3.22.4 \(\int \frac {A+B x}{\sqrt {a+b x} (d+e x)^{9/2}} \, dx\)

Optimal. Leaf size=198 \[ \frac {16 b^2 \sqrt {a+b x} (-7 a B e+6 A b e+b B d)}{105 e \sqrt {d+e x} (b d-a e)^4}+\frac {8 b \sqrt {a+b x} (-7 a B e+6 A b e+b B d)}{105 e (d+e x)^{3/2} (b d-a e)^3}+\frac {2 \sqrt {a+b x} (-7 a B e+6 A b e+b B d)}{35 e (d+e x)^{5/2} (b d-a e)^2}-\frac {2 \sqrt {a+b x} (B d-A e)}{7 e (d+e x)^{7/2} (b d-a e)} \]

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Rubi [A]  time = 0.13, antiderivative size = 198, 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 {16 b^2 \sqrt {a+b x} (-7 a B e+6 A b e+b B d)}{105 e \sqrt {d+e x} (b d-a e)^4}+\frac {8 b \sqrt {a+b x} (-7 a B e+6 A b e+b B d)}{105 e (d+e x)^{3/2} (b d-a e)^3}+\frac {2 \sqrt {a+b x} (-7 a B e+6 A b e+b B d)}{35 e (d+e x)^{5/2} (b d-a e)^2}-\frac {2 \sqrt {a+b x} (B d-A e)}{7 e (d+e x)^{7/2} (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(Sqrt[a + b*x]*(d + e*x)^(9/2)),x]

[Out]

(-2*(B*d - A*e)*Sqrt[a + b*x])/(7*e*(b*d - a*e)*(d + e*x)^(7/2)) + (2*(b*B*d + 6*A*b*e - 7*a*B*e)*Sqrt[a + b*x
])/(35*e*(b*d - a*e)^2*(d + e*x)^(5/2)) + (8*b*(b*B*d + 6*A*b*e - 7*a*B*e)*Sqrt[a + b*x])/(105*e*(b*d - a*e)^3
*(d + e*x)^(3/2)) + (16*b^2*(b*B*d + 6*A*b*e - 7*a*B*e)*Sqrt[a + b*x])/(105*e*(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}{\sqrt {a+b x} (d+e x)^{9/2}} \, dx &=-\frac {2 (B d-A e) \sqrt {a+b x}}{7 e (b d-a e) (d+e x)^{7/2}}+\frac {(b B d+6 A b e-7 a B e) \int \frac {1}{\sqrt {a+b x} (d+e x)^{7/2}} \, dx}{7 e (b d-a e)}\\ &=-\frac {2 (B d-A e) \sqrt {a+b x}}{7 e (b d-a e) (d+e x)^{7/2}}+\frac {2 (b B d+6 A b e-7 a B e) \sqrt {a+b x}}{35 e (b d-a e)^2 (d+e x)^{5/2}}+\frac {(4 b (b B d+6 A b e-7 a B e)) \int \frac {1}{\sqrt {a+b x} (d+e x)^{5/2}} \, dx}{35 e (b d-a e)^2}\\ &=-\frac {2 (B d-A e) \sqrt {a+b x}}{7 e (b d-a e) (d+e x)^{7/2}}+\frac {2 (b B d+6 A b e-7 a B e) \sqrt {a+b x}}{35 e (b d-a e)^2 (d+e x)^{5/2}}+\frac {8 b (b B d+6 A b e-7 a B e) \sqrt {a+b x}}{105 e (b d-a e)^3 (d+e x)^{3/2}}+\frac {\left (8 b^2 (b B d+6 A b e-7 a B e)\right ) \int \frac {1}{\sqrt {a+b x} (d+e x)^{3/2}} \, dx}{105 e (b d-a e)^3}\\ &=-\frac {2 (B d-A e) \sqrt {a+b x}}{7 e (b d-a e) (d+e x)^{7/2}}+\frac {2 (b B d+6 A b e-7 a B e) \sqrt {a+b x}}{35 e (b d-a e)^2 (d+e x)^{5/2}}+\frac {8 b (b B d+6 A b e-7 a B e) \sqrt {a+b x}}{105 e (b d-a e)^3 (d+e x)^{3/2}}+\frac {16 b^2 (b B d+6 A b e-7 a B e) \sqrt {a+b x}}{105 e (b d-a e)^4 \sqrt {d+e x}}\\ \end {align*}

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Mathematica [A]  time = 0.30, size = 113, normalized size = 0.57 \begin {gather*} \frac {2 \sqrt {a+b x} \left (15 (B d-A e)-\frac {(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 ) (-7 a B e+6 A b e+b B d)}{(b d-a e)^3}\right )}{105 e (d+e x)^{7/2} (a e-b d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(Sqrt[a + b*x]*(d + e*x)^(9/2)),x]

[Out]

(2*Sqrt[a + b*x]*(15*(B*d - A*e) - ((b*B*d + 6*A*b*e - 7*a*B*e)*(d + e*x)*(3*(b*d - a*e)^2 + 4*b*(d + e*x)*(3*
b*d - a*e + 2*b*e*x)))/(b*d - a*e)^3))/(105*e*(-(b*d) + a*e)*(d + e*x)^(7/2))

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IntegrateAlgebraic [A]  time = 0.17, size = 205, normalized size = 1.04 \begin {gather*} \frac {2 \sqrt {a+b x} \left (-\frac {105 A b^2 e (a+b x)}{d+e x}-\frac {15 A e^3 (a+b x)^3}{(d+e x)^3}+\frac {63 A b e^2 (a+b x)^2}{(d+e x)^2}+\frac {35 b^2 B d (a+b x)}{d+e x}-105 a b^2 B-\frac {21 a B e^2 (a+b x)^2}{(d+e x)^2}+\frac {15 B d e^2 (a+b x)^3}{(d+e x)^3}+\frac {70 a b B e (a+b x)}{d+e x}-\frac {42 b B d e (a+b x)^2}{(d+e x)^2}+105 A b^3\right )}{105 \sqrt {d+e x} (b d-a e)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(A + B*x)/(Sqrt[a + b*x]*(d + e*x)^(9/2)),x]

[Out]

(2*Sqrt[a + b*x]*(105*A*b^3 - 105*a*b^2*B + (15*B*d*e^2*(a + b*x)^3)/(d + e*x)^3 - (15*A*e^3*(a + b*x)^3)/(d +
e*x)^3 - (42*b*B*d*e*(a + b*x)^2)/(d + e*x)^2 + (63*A*b*e^2*(a + b*x)^2)/(d + e*x)^2 - (21*a*B*e^2*(a + b*x)^
2)/(d + e*x)^2 + (35*b^2*B*d*(a + b*x))/(d + e*x) - (105*A*b^2*e*(a + b*x))/(d + e*x) + (70*a*b*B*e*(a + b*x))
/(d + e*x)))/(105*(b*d - a*e)^4*Sqrt[d + e*x])

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fricas [B]  time = 37.27, size = 547, normalized size = 2.76 \begin {gather*} -\frac {2 \, {\left (15 \, A a^{3} e^{3} + 35 \, {\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} d^{3} - 7 \, {\left (4 \, B a^{2} b - 15 \, A a b^{2}\right )} d^{2} e + 3 \, {\left (2 \, B a^{3} - 21 \, A a^{2} b\right )} d e^{2} - 8 \, {\left (B b^{3} d e^{2} - {\left (7 \, B a b^{2} - 6 \, A b^{3}\right )} e^{3}\right )} x^{3} - 4 \, {\left (7 \, B b^{3} d^{2} e - 2 \, {\left (25 \, B a b^{2} - 21 \, A b^{3}\right )} d e^{2} + {\left (7 \, B a^{2} b - 6 \, A a b^{2}\right )} e^{3}\right )} x^{2} - {\left (35 \, B b^{3} d^{3} - 7 \, {\left (37 \, B a b^{2} - 30 \, A b^{3}\right )} d^{2} e + {\left (101 \, B a^{2} b - 84 \, A a b^{2}\right )} d e^{2} - 3 \, {\left (7 \, B a^{3} - 6 \, A a^{2} b\right )} e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{105 \, {\left (b^{4} d^{8} - 4 \, a b^{3} d^{7} e + 6 \, a^{2} b^{2} d^{6} e^{2} - 4 \, a^{3} b d^{5} e^{3} + a^{4} d^{4} e^{4} + {\left (b^{4} d^{4} e^{4} - 4 \, a b^{3} d^{3} e^{5} + 6 \, a^{2} b^{2} d^{2} e^{6} - 4 \, a^{3} b d e^{7} + a^{4} e^{8}\right )} x^{4} + 4 \, {\left (b^{4} d^{5} e^{3} - 4 \, a b^{3} d^{4} e^{4} + 6 \, a^{2} b^{2} d^{3} e^{5} - 4 \, a^{3} b d^{2} e^{6} + a^{4} d e^{7}\right )} x^{3} + 6 \, {\left (b^{4} d^{6} e^{2} - 4 \, a b^{3} d^{5} e^{3} + 6 \, a^{2} b^{2} d^{4} e^{4} - 4 \, a^{3} b d^{3} e^{5} + a^{4} d^{2} e^{6}\right )} x^{2} + 4 \, {\left (b^{4} d^{7} e - 4 \, a b^{3} d^{6} e^{2} + 6 \, a^{2} b^{2} d^{5} e^{3} - 4 \, a^{3} b d^{4} e^{4} + a^{4} d^{3} e^{5}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^(9/2)/(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

-2/105*(15*A*a^3*e^3 + 35*(2*B*a*b^2 - 3*A*b^3)*d^3 - 7*(4*B*a^2*b - 15*A*a*b^2)*d^2*e + 3*(2*B*a^3 - 21*A*a^2
*b)*d*e^2 - 8*(B*b^3*d*e^2 - (7*B*a*b^2 - 6*A*b^3)*e^3)*x^3 - 4*(7*B*b^3*d^2*e - 2*(25*B*a*b^2 - 21*A*b^3)*d*e
^2 + (7*B*a^2*b - 6*A*a*b^2)*e^3)*x^2 - (35*B*b^3*d^3 - 7*(37*B*a*b^2 - 30*A*b^3)*d^2*e + (101*B*a^2*b - 84*A*
a*b^2)*d*e^2 - 3*(7*B*a^3 - 6*A*a^2*b)*e^3)*x)*sqrt(b*x + a)*sqrt(e*x + d)/(b^4*d^8 - 4*a*b^3*d^7*e + 6*a^2*b^
2*d^6*e^2 - 4*a^3*b*d^5*e^3 + a^4*d^4*e^4 + (b^4*d^4*e^4 - 4*a*b^3*d^3*e^5 + 6*a^2*b^2*d^2*e^6 - 4*a^3*b*d*e^7
+ a^4*e^8)*x^4 + 4*(b^4*d^5*e^3 - 4*a*b^3*d^4*e^4 + 6*a^2*b^2*d^3*e^5 - 4*a^3*b*d^2*e^6 + a^4*d*e^7)*x^3 + 6*
(b^4*d^6*e^2 - 4*a*b^3*d^5*e^3 + 6*a^2*b^2*d^4*e^4 - 4*a^3*b*d^3*e^5 + a^4*d^2*e^6)*x^2 + 4*(b^4*d^7*e - 4*a*b
^3*d^6*e^2 + 6*a^2*b^2*d^5*e^3 - 4*a^3*b*d^4*e^4 + a^4*d^3*e^5)*x)

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giac [B]  time = 1.83, size = 579, normalized size = 2.92 \begin {gather*} \frac {2 \, {\left ({\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (B b^{8} d {\left | b \right |} e^{5} - 7 \, B a b^{7} {\left | b \right |} e^{6} + 6 \, A b^{8} {\left | b \right |} e^{6}\right )} {\left (b x + a\right )}}{b^{6} d^{4} e^{3} - 4 \, a b^{5} d^{3} e^{4} + 6 \, a^{2} b^{4} d^{2} e^{5} - 4 \, a^{3} b^{3} d e^{6} + a^{4} b^{2} e^{7}} + \frac {7 \, {\left (B b^{9} d^{2} {\left | b \right |} e^{4} - 8 \, B a b^{8} d {\left | b \right |} e^{5} + 6 \, A b^{9} d {\left | b \right |} e^{5} + 7 \, B a^{2} b^{7} {\left | b \right |} e^{6} - 6 \, A a b^{8} {\left | b \right |} e^{6}\right )}}{b^{6} d^{4} e^{3} - 4 \, a b^{5} d^{3} e^{4} + 6 \, a^{2} b^{4} d^{2} e^{5} - 4 \, a^{3} b^{3} d e^{6} + a^{4} b^{2} e^{7}}\right )} + \frac {35 \, {\left (B b^{10} d^{3} {\left | b \right |} e^{3} - 9 \, B a b^{9} d^{2} {\left | b \right |} e^{4} + 6 \, A b^{10} d^{2} {\left | b \right |} e^{4} + 15 \, B a^{2} b^{8} d {\left | b \right |} e^{5} - 12 \, A a b^{9} d {\left | b \right |} e^{5} - 7 \, B a^{3} b^{7} {\left | b \right |} e^{6} + 6 \, A a^{2} b^{8} {\left | b \right |} e^{6}\right )}}{b^{6} d^{4} e^{3} - 4 \, a b^{5} d^{3} e^{4} + 6 \, a^{2} b^{4} d^{2} e^{5} - 4 \, a^{3} b^{3} d e^{6} + a^{4} b^{2} e^{7}}\right )} {\left (b x + a\right )} - \frac {105 \, {\left (B a b^{10} d^{3} {\left | b \right |} e^{3} - A b^{11} d^{3} {\left | b \right |} e^{3} - 3 \, B a^{2} b^{9} d^{2} {\left | b \right |} e^{4} + 3 \, A a b^{10} d^{2} {\left | b \right |} e^{4} + 3 \, B a^{3} b^{8} d {\left | b \right |} e^{5} - 3 \, A a^{2} b^{9} d {\left | b \right |} e^{5} - B a^{4} b^{7} {\left | b \right |} e^{6} + A a^{3} b^{8} {\left | b \right |} e^{6}\right )}}{b^{6} d^{4} e^{3} - 4 \, a b^{5} d^{3} e^{4} + 6 \, a^{2} b^{4} d^{2} e^{5} - 4 \, a^{3} b^{3} d e^{6} + a^{4} b^{2} e^{7}}\right )} \sqrt {b x + a}}{105 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^(9/2)/(b*x+a)^(1/2),x, algorithm="giac")

[Out]

2/105*((4*(b*x + a)*(2*(B*b^8*d*abs(b)*e^5 - 7*B*a*b^7*abs(b)*e^6 + 6*A*b^8*abs(b)*e^6)*(b*x + a)/(b^6*d^4*e^3
- 4*a*b^5*d^3*e^4 + 6*a^2*b^4*d^2*e^5 - 4*a^3*b^3*d*e^6 + a^4*b^2*e^7) + 7*(B*b^9*d^2*abs(b)*e^4 - 8*B*a*b^8*
d*abs(b)*e^5 + 6*A*b^9*d*abs(b)*e^5 + 7*B*a^2*b^7*abs(b)*e^6 - 6*A*a*b^8*abs(b)*e^6)/(b^6*d^4*e^3 - 4*a*b^5*d^
3*e^4 + 6*a^2*b^4*d^2*e^5 - 4*a^3*b^3*d*e^6 + a^4*b^2*e^7)) + 35*(B*b^10*d^3*abs(b)*e^3 - 9*B*a*b^9*d^2*abs(b)
*e^4 + 6*A*b^10*d^2*abs(b)*e^4 + 15*B*a^2*b^8*d*abs(b)*e^5 - 12*A*a*b^9*d*abs(b)*e^5 - 7*B*a^3*b^7*abs(b)*e^6
+ 6*A*a^2*b^8*abs(b)*e^6)/(b^6*d^4*e^3 - 4*a*b^5*d^3*e^4 + 6*a^2*b^4*d^2*e^5 - 4*a^3*b^3*d*e^6 + a^4*b^2*e^7))
*(b*x + a) - 105*(B*a*b^10*d^3*abs(b)*e^3 - A*b^11*d^3*abs(b)*e^3 - 3*B*a^2*b^9*d^2*abs(b)*e^4 + 3*A*a*b^10*d^
2*abs(b)*e^4 + 3*B*a^3*b^8*d*abs(b)*e^5 - 3*A*a^2*b^9*d*abs(b)*e^5 - B*a^4*b^7*abs(b)*e^6 + A*a^3*b^8*abs(b)*e
^6)/(b^6*d^4*e^3 - 4*a*b^5*d^3*e^4 + 6*a^2*b^4*d^2*e^5 - 4*a^3*b^3*d*e^6 + a^4*b^2*e^7))*sqrt(b*x + a)/(b^2*d
+ (b*x + a)*b*e - a*b*e)^(7/2)

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maple [A]  time = 0.01, size = 322, normalized size = 1.63 \begin {gather*} -\frac {2 \sqrt {b x +a}\, \left (-48 A \,b^{3} e^{3} x^{3}+56 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}-168 A \,b^{3} d \,e^{2} x^{2}-28 B \,a^{2} b \,e^{3} x^{2}+200 B a \,b^{2} d \,e^{2} x^{2}-28 B \,b^{3} d^{2} e \,x^{2}-18 A \,a^{2} b \,e^{3} x +84 A a \,b^{2} d \,e^{2} x -210 A \,b^{3} d^{2} e x +21 B \,a^{3} e^{3} x -101 B \,a^{2} b d \,e^{2} x +259 B a \,b^{2} d^{2} e x -35 B \,b^{3} d^{3} x +15 A \,a^{3} e^{3}-63 A \,a^{2} b d \,e^{2}+105 A a \,b^{2} d^{2} e -105 A \,b^{3} d^{3}+6 B \,a^{3} d \,e^{2}-28 B \,a^{2} b \,d^{2} e +70 B a \,b^{2} d^{3}\right )}{105 \left (e x +d \right )^{\frac {7}{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)/(e*x+d)^(9/2)/(b*x+a)^(1/2),x)

[Out]

-2/105*(b*x+a)^(1/2)*(-48*A*b^3*e^3*x^3+56*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^2-168*A*b^3*d*e^
2*x^2-28*B*a^2*b*e^3*x^2+200*B*a*b^2*d*e^2*x^2-28*B*b^3*d^2*e*x^2-18*A*a^2*b*e^3*x+84*A*a*b^2*d*e^2*x-210*A*b^
3*d^2*e*x+21*B*a^3*e^3*x-101*B*a^2*b*d*e^2*x+259*B*a*b^2*d^2*e*x-35*B*b^3*d^3*x+15*A*a^3*e^3-63*A*a^2*b*d*e^2+
105*A*a*b^2*d^2*e-105*A*b^3*d^3+6*B*a^3*d*e^2-28*B*a^2*b*d^2*e+70*B*a*b^2*d^3)/(e*x+d)^(7/2)/(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)/(e*x+d)^(9/2)/(b*x+a)^(1/2),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 = 2.49, size = 419, normalized size = 2.12 \begin {gather*} \frac {\sqrt {d+e\,x}\,\left (\frac {x\,\left (-42\,B\,a^4\,e^3+190\,B\,a^3\,b\,d\,e^2+6\,A\,a^3\,b\,e^3-462\,B\,a^2\,b^2\,d^2\,e-42\,A\,a^2\,b^2\,d\,e^2-70\,B\,a\,b^3\,d^3+210\,A\,a\,b^3\,d^2\,e+210\,A\,b^4\,d^3\right )}{105\,e^4\,{\left (a\,e-b\,d\right )}^4}-\frac {12\,B\,a^4\,d\,e^2+30\,A\,a^4\,e^3-56\,B\,a^3\,b\,d^2\,e-126\,A\,a^3\,b\,d\,e^2+140\,B\,a^2\,b^2\,d^3+210\,A\,a^2\,b^2\,d^2\,e-210\,A\,a\,b^3\,d^3}{105\,e^4\,{\left (a\,e-b\,d\right )}^4}+\frac {16\,b^3\,x^4\,\left (6\,A\,b\,e-7\,B\,a\,e+B\,b\,d\right )}{105\,e^2\,{\left (a\,e-b\,d\right )}^4}+\frac {8\,b^2\,x^3\,\left (a\,e+7\,b\,d\right )\,\left (6\,A\,b\,e-7\,B\,a\,e+B\,b\,d\right )}{105\,e^3\,{\left (a\,e-b\,d\right )}^4}+\frac {2\,b\,x^2\,\left (-a^2\,e^2+14\,a\,b\,d\,e+35\,b^2\,d^2\right )\,\left (6\,A\,b\,e-7\,B\,a\,e+B\,b\,d\right )}{105\,e^4\,{\left (a\,e-b\,d\right )}^4}\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)^(1/2)*(d + e*x)^(9/2)),x)

[Out]

((d + e*x)^(1/2)*((x*(210*A*b^4*d^3 - 42*B*a^4*e^3 + 6*A*a^3*b*e^3 - 70*B*a*b^3*d^3 - 42*A*a^2*b^2*d*e^2 - 462
*B*a^2*b^2*d^2*e + 210*A*a*b^3*d^2*e + 190*B*a^3*b*d*e^2))/(105*e^4*(a*e - b*d)^4) - (30*A*a^4*e^3 - 210*A*a*b
^3*d^3 + 12*B*a^4*d*e^2 + 140*B*a^2*b^2*d^3 + 210*A*a^2*b^2*d^2*e - 126*A*a^3*b*d*e^2 - 56*B*a^3*b*d^2*e)/(105
*e^4*(a*e - b*d)^4) + (16*b^3*x^4*(6*A*b*e - 7*B*a*e + B*b*d))/(105*e^2*(a*e - b*d)^4) + (8*b^2*x^3*(a*e + 7*b
*d)*(6*A*b*e - 7*B*a*e + B*b*d))/(105*e^3*(a*e - b*d)^4) + (2*b*x^2*(35*b^2*d^2 - a^2*e^2 + 14*a*b*d*e)*(6*A*b
*e - 7*B*a*e + B*b*d))/(105*e^4*(a*e - b*d)^4)))/(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)/(e*x+d)**(9/2)/(b*x+a)**(1/2),x)

[Out]

Timed out

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