3.22.12 \(\int \frac {A+B x}{(a+b x)^{3/2} (d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=187 \[ -\frac {2 (A b-a B)}{b \sqrt {a+b x} (d+e x)^{5/2} (b d-a e)}+\frac {16 b \sqrt {a+b x} (5 a B e-6 A b e+b B d)}{15 \sqrt {d+e x} (b d-a e)^4}+\frac {8 \sqrt {a+b x} (5 a B e-6 A b e+b B d)}{15 (d+e x)^{3/2} (b d-a e)^3}+\frac {2 \sqrt {a+b x} (5 a B e-6 A b e+b B d)}{5 b (d+e x)^{5/2} (b d-a e)^2} \]

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Rubi [A]  time = 0.11, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 4, 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 {2 (A b-a B)}{b \sqrt {a+b x} (d+e x)^{5/2} (b d-a e)}+\frac {16 b \sqrt {a+b x} (5 a B e-6 A b e+b B d)}{15 \sqrt {d+e x} (b d-a e)^4}+\frac {8 \sqrt {a+b x} (5 a B e-6 A b e+b B d)}{15 (d+e x)^{3/2} (b d-a e)^3}+\frac {2 \sqrt {a+b x} (5 a B e-6 A b e+b B d)}{5 b (d+e x)^{5/2} (b d-a e)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/((a + b*x)^(3/2)*(d + e*x)^(7/2)),x]

[Out]

(-2*(A*b - a*B))/(b*(b*d - a*e)*Sqrt[a + b*x]*(d + e*x)^(5/2)) + (2*(b*B*d - 6*A*b*e + 5*a*B*e)*Sqrt[a + b*x])
/(5*b*(b*d - a*e)^2*(d + e*x)^(5/2)) + (8*(b*B*d - 6*A*b*e + 5*a*B*e)*Sqrt[a + b*x])/(15*(b*d - a*e)^3*(d + e*
x)^(3/2)) + (16*b*(b*B*d - 6*A*b*e + 5*a*B*e)*Sqrt[a + b*x])/(15*(b*d - a*e)^4*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)^{7/2}} \, dx &=-\frac {2 (A b-a B)}{b (b d-a e) \sqrt {a+b x} (d+e x)^{5/2}}+\frac {(b B d-6 A b e+5 a B e) \int \frac {1}{\sqrt {a+b x} (d+e x)^{7/2}} \, dx}{b (b d-a e)}\\ &=-\frac {2 (A b-a B)}{b (b d-a e) \sqrt {a+b x} (d+e x)^{5/2}}+\frac {2 (b B d-6 A b e+5 a B e) \sqrt {a+b x}}{5 b (b d-a e)^2 (d+e x)^{5/2}}+\frac {(4 (b B d-6 A b e+5 a B e)) \int \frac {1}{\sqrt {a+b x} (d+e x)^{5/2}} \, dx}{5 (b d-a e)^2}\\ &=-\frac {2 (A b-a B)}{b (b d-a e) \sqrt {a+b x} (d+e x)^{5/2}}+\frac {2 (b B d-6 A b e+5 a B e) \sqrt {a+b x}}{5 b (b d-a e)^2 (d+e x)^{5/2}}+\frac {8 (b B d-6 A b e+5 a B e) \sqrt {a+b x}}{15 (b d-a e)^3 (d+e x)^{3/2}}+\frac {(8 b (b B d-6 A b e+5 a B e)) \int \frac {1}{\sqrt {a+b x} (d+e x)^{3/2}} \, dx}{15 (b d-a e)^3}\\ &=-\frac {2 (A b-a B)}{b (b d-a e) \sqrt {a+b x} (d+e x)^{5/2}}+\frac {2 (b B d-6 A b e+5 a B e) \sqrt {a+b x}}{5 b (b d-a e)^2 (d+e x)^{5/2}}+\frac {8 (b B d-6 A b e+5 a B e) \sqrt {a+b x}}{15 (b d-a e)^3 (d+e x)^{3/2}}+\frac {16 b (b B d-6 A b e+5 a B e) \sqrt {a+b x}}{15 (b d-a e)^4 \sqrt {d+e x}}\\ \end {align*}

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Mathematica [A]  time = 0.19, size = 114, normalized size = 0.61 \begin {gather*} \frac {2 \left (-(a+b x) \left (4 b (d+e x) (-a e+3 b d+2 b e x)+3 (b d-a e)^2\right ) (-5 a B e+6 A b e-b B d)-15 (A b-a B) (b d-a e)^3\right )}{15 b \sqrt {a+b x} (d+e x)^{5/2} (b d-a e)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/((a + b*x)^(3/2)*(d + e*x)^(7/2)),x]

[Out]

(2*(-15*(A*b - a*B)*(b*d - a*e)^3 - (-(b*B*d) + 6*A*b*e - 5*a*B*e)*(a + b*x)*(3*(b*d - a*e)^2 + 4*b*(d + e*x)*
(3*b*d - a*e + 2*b*e*x))))/(15*b*(b*d - a*e)^4*Sqrt[a + b*x]*(d + e*x)^(5/2))

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IntegrateAlgebraic [A]  time = 0.20, size = 205, normalized size = 1.10 \begin {gather*} \frac {2 (a+b x)^{5/2} \left (-\frac {15 A b^3 (d+e x)^3}{(a+b x)^3}-\frac {45 A b^2 e (d+e x)^2}{(a+b x)^2}+\frac {15 A b e^2 (d+e x)}{a+b x}+\frac {15 a b^2 B (d+e x)^3}{(a+b x)^3}+\frac {15 b^2 B d (d+e x)^2}{(a+b x)^2}-\frac {5 a B e^2 (d+e x)}{a+b x}+\frac {30 a b B e (d+e x)^2}{(a+b x)^2}-\frac {10 b B d e (d+e x)}{a+b x}-3 A e^3+3 B d e^2\right )}{15 (d+e x)^{5/2} (b d-a e)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(A + B*x)/((a + b*x)^(3/2)*(d + e*x)^(7/2)),x]

[Out]

(2*(a + b*x)^(5/2)*(3*B*d*e^2 - 3*A*e^3 - (10*b*B*d*e*(d + e*x))/(a + b*x) + (15*A*b*e^2*(d + e*x))/(a + b*x)
- (5*a*B*e^2*(d + e*x))/(a + b*x) + (15*b^2*B*d*(d + e*x)^2)/(a + b*x)^2 - (45*A*b^2*e*(d + e*x)^2)/(a + b*x)^
2 + (30*a*b*B*e*(d + e*x)^2)/(a + b*x)^2 - (15*A*b^3*(d + e*x)^3)/(a + b*x)^3 + (15*a*b^2*B*(d + e*x)^3)/(a +
b*x)^3))/(15*(b*d - a*e)^4*(d + e*x)^(5/2))

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fricas [B]  time = 22.42, size = 583, normalized size = 3.12 \begin {gather*} -\frac {2 \, {\left (3 \, A a^{3} e^{3} - 15 \, {\left (2 \, B a b^{2} - A b^{3}\right )} d^{3} - 5 \, {\left (4 \, B a^{2} b - 9 \, A a b^{2}\right )} d^{2} e + {\left (2 \, B a^{3} - 15 \, A a^{2} b\right )} d e^{2} - 8 \, {\left (B b^{3} d e^{2} + {\left (5 \, B a b^{2} - 6 \, A b^{3}\right )} e^{3}\right )} x^{3} - 4 \, {\left (5 \, B b^{3} d^{2} e + 2 \, {\left (13 \, B a b^{2} - 15 \, A b^{3}\right )} d e^{2} + {\left (5 \, B a^{2} b - 6 \, A a b^{2}\right )} e^{3}\right )} x^{2} - {\left (15 \, B b^{3} d^{3} + 5 \, {\left (17 \, B a b^{2} - 18 \, A b^{3}\right )} d^{2} e + {\left (49 \, B a^{2} b - 60 \, A a b^{2}\right )} d e^{2} - {\left (5 \, B a^{3} - 6 \, A a^{2} b\right )} e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{15 \, {\left (a b^{4} d^{7} - 4 \, a^{2} b^{3} d^{6} e + 6 \, a^{3} b^{2} d^{5} e^{2} - 4 \, a^{4} b d^{4} e^{3} + a^{5} d^{3} e^{4} + {\left (b^{5} d^{4} e^{3} - 4 \, a b^{4} d^{3} e^{4} + 6 \, a^{2} b^{3} d^{2} e^{5} - 4 \, a^{3} b^{2} d e^{6} + a^{4} b e^{7}\right )} x^{4} + {\left (3 \, b^{5} d^{5} e^{2} - 11 \, a b^{4} d^{4} e^{3} + 14 \, a^{2} b^{3} d^{3} e^{4} - 6 \, a^{3} b^{2} d^{2} e^{5} - a^{4} b d e^{6} + a^{5} e^{7}\right )} x^{3} + 3 \, {\left (b^{5} d^{6} e - 3 \, a b^{4} d^{5} e^{2} + 2 \, a^{2} b^{3} d^{4} e^{3} + 2 \, a^{3} b^{2} d^{3} e^{4} - 3 \, a^{4} b d^{2} e^{5} + a^{5} d e^{6}\right )} x^{2} + {\left (b^{5} d^{7} - a b^{4} d^{6} e - 6 \, a^{2} b^{3} d^{5} e^{2} + 14 \, a^{3} b^{2} d^{4} e^{3} - 11 \, a^{4} b d^{3} e^{4} + 3 \, a^{5} d^{2} e^{5}\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)^(7/2),x, algorithm="fricas")

[Out]

-2/15*(3*A*a^3*e^3 - 15*(2*B*a*b^2 - A*b^3)*d^3 - 5*(4*B*a^2*b - 9*A*a*b^2)*d^2*e + (2*B*a^3 - 15*A*a^2*b)*d*e
^2 - 8*(B*b^3*d*e^2 + (5*B*a*b^2 - 6*A*b^3)*e^3)*x^3 - 4*(5*B*b^3*d^2*e + 2*(13*B*a*b^2 - 15*A*b^3)*d*e^2 + (5
*B*a^2*b - 6*A*a*b^2)*e^3)*x^2 - (15*B*b^3*d^3 + 5*(17*B*a*b^2 - 18*A*b^3)*d^2*e + (49*B*a^2*b - 60*A*a*b^2)*d
*e^2 - (5*B*a^3 - 6*A*a^2*b)*e^3)*x)*sqrt(b*x + a)*sqrt(e*x + d)/(a*b^4*d^7 - 4*a^2*b^3*d^6*e + 6*a^3*b^2*d^5*
e^2 - 4*a^4*b*d^4*e^3 + a^5*d^3*e^4 + (b^5*d^4*e^3 - 4*a*b^4*d^3*e^4 + 6*a^2*b^3*d^2*e^5 - 4*a^3*b^2*d*e^6 + a
^4*b*e^7)*x^4 + (3*b^5*d^5*e^2 - 11*a*b^4*d^4*e^3 + 14*a^2*b^3*d^3*e^4 - 6*a^3*b^2*d^2*e^5 - a^4*b*d*e^6 + a^5
*e^7)*x^3 + 3*(b^5*d^6*e - 3*a*b^4*d^5*e^2 + 2*a^2*b^3*d^4*e^3 + 2*a^3*b^2*d^3*e^4 - 3*a^4*b*d^2*e^5 + a^5*d*e
^6)*x^2 + (b^5*d^7 - a*b^4*d^6*e - 6*a^2*b^3*d^5*e^2 + 14*a^3*b^2*d^4*e^3 - 11*a^4*b*d^3*e^4 + 3*a^5*d^2*e^5)*
x)

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giac [B]  time = 3.15, size = 1266, normalized size = 6.77

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)^(7/2),x, algorithm="giac")

[Out]

4*(B^2*a^2*b^7*e - 2*A*B*a*b^8*e + A^2*b^9*e)/((B*a*b^(11/2)*d*e^(1/2) - A*b^(13/2)*d*e^(1/2) - B*a^2*b^(9/2)*
e^(3/2) + A*a*b^(11/2)*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
^(7/2)*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^(9/2)*e^(1/2))*(b
^3*d^3*abs(b) - 3*a*b^2*d^2*abs(b)*e + 3*a^2*b*d*abs(b)*e^2 - a^3*abs(b)*e^3)) + 2/15*((b*x + a)*((8*B*b^13*d^
6*e^4 - 15*B*a*b^12*d^5*e^5 - 33*A*b^13*d^5*e^5 - 45*B*a^2*b^11*d^4*e^6 + 165*A*a*b^12*d^4*e^6 + 170*B*a^3*b^1
0*d^3*e^7 - 330*A*a^2*b^11*d^3*e^7 - 210*B*a^4*b^9*d^2*e^8 + 330*A*a^3*b^10*d^2*e^8 + 117*B*a^5*b^8*d*e^9 - 16
5*A*a^4*b^9*d*e^9 - 25*B*a^6*b^7*e^10 + 33*A*a^5*b^8*e^10)*(b*x + a)/(b^11*d^9*abs(b)*e^2 - 9*a*b^10*d^8*abs(b
)*e^3 + 36*a^2*b^9*d^7*abs(b)*e^4 - 84*a^3*b^8*d^6*abs(b)*e^5 + 126*a^4*b^7*d^5*abs(b)*e^6 - 126*a^5*b^6*d^4*a
bs(b)*e^7 + 84*a^6*b^5*d^3*abs(b)*e^8 - 36*a^7*b^4*d^2*abs(b)*e^9 + 9*a^8*b^3*d*abs(b)*e^10 - a^9*b^2*abs(b)*e
^11) + 5*(4*B*b^14*d^7*e^3 - 13*B*a*b^13*d^6*e^4 - 15*A*b^14*d^6*e^4 - 6*B*a^2*b^12*d^5*e^5 + 90*A*a*b^13*d^5*
e^5 + 85*B*a^3*b^11*d^4*e^6 - 225*A*a^2*b^12*d^4*e^6 - 160*B*a^4*b^10*d^3*e^7 + 300*A*a^3*b^11*d^3*e^7 + 141*B
*a^5*b^9*d^2*e^8 - 225*A*a^4*b^10*d^2*e^8 - 62*B*a^6*b^8*d*e^9 + 90*A*a^5*b^9*d*e^9 + 11*B*a^7*b^7*e^10 - 15*A
*a^6*b^8*e^10)/(b^11*d^9*abs(b)*e^2 - 9*a*b^10*d^8*abs(b)*e^3 + 36*a^2*b^9*d^7*abs(b)*e^4 - 84*a^3*b^8*d^6*abs
(b)*e^5 + 126*a^4*b^7*d^5*abs(b)*e^6 - 126*a^5*b^6*d^4*abs(b)*e^7 + 84*a^6*b^5*d^3*abs(b)*e^8 - 36*a^7*b^4*d^2
*abs(b)*e^9 + 9*a^8*b^3*d*abs(b)*e^10 - a^9*b^2*abs(b)*e^11)) + 15*(B*b^15*d^8*e^2 - 5*B*a*b^14*d^7*e^3 - 3*A*
b^15*d^7*e^3 + 7*B*a^2*b^13*d^6*e^4 + 21*A*a*b^14*d^6*e^4 + 7*B*a^3*b^12*d^5*e^5 - 63*A*a^2*b^13*d^5*e^5 - 35*
B*a^4*b^11*d^4*e^6 + 105*A*a^3*b^12*d^4*e^6 + 49*B*a^5*b^10*d^3*e^7 - 105*A*a^4*b^11*d^3*e^7 - 35*B*a^6*b^9*d^
2*e^8 + 63*A*a^5*b^10*d^2*e^8 + 13*B*a^7*b^8*d*e^9 - 21*A*a^6*b^9*d*e^9 - 2*B*a^8*b^7*e^10 + 3*A*a^7*b^8*e^10)
/(b^11*d^9*abs(b)*e^2 - 9*a*b^10*d^8*abs(b)*e^3 + 36*a^2*b^9*d^7*abs(b)*e^4 - 84*a^3*b^8*d^6*abs(b)*e^5 + 126*
a^4*b^7*d^5*abs(b)*e^6 - 126*a^5*b^6*d^4*abs(b)*e^7 + 84*a^6*b^5*d^3*abs(b)*e^8 - 36*a^7*b^4*d^2*abs(b)*e^9 +
9*a^8*b^3*d*abs(b)*e^10 - a^9*b^2*abs(b)*e^11))*sqrt(b*x + a)/(b^2*d + (b*x + a)*b*e - a*b*e)^(5/2)

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maple [A]  time = 0.01, size = 322, normalized size = 1.72 \begin {gather*} -\frac {2 \left (48 A \,b^{3} e^{3} x^{3}-40 B a \,b^{2} e^{3} x^{3}-8 B \,b^{3} d \,e^{2} x^{3}+24 A a \,b^{2} e^{3} x^{2}+120 A \,b^{3} d \,e^{2} x^{2}-20 B \,a^{2} b \,e^{3} x^{2}-104 B a \,b^{2} d \,e^{2} x^{2}-20 B \,b^{3} d^{2} e \,x^{2}-6 A \,a^{2} b \,e^{3} x +60 A a \,b^{2} d \,e^{2} x +90 A \,b^{3} d^{2} e x +5 B \,a^{3} e^{3} x -49 B \,a^{2} b d \,e^{2} x -85 B a \,b^{2} d^{2} e x -15 B \,b^{3} d^{3} x +3 A \,a^{3} e^{3}-15 A \,a^{2} b d \,e^{2}+45 A a \,b^{2} d^{2} e +15 A \,b^{3} d^{3}+2 B \,a^{3} d \,e^{2}-20 B \,a^{2} b \,d^{2} e -30 B a \,b^{2} d^{3}\right )}{15 \sqrt {b x +a}\, \left (e x +d \right )^{\frac {5}{2}} \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\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)^(7/2),x)

[Out]

-2/15*(48*A*b^3*e^3*x^3-40*B*a*b^2*e^3*x^3-8*B*b^3*d*e^2*x^3+24*A*a*b^2*e^3*x^2+120*A*b^3*d*e^2*x^2-20*B*a^2*b
*e^3*x^2-104*B*a*b^2*d*e^2*x^2-20*B*b^3*d^2*e*x^2-6*A*a^2*b*e^3*x+60*A*a*b^2*d*e^2*x+90*A*b^3*d^2*e*x+5*B*a^3*
e^3*x-49*B*a^2*b*d*e^2*x-85*B*a*b^2*d^2*e*x-15*B*b^3*d^3*x+3*A*a^3*e^3-15*A*a^2*b*d*e^2+45*A*a*b^2*d^2*e+15*A*
b^3*d^3+2*B*a^3*d*e^2-20*B*a^2*b*d^2*e-30*B*a*b^2*d^3)/(b*x+a)^(1/2)/(e*x+d)^(5/2)/(a^4*e^4-4*a^3*b*d*e^3+6*a^
2*b^2*d^2*e^2-4*a*b^3*d^3*e+b^4*d^4)

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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)^(7/2),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 = 2.46, size = 288, normalized size = 1.54 \begin {gather*} \frac {\sqrt {d+e\,x}\,\left (\frac {2\,x\,\left (-a^2\,e^2+10\,a\,b\,d\,e+15\,b^2\,d^2\right )\,\left (5\,B\,a\,e-6\,A\,b\,e+B\,b\,d\right )}{15\,e^3\,{\left (a\,e-b\,d\right )}^4}-\frac {4\,B\,a^3\,d\,e^2+6\,A\,a^3\,e^3-40\,B\,a^2\,b\,d^2\,e-30\,A\,a^2\,b\,d\,e^2-60\,B\,a\,b^2\,d^3+90\,A\,a\,b^2\,d^2\,e+30\,A\,b^3\,d^3}{15\,e^3\,{\left (a\,e-b\,d\right )}^4}+\frac {16\,b^2\,x^3\,\left (5\,B\,a\,e-6\,A\,b\,e+B\,b\,d\right )}{15\,e\,{\left (a\,e-b\,d\right )}^4}+\frac {8\,b\,x^2\,\left (a\,e+5\,b\,d\right )\,\left (5\,B\,a\,e-6\,A\,b\,e+B\,b\,d\right )}{15\,e^2\,{\left (a\,e-b\,d\right )}^4}\right )}{x^3\,\sqrt {a+b\,x}+\frac {d^3\,\sqrt {a+b\,x}}{e^3}+\frac {3\,d\,x^2\,\sqrt {a+b\,x}}{e}+\frac {3\,d^2\,x\,\sqrt {a+b\,x}}{e^2}} \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)^(7/2)),x)

[Out]

((d + e*x)^(1/2)*((2*x*(15*b^2*d^2 - a^2*e^2 + 10*a*b*d*e)*(5*B*a*e - 6*A*b*e + B*b*d))/(15*e^3*(a*e - b*d)^4)
- (6*A*a^3*e^3 + 30*A*b^3*d^3 - 60*B*a*b^2*d^3 + 4*B*a^3*d*e^2 + 90*A*a*b^2*d^2*e - 30*A*a^2*b*d*e^2 - 40*B*a
^2*b*d^2*e)/(15*e^3*(a*e - b*d)^4) + (16*b^2*x^3*(5*B*a*e - 6*A*b*e + B*b*d))/(15*e*(a*e - b*d)^4) + (8*b*x^2*
(a*e + 5*b*d)*(5*B*a*e - 6*A*b*e + B*b*d))/(15*e^2*(a*e - b*d)^4)))/(x^3*(a + b*x)^(1/2) + (d^3*(a + b*x)^(1/2
))/e^3 + (3*d*x^2*(a + b*x)^(1/2))/e + (3*d^2*x*(a + b*x)^(1/2))/e^2)

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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)**(7/2),x)

[Out]

Timed out

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