\,e^2-56\,A\,a^3\,b\,d\,e^3+140\,B\,a^2\,b^2\,d^3\,e+140\,A\,a^2\,b^2\,d^2\,e^2+140\,B\,a\,b^3\,d^4-280\,A\,a\,b^3\,d^3\,e-70\,A\,b^4\,d^4}{35\,e^4\,{\left (a\,e-b\,d\right )}^5}+\frac {32\,b^3\,x^4\,\left (7\,B\,a\,e-8\,A\,b\,e+B\,b\,d\right )}{35\,e\,{\left (a\,e-b\,d\right )}^5}+\frac {2\,x\,\left (7\,B\,a\,e-8\,A\,b\,e+B\,b\,d\right )\,\left (a^3\,e^3-7\,a^2\,b\,d\,e^2+35\,a\,b^2\,d^2\,e+35\,b^3\,d^3\right )}{35\,e^4\,{\left (a\,e-b\,d\right )}^5}+\frac {16\,b^2\,x^3\,\left (a\,e+7\,b\,d\right )\,\left (7\,B\,a\,e-8\,A\,b\,e+B\,b\,d\right )}{35\,e^2\,{\left (a\,e-b\,d\right )}^5}+\frac {4\,b\,x^2\,\left (-a^2\,e^2+14\,a\,b\,d\,e+35\,b^2\,d^2\right )\,\left (7\,B\,a\,e-8\,A\,b\,e+B\,b\,d\right )}{35\,e^3\,{\left (a\,e-b\,d\right )}^5}\right )}{x^4\,\sqrt {a+b\,x}+\frac {d^4\,\sqrt {a+b\,x}}{e^4}+\frac {6\,d^2\,x^2\,\sqrt {a+b\,x}}{e^2}+\frac {4\,d\,x^3\,\sqrt {a+b\,x}}{e}+\frac {4\,d^3\,x\,\sqrt {a+b\,x}}{e^3}} \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)^(9/2)),x)
[Out]
-((d + e*x)^(1/2)*((10*A*a^4*e^4 - 70*A*b^4*d^4 + 140*B*a*b^3*d^4 + 4*B*a^4*d*e^3 + 140*B*a^2*b^2*d^3*e - 28*B
*a^3*b*d^2*e^2 + 140*A*a^2*b^2*d^2*e^2 - 280*A*a*b^3*d^3*e - 56*A*a^3*b*d*e^3)/(35*e^4*(a*e - b*d)^5) + (32*b^
3*x^4*(7*B*a*e - 8*A*b*e + B*b*d))/(35*e*(a*e - b*d)^5) + (2*x*(7*B*a*e - 8*A*b*e + B*b*d)*(a^3*e^3 + 35*b^3*d
^3 + 35*a*b^2*d^2*e - 7*a^2*b*d*e^2))/(35*e^4*(a*e - b*d)^5) + (16*b^2*x^3*(a*e + 7*b*d)*(7*B*a*e - 8*A*b*e +
B*b*d))/(35*e^2*(a*e - b*d)^5) + (4*b*x^2*(35*b^2*d^2 - a^2*e^2 + 14*a*b*d*e)*(7*B*a*e - 8*A*b*e + B*b*d))/(35
*e^3*(a*e - b*d)^5)))/(x^4*(a + b*x)^(1/2) + (d^4*(a + b*x)^(1/2))/e^4 + (6*d^2*x^2*(a + b*x)^(1/2))/e^2 + (4*
d*x^3*(a + b*x)^(1/2))/e + (4*d^3*x*(a + b*x)^(1/2))/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)**(3/2)/(e*x+d)**(9/2),x)
[Out]
Timed out
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