3.22.5 \(\int \frac {A+B x}{\sqrt {a+b x} (d+e x)^{11/2}} \, dx\)

Optimal. Leaf size=251 \[ \frac {32 b^3 \sqrt {a+b x} (-9 a B e+8 A b e+b B d)}{315 e \sqrt {d+e x} (b d-a e)^5}+\frac {16 b^2 \sqrt {a+b x} (-9 a B e+8 A b e+b B d)}{315 e (d+e x)^{3/2} (b d-a e)^4}+\frac {4 b \sqrt {a+b x} (-9 a B e+8 A b e+b B d)}{105 e (d+e x)^{5/2} (b d-a e)^3}+\frac {2 \sqrt {a+b x} (-9 a B e+8 A b e+b B d)}{63 e (d+e x)^{7/2} (b d-a e)^2}-\frac {2 \sqrt {a+b x} (B d-A e)}{9 e (d+e x)^{9/2} (b d-a e)} \]

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Rubi [A]  time = 0.16, antiderivative size = 251, 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^3 \sqrt {a+b x} (-9 a B e+8 A b e+b B d)}{315 e \sqrt {d+e x} (b d-a e)^5}+\frac {16 b^2 \sqrt {a+b x} (-9 a B e+8 A b e+b B d)}{315 e (d+e x)^{3/2} (b d-a e)^4}+\frac {4 b \sqrt {a+b x} (-9 a B e+8 A b e+b B d)}{105 e (d+e x)^{5/2} (b d-a e)^3}+\frac {2 \sqrt {a+b x} (-9 a B e+8 A b e+b B d)}{63 e (d+e x)^{7/2} (b d-a e)^2}-\frac {2 \sqrt {a+b x} (B d-A e)}{9 e (d+e x)^{9/2} (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(Sqrt[a + b*x]*(d + e*x)^(11/2)),x]

[Out]

(-2*(B*d - A*e)*Sqrt[a + b*x])/(9*e*(b*d - a*e)*(d + e*x)^(9/2)) + (2*(b*B*d + 8*A*b*e - 9*a*B*e)*Sqrt[a + b*x
])/(63*e*(b*d - a*e)^2*(d + e*x)^(7/2)) + (4*b*(b*B*d + 8*A*b*e - 9*a*B*e)*Sqrt[a + b*x])/(105*e*(b*d - a*e)^3
*(d + e*x)^(5/2)) + (16*b^2*(b*B*d + 8*A*b*e - 9*a*B*e)*Sqrt[a + b*x])/(315*e*(b*d - a*e)^4*(d + e*x)^(3/2)) +
(32*b^3*(b*B*d + 8*A*b*e - 9*a*B*e)*Sqrt[a + b*x])/(315*e*(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}{\sqrt {a+b x} (d+e x)^{11/2}} \, dx &=-\frac {2 (B d-A e) \sqrt {a+b x}}{9 e (b d-a e) (d+e x)^{9/2}}+\frac {(b B d+8 A b e-9 a B e) \int \frac {1}{\sqrt {a+b x} (d+e x)^{9/2}} \, dx}{9 e (b d-a e)}\\ &=-\frac {2 (B d-A e) \sqrt {a+b x}}{9 e (b d-a e) (d+e x)^{9/2}}+\frac {2 (b B d+8 A b e-9 a B e) \sqrt {a+b x}}{63 e (b d-a e)^2 (d+e x)^{7/2}}+\frac {(2 b (b B d+8 A b e-9 a B e)) \int \frac {1}{\sqrt {a+b x} (d+e x)^{7/2}} \, dx}{21 e (b d-a e)^2}\\ &=-\frac {2 (B d-A e) \sqrt {a+b x}}{9 e (b d-a e) (d+e x)^{9/2}}+\frac {2 (b B d+8 A b e-9 a B e) \sqrt {a+b x}}{63 e (b d-a e)^2 (d+e x)^{7/2}}+\frac {4 b (b B d+8 A b e-9 a B e) \sqrt {a+b x}}{105 e (b d-a e)^3 (d+e x)^{5/2}}+\frac {\left (8 b^2 (b B d+8 A b e-9 a B e)\right ) \int \frac {1}{\sqrt {a+b x} (d+e x)^{5/2}} \, dx}{105 e (b d-a e)^3}\\ &=-\frac {2 (B d-A e) \sqrt {a+b x}}{9 e (b d-a e) (d+e x)^{9/2}}+\frac {2 (b B d+8 A b e-9 a B e) \sqrt {a+b x}}{63 e (b d-a e)^2 (d+e x)^{7/2}}+\frac {4 b (b B d+8 A b e-9 a B e) \sqrt {a+b x}}{105 e (b d-a e)^3 (d+e x)^{5/2}}+\frac {16 b^2 (b B d+8 A b e-9 a B e) \sqrt {a+b x}}{315 e (b d-a e)^4 (d+e x)^{3/2}}+\frac {\left (16 b^3 (b B d+8 A b e-9 a B e)\right ) \int \frac {1}{\sqrt {a+b x} (d+e x)^{3/2}} \, dx}{315 e (b d-a e)^4}\\ &=-\frac {2 (B d-A e) \sqrt {a+b x}}{9 e (b d-a e) (d+e x)^{9/2}}+\frac {2 (b B d+8 A b e-9 a B e) \sqrt {a+b x}}{63 e (b d-a e)^2 (d+e x)^{7/2}}+\frac {4 b (b B d+8 A b e-9 a B e) \sqrt {a+b x}}{105 e (b d-a e)^3 (d+e x)^{5/2}}+\frac {16 b^2 (b B d+8 A b e-9 a B e) \sqrt {a+b x}}{315 e (b d-a e)^4 (d+e x)^{3/2}}+\frac {32 b^3 (b B d+8 A b e-9 a B e) \sqrt {a+b x}}{315 e (b d-a e)^5 \sqrt {d+e x}}\\ \end {align*}

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Mathematica [A]  time = 0.38, size = 134, normalized size = 0.53 \begin {gather*} \frac {2 \sqrt {a+b x} \left (35 (B d-A e)-\frac {(d+e 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 ) (-9 a B e+8 A b e+b B d)}{(b d-a e)^4}\right )}{315 e (d+e x)^{9/2} (a e-b d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(Sqrt[a + b*x]*(d + e*x)^(11/2)),x]

[Out]

(2*Sqrt[a + b*x]*(35*(B*d - A*e) - ((b*B*d + 8*A*b*e - 9*a*B*e)*(d + e*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))))/(b*d - a*e)^4))/(315*e*(-(b*d) + a*e)*(d + e*x)^(9/2)
)

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IntegrateAlgebraic [A]  time = 0.18, size = 344, normalized size = 1.37 \begin {gather*} \frac {2 \left (\frac {315 A b^4 \sqrt {a+b x}}{\sqrt {d+e x}}-\frac {420 A b^3 e (a+b x)^{3/2}}{(d+e x)^{3/2}}+\frac {378 A b^2 e^2 (a+b x)^{5/2}}{(d+e x)^{5/2}}+\frac {35 A e^4 (a+b x)^{9/2}}{(d+e x)^{9/2}}-\frac {180 A b e^3 (a+b x)^{7/2}}{(d+e x)^{7/2}}+\frac {105 b^3 B d (a+b x)^{3/2}}{(d+e x)^{3/2}}-\frac {315 a b^3 B \sqrt {a+b x}}{\sqrt {d+e x}}-\frac {189 b^2 B d e (a+b x)^{5/2}}{(d+e x)^{5/2}}+\frac {315 a b^2 B e (a+b x)^{3/2}}{(d+e x)^{3/2}}-\frac {35 B d e^3 (a+b x)^{9/2}}{(d+e x)^{9/2}}+\frac {45 a B e^3 (a+b x)^{7/2}}{(d+e x)^{7/2}}+\frac {135 b B d e^2 (a+b x)^{7/2}}{(d+e x)^{7/2}}-\frac {189 a b B e^2 (a+b x)^{5/2}}{(d+e x)^{5/2}}\right )}{315 (b d-a e)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(A + B*x)/(Sqrt[a + b*x]*(d + e*x)^(11/2)),x]

[Out]

(2*((-35*B*d*e^3*(a + b*x)^(9/2))/(d + e*x)^(9/2) + (35*A*e^4*(a + b*x)^(9/2))/(d + e*x)^(9/2) + (135*b*B*d*e^
2*(a + b*x)^(7/2))/(d + e*x)^(7/2) - (180*A*b*e^3*(a + b*x)^(7/2))/(d + e*x)^(7/2) + (45*a*B*e^3*(a + b*x)^(7/
2))/(d + e*x)^(7/2) - (189*b^2*B*d*e*(a + b*x)^(5/2))/(d + e*x)^(5/2) + (378*A*b^2*e^2*(a + b*x)^(5/2))/(d + e
*x)^(5/2) - (189*a*b*B*e^2*(a + b*x)^(5/2))/(d + e*x)^(5/2) + (105*b^3*B*d*(a + b*x)^(3/2))/(d + e*x)^(3/2) -
(420*A*b^3*e*(a + b*x)^(3/2))/(d + e*x)^(3/2) + (315*a*b^2*B*e*(a + b*x)^(3/2))/(d + e*x)^(3/2) + (315*A*b^4*S
qrt[a + b*x])/Sqrt[d + e*x] - (315*a*b^3*B*Sqrt[a + b*x])/Sqrt[d + e*x]))/(315*(b*d - a*e)^5)

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fricas [B]  time = 160.58, size = 839, normalized size = 3.34 \begin {gather*} \frac {2 \, {\left (35 \, A a^{4} e^{4} - 105 \, {\left (2 \, B a b^{3} - 3 \, A b^{4}\right )} d^{4} + 42 \, {\left (3 \, B a^{2} b^{2} - 10 \, A a b^{3}\right )} d^{3} e - 54 \, {\left (B a^{3} b - 7 \, A a^{2} b^{2}\right )} d^{2} e^{2} + 10 \, {\left (B a^{4} - 18 \, A a^{3} b\right )} d e^{3} + 16 \, {\left (B b^{4} d e^{3} - {\left (9 \, B a b^{3} - 8 \, A b^{4}\right )} e^{4}\right )} x^{4} + 8 \, {\left (9 \, B b^{4} d^{2} e^{2} - 2 \, {\left (41 \, B a b^{3} - 36 \, A b^{4}\right )} d e^{3} + {\left (9 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} e^{4}\right )} x^{3} + 6 \, {\left (21 \, B b^{4} d^{3} e - 3 \, {\left (65 \, B a b^{3} - 56 \, A b^{4}\right )} d^{2} e^{2} + {\left (55 \, B a^{2} b^{2} - 48 \, A a b^{3}\right )} d e^{3} - {\left (9 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} e^{4}\right )} x^{2} + {\left (105 \, B b^{4} d^{4} - 168 \, {\left (6 \, B a b^{3} - 5 \, A b^{4}\right )} d^{3} e + 18 \, {\left (33 \, B a^{2} b^{2} - 28 \, A a b^{3}\right )} d^{2} e^{2} - 8 \, {\left (31 \, B a^{3} b - 27 \, A a^{2} b^{2}\right )} d e^{3} + 5 \, {\left (9 \, B a^{4} - 8 \, A a^{3} b\right )} e^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{315 \, {\left (b^{5} d^{10} - 5 \, a b^{4} d^{9} e + 10 \, a^{2} b^{3} d^{8} e^{2} - 10 \, a^{3} b^{2} d^{7} e^{3} + 5 \, a^{4} b d^{6} e^{4} - a^{5} d^{5} e^{5} + {\left (b^{5} d^{5} e^{5} - 5 \, a b^{4} d^{4} e^{6} + 10 \, a^{2} b^{3} d^{3} e^{7} - 10 \, a^{3} b^{2} d^{2} e^{8} + 5 \, a^{4} b d e^{9} - a^{5} e^{10}\right )} x^{5} + 5 \, {\left (b^{5} d^{6} e^{4} - 5 \, a b^{4} d^{5} e^{5} + 10 \, a^{2} b^{3} d^{4} e^{6} - 10 \, a^{3} b^{2} d^{3} e^{7} + 5 \, a^{4} b d^{2} e^{8} - a^{5} d e^{9}\right )} x^{4} + 10 \, {\left (b^{5} d^{7} e^{3} - 5 \, a b^{4} d^{6} e^{4} + 10 \, a^{2} b^{3} d^{5} e^{5} - 10 \, a^{3} b^{2} d^{4} e^{6} + 5 \, a^{4} b d^{3} e^{7} - a^{5} d^{2} e^{8}\right )} x^{3} + 10 \, {\left (b^{5} d^{8} e^{2} - 5 \, a b^{4} d^{7} e^{3} + 10 \, a^{2} b^{3} d^{6} e^{4} - 10 \, a^{3} b^{2} d^{5} e^{5} + 5 \, a^{4} b d^{4} e^{6} - a^{5} d^{3} e^{7}\right )} x^{2} + 5 \, {\left (b^{5} d^{9} e - 5 \, a b^{4} d^{8} e^{2} + 10 \, a^{2} b^{3} d^{7} e^{3} - 10 \, a^{3} b^{2} d^{6} e^{4} + 5 \, a^{4} b d^{5} e^{5} - a^{5} d^{4} e^{6}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^(11/2)/(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

2/315*(35*A*a^4*e^4 - 105*(2*B*a*b^3 - 3*A*b^4)*d^4 + 42*(3*B*a^2*b^2 - 10*A*a*b^3)*d^3*e - 54*(B*a^3*b - 7*A*
a^2*b^2)*d^2*e^2 + 10*(B*a^4 - 18*A*a^3*b)*d*e^3 + 16*(B*b^4*d*e^3 - (9*B*a*b^3 - 8*A*b^4)*e^4)*x^4 + 8*(9*B*b
^4*d^2*e^2 - 2*(41*B*a*b^3 - 36*A*b^4)*d*e^3 + (9*B*a^2*b^2 - 8*A*a*b^3)*e^4)*x^3 + 6*(21*B*b^4*d^3*e - 3*(65*
B*a*b^3 - 56*A*b^4)*d^2*e^2 + (55*B*a^2*b^2 - 48*A*a*b^3)*d*e^3 - (9*B*a^3*b - 8*A*a^2*b^2)*e^4)*x^2 + (105*B*
b^4*d^4 - 168*(6*B*a*b^3 - 5*A*b^4)*d^3*e + 18*(33*B*a^2*b^2 - 28*A*a*b^3)*d^2*e^2 - 8*(31*B*a^3*b - 27*A*a^2*
b^2)*d*e^3 + 5*(9*B*a^4 - 8*A*a^3*b)*e^4)*x)*sqrt(b*x + a)*sqrt(e*x + d)/(b^5*d^10 - 5*a*b^4*d^9*e + 10*a^2*b^
3*d^8*e^2 - 10*a^3*b^2*d^7*e^3 + 5*a^4*b*d^6*e^4 - a^5*d^5*e^5 + (b^5*d^5*e^5 - 5*a*b^4*d^4*e^6 + 10*a^2*b^3*d
^3*e^7 - 10*a^3*b^2*d^2*e^8 + 5*a^4*b*d*e^9 - a^5*e^10)*x^5 + 5*(b^5*d^6*e^4 - 5*a*b^4*d^5*e^5 + 10*a^2*b^3*d^
4*e^6 - 10*a^3*b^2*d^3*e^7 + 5*a^4*b*d^2*e^8 - a^5*d*e^9)*x^4 + 10*(b^5*d^7*e^3 - 5*a*b^4*d^6*e^4 + 10*a^2*b^3
*d^5*e^5 - 10*a^3*b^2*d^4*e^6 + 5*a^4*b*d^3*e^7 - a^5*d^2*e^8)*x^3 + 10*(b^5*d^8*e^2 - 5*a*b^4*d^7*e^3 + 10*a^
2*b^3*d^6*e^4 - 10*a^3*b^2*d^5*e^5 + 5*a^4*b*d^4*e^6 - a^5*d^3*e^7)*x^2 + 5*(b^5*d^9*e - 5*a*b^4*d^8*e^2 + 10*
a^2*b^3*d^7*e^3 - 10*a^3*b^2*d^6*e^4 + 5*a^4*b*d^5*e^5 - a^5*d^4*e^6)*x)

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giac [B]  time = 2.78, size = 871, normalized size = 3.47 \begin {gather*} \frac {2 \, {\left ({\left (2 \, {\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (B b^{10} d {\left | b \right |} e^{7} - 9 \, B a b^{9} {\left | b \right |} e^{8} + 8 \, A b^{10} {\left | b \right |} e^{8}\right )} {\left (b x + a\right )}}{b^{7} d^{5} e^{4} - 5 \, a b^{6} d^{4} e^{5} + 10 \, a^{2} b^{5} d^{3} e^{6} - 10 \, a^{3} b^{4} d^{2} e^{7} + 5 \, a^{4} b^{3} d e^{8} - a^{5} b^{2} e^{9}} + \frac {9 \, {\left (B b^{11} d^{2} {\left | b \right |} e^{6} - 10 \, B a b^{10} d {\left | b \right |} e^{7} + 8 \, A b^{11} d {\left | b \right |} e^{7} + 9 \, B a^{2} b^{9} {\left | b \right |} e^{8} - 8 \, A a b^{10} {\left | b \right |} e^{8}\right )}}{b^{7} d^{5} e^{4} - 5 \, a b^{6} d^{4} e^{5} + 10 \, a^{2} b^{5} d^{3} e^{6} - 10 \, a^{3} b^{4} d^{2} e^{7} + 5 \, a^{4} b^{3} d e^{8} - a^{5} b^{2} e^{9}}\right )} + \frac {63 \, {\left (B b^{12} d^{3} {\left | b \right |} e^{5} - 11 \, B a b^{11} d^{2} {\left | b \right |} e^{6} + 8 \, A b^{12} d^{2} {\left | b \right |} e^{6} + 19 \, B a^{2} b^{10} d {\left | b \right |} e^{7} - 16 \, A a b^{11} d {\left | b \right |} e^{7} - 9 \, B a^{3} b^{9} {\left | b \right |} e^{8} + 8 \, A a^{2} b^{10} {\left | b \right |} e^{8}\right )}}{b^{7} d^{5} e^{4} - 5 \, a b^{6} d^{4} e^{5} + 10 \, a^{2} b^{5} d^{3} e^{6} - 10 \, a^{3} b^{4} d^{2} e^{7} + 5 \, a^{4} b^{3} d e^{8} - a^{5} b^{2} e^{9}}\right )} {\left (b x + a\right )} + \frac {105 \, {\left (B b^{13} d^{4} {\left | b \right |} e^{4} - 12 \, B a b^{12} d^{3} {\left | b \right |} e^{5} + 8 \, A b^{13} d^{3} {\left | b \right |} e^{5} + 30 \, B a^{2} b^{11} d^{2} {\left | b \right |} e^{6} - 24 \, A a b^{12} d^{2} {\left | b \right |} e^{6} - 28 \, B a^{3} b^{10} d {\left | b \right |} e^{7} + 24 \, A a^{2} b^{11} d {\left | b \right |} e^{7} + 9 \, B a^{4} b^{9} {\left | b \right |} e^{8} - 8 \, A a^{3} b^{10} {\left | b \right |} e^{8}\right )}}{b^{7} d^{5} e^{4} - 5 \, a b^{6} d^{4} e^{5} + 10 \, a^{2} b^{5} d^{3} e^{6} - 10 \, a^{3} b^{4} d^{2} e^{7} + 5 \, a^{4} b^{3} d e^{8} - a^{5} b^{2} e^{9}}\right )} {\left (b x + a\right )} - \frac {315 \, {\left (B a b^{13} d^{4} {\left | b \right |} e^{4} - A b^{14} d^{4} {\left | b \right |} e^{4} - 4 \, B a^{2} b^{12} d^{3} {\left | b \right |} e^{5} + 4 \, A a b^{13} d^{3} {\left | b \right |} e^{5} + 6 \, B a^{3} b^{11} d^{2} {\left | b \right |} e^{6} - 6 \, A a^{2} b^{12} d^{2} {\left | b \right |} e^{6} - 4 \, B a^{4} b^{10} d {\left | b \right |} e^{7} + 4 \, A a^{3} b^{11} d {\left | b \right |} e^{7} + B a^{5} b^{9} {\left | b \right |} e^{8} - A a^{4} b^{10} {\left | b \right |} e^{8}\right )}}{b^{7} d^{5} e^{4} - 5 \, a b^{6} d^{4} e^{5} + 10 \, a^{2} b^{5} d^{3} e^{6} - 10 \, a^{3} b^{4} d^{2} e^{7} + 5 \, a^{4} b^{3} d e^{8} - a^{5} b^{2} e^{9}}\right )} \sqrt {b x + a}}{315 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {9}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^(11/2)/(b*x+a)^(1/2),x, algorithm="giac")

[Out]

2/315*((2*(4*(b*x + a)*(2*(B*b^10*d*abs(b)*e^7 - 9*B*a*b^9*abs(b)*e^8 + 8*A*b^10*abs(b)*e^8)*(b*x + a)/(b^7*d^
5*e^4 - 5*a*b^6*d^4*e^5 + 10*a^2*b^5*d^3*e^6 - 10*a^3*b^4*d^2*e^7 + 5*a^4*b^3*d*e^8 - a^5*b^2*e^9) + 9*(B*b^11
*d^2*abs(b)*e^6 - 10*B*a*b^10*d*abs(b)*e^7 + 8*A*b^11*d*abs(b)*e^7 + 9*B*a^2*b^9*abs(b)*e^8 - 8*A*a*b^10*abs(b
)*e^8)/(b^7*d^5*e^4 - 5*a*b^6*d^4*e^5 + 10*a^2*b^5*d^3*e^6 - 10*a^3*b^4*d^2*e^7 + 5*a^4*b^3*d*e^8 - a^5*b^2*e^
9)) + 63*(B*b^12*d^3*abs(b)*e^5 - 11*B*a*b^11*d^2*abs(b)*e^6 + 8*A*b^12*d^2*abs(b)*e^6 + 19*B*a^2*b^10*d*abs(b
)*e^7 - 16*A*a*b^11*d*abs(b)*e^7 - 9*B*a^3*b^9*abs(b)*e^8 + 8*A*a^2*b^10*abs(b)*e^8)/(b^7*d^5*e^4 - 5*a*b^6*d^
4*e^5 + 10*a^2*b^5*d^3*e^6 - 10*a^3*b^4*d^2*e^7 + 5*a^4*b^3*d*e^8 - a^5*b^2*e^9))*(b*x + a) + 105*(B*b^13*d^4*
abs(b)*e^4 - 12*B*a*b^12*d^3*abs(b)*e^5 + 8*A*b^13*d^3*abs(b)*e^5 + 30*B*a^2*b^11*d^2*abs(b)*e^6 - 24*A*a*b^12
*d^2*abs(b)*e^6 - 28*B*a^3*b^10*d*abs(b)*e^7 + 24*A*a^2*b^11*d*abs(b)*e^7 + 9*B*a^4*b^9*abs(b)*e^8 - 8*A*a^3*b
^10*abs(b)*e^8)/(b^7*d^5*e^4 - 5*a*b^6*d^4*e^5 + 10*a^2*b^5*d^3*e^6 - 10*a^3*b^4*d^2*e^7 + 5*a^4*b^3*d*e^8 - a
^5*b^2*e^9))*(b*x + a) - 315*(B*a*b^13*d^4*abs(b)*e^4 - A*b^14*d^4*abs(b)*e^4 - 4*B*a^2*b^12*d^3*abs(b)*e^5 +
4*A*a*b^13*d^3*abs(b)*e^5 + 6*B*a^3*b^11*d^2*abs(b)*e^6 - 6*A*a^2*b^12*d^2*abs(b)*e^6 - 4*B*a^4*b^10*d*abs(b)*
e^7 + 4*A*a^3*b^11*d*abs(b)*e^7 + B*a^5*b^9*abs(b)*e^8 - A*a^4*b^10*abs(b)*e^8)/(b^7*d^5*e^4 - 5*a*b^6*d^4*e^5
+ 10*a^2*b^5*d^3*e^6 - 10*a^3*b^4*d^2*e^7 + 5*a^4*b^3*d*e^8 - a^5*b^2*e^9))*sqrt(b*x + a)/(b^2*d + (b*x + a)*
b*e - a*b*e)^(9/2)

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maple [B]  time = 0.01, size = 505, normalized size = 2.01 \begin {gather*} -\frac {2 \sqrt {b x +a}\, \left (128 A \,b^{4} e^{4} x^{4}-144 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}+576 A \,b^{4} d \,e^{3} x^{3}+72 B \,a^{2} b^{2} e^{4} x^{3}-656 B a \,b^{3} d \,e^{3} x^{3}+72 B \,b^{4} d^{2} e^{2} x^{3}+48 A \,a^{2} b^{2} e^{4} x^{2}-288 A a \,b^{3} d \,e^{3} x^{2}+1008 A \,b^{4} d^{2} e^{2} x^{2}-54 B \,a^{3} b \,e^{4} x^{2}+330 B \,a^{2} b^{2} d \,e^{3} x^{2}-1170 B a \,b^{3} d^{2} e^{2} x^{2}+126 B \,b^{4} d^{3} e \,x^{2}-40 A \,a^{3} b \,e^{4} x +216 A \,a^{2} b^{2} d \,e^{3} x -504 A a \,b^{3} d^{2} e^{2} x +840 A \,b^{4} d^{3} e x +45 B \,a^{4} e^{4} x -248 B \,a^{3} b d \,e^{3} x +594 B \,a^{2} b^{2} d^{2} e^{2} x -1008 B a \,b^{3} d^{3} e x +105 B \,b^{4} d^{4} x +35 A \,a^{4} e^{4}-180 A \,a^{3} b d \,e^{3}+378 A \,a^{2} b^{2} d^{2} e^{2}-420 A a \,b^{3} d^{3} e +315 A \,b^{4} d^{4}+10 B \,a^{4} d \,e^{3}-54 B \,a^{3} b \,d^{2} e^{2}+126 B \,a^{2} b^{2} d^{3} e -210 B a \,b^{3} d^{4}\right )}{315 \left (e x +d \right )^{\frac {9}{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)/(e*x+d)^(11/2)/(b*x+a)^(1/2),x)

[Out]

-2/315*(b*x+a)^(1/2)*(128*A*b^4*e^4*x^4-144*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+576*A*b^4*d*
e^3*x^3+72*B*a^2*b^2*e^4*x^3-656*B*a*b^3*d*e^3*x^3+72*B*b^4*d^2*e^2*x^3+48*A*a^2*b^2*e^4*x^2-288*A*a*b^3*d*e^3
*x^2+1008*A*b^4*d^2*e^2*x^2-54*B*a^3*b*e^4*x^2+330*B*a^2*b^2*d*e^3*x^2-1170*B*a*b^3*d^2*e^2*x^2+126*B*b^4*d^3*
e*x^2-40*A*a^3*b*e^4*x+216*A*a^2*b^2*d*e^3*x-504*A*a*b^3*d^2*e^2*x+840*A*b^4*d^3*e*x+45*B*a^4*e^4*x-248*B*a^3*
b*d*e^3*x+594*B*a^2*b^2*d^2*e^2*x-1008*B*a*b^3*d^3*e*x+105*B*b^4*d^4*x+35*A*a^4*e^4-180*A*a^3*b*d*e^3+378*A*a^
2*b^2*d^2*e^2-420*A*a*b^3*d^3*e+315*A*b^4*d^4+10*B*a^4*d*e^3-54*B*a^3*b*d^2*e^2+126*B*a^2*b^2*d^3*e-210*B*a*b^
3*d^4)/(e*x+d)^(9/2)/(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)/(e*x+d)^(11/2)/(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 = 2.75, size = 570, normalized size = 2.27 \begin {gather*} -\frac {\sqrt {d+e\,x}\,\left (\frac {20\,B\,a^5\,d\,e^3+70\,A\,a^5\,e^4-108\,B\,a^4\,b\,d^2\,e^2-360\,A\,a^4\,b\,d\,e^3+252\,B\,a^3\,b^2\,d^3\,e+756\,A\,a^3\,b^2\,d^2\,e^2-420\,B\,a^2\,b^3\,d^4-840\,A\,a^2\,b^3\,d^3\,e+630\,A\,a\,b^4\,d^4}{315\,e^5\,{\left (a\,e-b\,d\right )}^5}+\frac {x\,\left (90\,B\,a^5\,e^4-476\,B\,a^4\,b\,d\,e^3-10\,A\,a^4\,b\,e^4+1080\,B\,a^3\,b^2\,d^2\,e^2+72\,A\,a^3\,b^2\,d\,e^3-1764\,B\,a^2\,b^3\,d^3\,e-252\,A\,a^2\,b^3\,d^2\,e^2-210\,B\,a\,b^4\,d^4+840\,A\,a\,b^4\,d^3\,e+630\,A\,b^5\,d^4\right )}{315\,e^5\,{\left (a\,e-b\,d\right )}^5}+\frac {32\,b^4\,x^5\,\left (8\,A\,b\,e-9\,B\,a\,e+B\,b\,d\right )}{315\,e^2\,{\left (a\,e-b\,d\right )}^5}+\frac {16\,b^3\,x^4\,\left (a\,e+9\,b\,d\right )\,\left (8\,A\,b\,e-9\,B\,a\,e+B\,b\,d\right )}{315\,e^3\,{\left (a\,e-b\,d\right )}^5}+\frac {4\,b^2\,x^3\,\left (-a^2\,e^2+18\,a\,b\,d\,e+63\,b^2\,d^2\right )\,\left (8\,A\,b\,e-9\,B\,a\,e+B\,b\,d\right )}{315\,e^4\,{\left (a\,e-b\,d\right )}^5}+\frac {2\,b\,x^2\,\left (8\,A\,b\,e-9\,B\,a\,e+B\,b\,d\right )\,\left (a^3\,e^3-9\,a^2\,b\,d\,e^2+63\,a\,b^2\,d^2\,e+105\,b^3\,d^3\right )}{315\,e^5\,{\left (a\,e-b\,d\right )}^5}\right )}{x^5\,\sqrt {a+b\,x}+\frac {d^5\,\sqrt {a+b\,x}}{e^5}+\frac {10\,d^2\,x^3\,\sqrt {a+b\,x}}{e^2}+\frac {10\,d^3\,x^2\,\sqrt {a+b\,x}}{e^3}+\frac {5\,d\,x^4\,\sqrt {a+b\,x}}{e}+\frac {5\,d^4\,x\,\sqrt {a+b\,x}}{e^4}} \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)^(11/2)),x)

[Out]

-((d + e*x)^(1/2)*((70*A*a^5*e^4 + 630*A*a*b^4*d^4 + 20*B*a^5*d*e^3 - 420*B*a^2*b^3*d^4 - 840*A*a^2*b^3*d^3*e
+ 252*B*a^3*b^2*d^3*e - 108*B*a^4*b*d^2*e^2 + 756*A*a^3*b^2*d^2*e^2 - 360*A*a^4*b*d*e^3)/(315*e^5*(a*e - b*d)^
5) + (x*(630*A*b^5*d^4 + 90*B*a^5*e^4 - 10*A*a^4*b*e^4 - 210*B*a*b^4*d^4 + 72*A*a^3*b^2*d*e^3 - 1764*B*a^2*b^3
*d^3*e - 252*A*a^2*b^3*d^2*e^2 + 1080*B*a^3*b^2*d^2*e^2 + 840*A*a*b^4*d^3*e - 476*B*a^4*b*d*e^3))/(315*e^5*(a*
e - b*d)^5) + (32*b^4*x^5*(8*A*b*e - 9*B*a*e + B*b*d))/(315*e^2*(a*e - b*d)^5) + (16*b^3*x^4*(a*e + 9*b*d)*(8*
A*b*e - 9*B*a*e + B*b*d))/(315*e^3*(a*e - b*d)^5) + (4*b^2*x^3*(63*b^2*d^2 - a^2*e^2 + 18*a*b*d*e)*(8*A*b*e -
9*B*a*e + B*b*d))/(315*e^4*(a*e - b*d)^5) + (2*b*x^2*(8*A*b*e - 9*B*a*e + B*b*d)*(a^3*e^3 + 105*b^3*d^3 + 63*a
*b^2*d^2*e - 9*a^2*b*d*e^2))/(315*e^5*(a*e - b*d)^5)))/(x^5*(a + b*x)^(1/2) + (d^5*(a + b*x)^(1/2))/e^5 + (10*
d^2*x^3*(a + b*x)^(1/2))/e^2 + (10*d^3*x^2*(a + b*x)^(1/2))/e^3 + (5*d*x^4*(a + b*x)^(1/2))/e + (5*d^4*x*(a +
b*x)^(1/2))/e^4)

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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)**(11/2)/(b*x+a)**(1/2),x)

[Out]

Timed out

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