3.21.70 \(\int \frac {\sqrt {a+b x} (A+B x)}{(d+e x)^{13/2}} \, dx\)

Optimal. Leaf size=255 \[ \frac {32 b^3 (a+b x)^{3/2} (-11 a B e+8 A b e+3 b B d)}{3465 e (d+e x)^{3/2} (b d-a e)^5}+\frac {16 b^2 (a+b x)^{3/2} (-11 a B e+8 A b e+3 b B d)}{1155 e (d+e x)^{5/2} (b d-a e)^4}+\frac {4 b (a+b x)^{3/2} (-11 a B e+8 A b e+3 b B d)}{231 e (d+e x)^{7/2} (b d-a e)^3}+\frac {2 (a+b x)^{3/2} (-11 a B e+8 A b e+3 b B d)}{99 e (d+e x)^{9/2} (b d-a e)^2}-\frac {2 (a+b x)^{3/2} (B d-A e)}{11 e (d+e x)^{11/2} (b d-a e)} \]

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Rubi [A]  time = 0.17, antiderivative size = 255, 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 (a+b x)^{3/2} (-11 a B e+8 A b e+3 b B d)}{3465 e (d+e x)^{3/2} (b d-a e)^5}+\frac {16 b^2 (a+b x)^{3/2} (-11 a B e+8 A b e+3 b B d)}{1155 e (d+e x)^{5/2} (b d-a e)^4}+\frac {4 b (a+b x)^{3/2} (-11 a B e+8 A b e+3 b B d)}{231 e (d+e x)^{7/2} (b d-a e)^3}+\frac {2 (a+b x)^{3/2} (-11 a B e+8 A b e+3 b B d)}{99 e (d+e x)^{9/2} (b d-a e)^2}-\frac {2 (a+b x)^{3/2} (B d-A e)}{11 e (d+e x)^{11/2} (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sqrt[a + b*x]*(A + B*x))/(d + e*x)^(13/2),x]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(3/2))/(11*e*(b*d - a*e)*(d + e*x)^(11/2)) + (2*(3*b*B*d + 8*A*b*e - 11*a*B*e)*(a +
b*x)^(3/2))/(99*e*(b*d - a*e)^2*(d + e*x)^(9/2)) + (4*b*(3*b*B*d + 8*A*b*e - 11*a*B*e)*(a + b*x)^(3/2))/(231*e
*(b*d - a*e)^3*(d + e*x)^(7/2)) + (16*b^2*(3*b*B*d + 8*A*b*e - 11*a*B*e)*(a + b*x)^(3/2))/(1155*e*(b*d - a*e)^
4*(d + e*x)^(5/2)) + (32*b^3*(3*b*B*d + 8*A*b*e - 11*a*B*e)*(a + b*x)^(3/2))/(3465*e*(b*d - a*e)^5*(d + e*x)^(
3/2))

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 {\sqrt {a+b x} (A+B x)}{(d+e x)^{13/2}} \, dx &=-\frac {2 (B d-A e) (a+b x)^{3/2}}{11 e (b d-a e) (d+e x)^{11/2}}+\frac {(3 b B d+8 A b e-11 a B e) \int \frac {\sqrt {a+b x}}{(d+e x)^{11/2}} \, dx}{11 e (b d-a e)}\\ &=-\frac {2 (B d-A e) (a+b x)^{3/2}}{11 e (b d-a e) (d+e x)^{11/2}}+\frac {2 (3 b B d+8 A b e-11 a B e) (a+b x)^{3/2}}{99 e (b d-a e)^2 (d+e x)^{9/2}}+\frac {(2 b (3 b B d+8 A b e-11 a B e)) \int \frac {\sqrt {a+b x}}{(d+e x)^{9/2}} \, dx}{33 e (b d-a e)^2}\\ &=-\frac {2 (B d-A e) (a+b x)^{3/2}}{11 e (b d-a e) (d+e x)^{11/2}}+\frac {2 (3 b B d+8 A b e-11 a B e) (a+b x)^{3/2}}{99 e (b d-a e)^2 (d+e x)^{9/2}}+\frac {4 b (3 b B d+8 A b e-11 a B e) (a+b x)^{3/2}}{231 e (b d-a e)^3 (d+e x)^{7/2}}+\frac {\left (8 b^2 (3 b B d+8 A b e-11 a B e)\right ) \int \frac {\sqrt {a+b x}}{(d+e x)^{7/2}} \, dx}{231 e (b d-a e)^3}\\ &=-\frac {2 (B d-A e) (a+b x)^{3/2}}{11 e (b d-a e) (d+e x)^{11/2}}+\frac {2 (3 b B d+8 A b e-11 a B e) (a+b x)^{3/2}}{99 e (b d-a e)^2 (d+e x)^{9/2}}+\frac {4 b (3 b B d+8 A b e-11 a B e) (a+b x)^{3/2}}{231 e (b d-a e)^3 (d+e x)^{7/2}}+\frac {16 b^2 (3 b B d+8 A b e-11 a B e) (a+b x)^{3/2}}{1155 e (b d-a e)^4 (d+e x)^{5/2}}+\frac {\left (16 b^3 (3 b B d+8 A b e-11 a B e)\right ) \int \frac {\sqrt {a+b x}}{(d+e x)^{5/2}} \, dx}{1155 e (b d-a e)^4}\\ &=-\frac {2 (B d-A e) (a+b x)^{3/2}}{11 e (b d-a e) (d+e x)^{11/2}}+\frac {2 (3 b B d+8 A b e-11 a B e) (a+b x)^{3/2}}{99 e (b d-a e)^2 (d+e x)^{9/2}}+\frac {4 b (3 b B d+8 A b e-11 a B e) (a+b x)^{3/2}}{231 e (b d-a e)^3 (d+e x)^{7/2}}+\frac {16 b^2 (3 b B d+8 A b e-11 a B e) (a+b x)^{3/2}}{1155 e (b d-a e)^4 (d+e x)^{5/2}}+\frac {32 b^3 (3 b B d+8 A b e-11 a B e) (a+b x)^{3/2}}{3465 e (b d-a e)^5 (d+e x)^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.43, size = 135, normalized size = 0.53 \begin {gather*} \frac {2 (a+b x)^{3/2} \left (315 (B d-A e)-\frac {(d+e x) \left (2 b (d+e x) \left (4 b (d+e x) (-3 a e+5 b d+2 b e x)+15 (b d-a e)^2\right )+35 (b d-a e)^3\right ) (-11 a B e+8 A b e+3 b B d)}{(b d-a e)^4}\right )}{3465 e (d+e x)^{11/2} (a e-b d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[a + b*x]*(A + B*x))/(d + e*x)^(13/2),x]

[Out]

(2*(a + b*x)^(3/2)*(315*(B*d - A*e) - ((3*b*B*d + 8*A*b*e - 11*a*B*e)*(d + e*x)*(35*(b*d - a*e)^3 + 2*b*(d + e
*x)*(15*(b*d - a*e)^2 + 4*b*(d + e*x)*(5*b*d - 3*a*e + 2*b*e*x))))/(b*d - a*e)^4))/(3465*e*(-(b*d) + a*e)*(d +
e*x)^(11/2))

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IntegrateAlgebraic [A]  time = 0.26, size = 276, normalized size = 1.08 \begin {gather*} \frac {2 (a+b x)^{3/2} \left (-\frac {2772 A b^3 e (a+b x)}{d+e x}+\frac {2970 A b^2 e^2 (a+b x)^2}{(d+e x)^2}+\frac {315 A e^4 (a+b x)^4}{(d+e x)^4}-\frac {1540 A b e^3 (a+b x)^3}{(d+e x)^3}+\frac {693 b^3 B d (a+b x)}{d+e x}-1155 a b^3 B+\frac {2079 a b^2 B e (a+b x)}{d+e x}-\frac {1485 b^2 B d e (a+b x)^2}{(d+e x)^2}+\frac {385 a B e^3 (a+b x)^3}{(d+e x)^3}-\frac {315 B d e^3 (a+b x)^4}{(d+e x)^4}-\frac {1485 a b B e^2 (a+b x)^2}{(d+e x)^2}+\frac {1155 b B d e^2 (a+b x)^3}{(d+e x)^3}+1155 A b^4\right )}{3465 (d+e x)^{3/2} (b d-a e)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(Sqrt[a + b*x]*(A + B*x))/(d + e*x)^(13/2),x]

[Out]

(2*(a + b*x)^(3/2)*(1155*A*b^4 - 1155*a*b^3*B - (315*B*d*e^3*(a + b*x)^4)/(d + e*x)^4 + (315*A*e^4*(a + b*x)^4
)/(d + e*x)^4 + (1155*b*B*d*e^2*(a + b*x)^3)/(d + e*x)^3 - (1540*A*b*e^3*(a + b*x)^3)/(d + e*x)^3 + (385*a*B*e
^3*(a + b*x)^3)/(d + e*x)^3 - (1485*b^2*B*d*e*(a + b*x)^2)/(d + e*x)^2 + (2970*A*b^2*e^2*(a + b*x)^2)/(d + e*x
)^2 - (1485*a*b*B*e^2*(a + b*x)^2)/(d + e*x)^2 + (693*b^3*B*d*(a + b*x))/(d + e*x) - (2772*A*b^3*e*(a + b*x))/
(d + e*x) + (2079*a*b^2*B*e*(a + b*x))/(d + e*x)))/(3465*(b*d - a*e)^5*(d + e*x)^(3/2))

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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)^(1/2)/(e*x+d)^(13/2),x, algorithm="fricas")

[Out]

Timed out

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giac [B]  time = 4.34, size = 875, normalized size = 3.43 \begin {gather*} \frac {2 \, {\left ({\left (2 \, {\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (3 \, B b^{12} d {\left | b \right |} e^{8} - 11 \, B a b^{11} {\left | b \right |} e^{9} + 8 \, A b^{12} {\left | b \right |} e^{9}\right )} {\left (b x + a\right )}}{b^{7} d^{5} e^{5} - 5 \, a b^{6} d^{4} e^{6} + 10 \, a^{2} b^{5} d^{3} e^{7} - 10 \, a^{3} b^{4} d^{2} e^{8} + 5 \, a^{4} b^{3} d e^{9} - a^{5} b^{2} e^{10}} + \frac {11 \, {\left (3 \, B b^{13} d^{2} {\left | b \right |} e^{7} - 14 \, B a b^{12} d {\left | b \right |} e^{8} + 8 \, A b^{13} d {\left | b \right |} e^{8} + 11 \, B a^{2} b^{11} {\left | b \right |} e^{9} - 8 \, A a b^{12} {\left | b \right |} e^{9}\right )}}{b^{7} d^{5} e^{5} - 5 \, a b^{6} d^{4} e^{6} + 10 \, a^{2} b^{5} d^{3} e^{7} - 10 \, a^{3} b^{4} d^{2} e^{8} + 5 \, a^{4} b^{3} d e^{9} - a^{5} b^{2} e^{10}}\right )} + \frac {99 \, {\left (3 \, B b^{14} d^{3} {\left | b \right |} e^{6} - 17 \, B a b^{13} d^{2} {\left | b \right |} e^{7} + 8 \, A b^{14} d^{2} {\left | b \right |} e^{7} + 25 \, B a^{2} b^{12} d {\left | b \right |} e^{8} - 16 \, A a b^{13} d {\left | b \right |} e^{8} - 11 \, B a^{3} b^{11} {\left | b \right |} e^{9} + 8 \, A a^{2} b^{12} {\left | b \right |} e^{9}\right )}}{b^{7} d^{5} e^{5} - 5 \, a b^{6} d^{4} e^{6} + 10 \, a^{2} b^{5} d^{3} e^{7} - 10 \, a^{3} b^{4} d^{2} e^{8} + 5 \, a^{4} b^{3} d e^{9} - a^{5} b^{2} e^{10}}\right )} {\left (b x + a\right )} + \frac {231 \, {\left (3 \, B b^{15} d^{4} {\left | b \right |} e^{5} - 20 \, B a b^{14} d^{3} {\left | b \right |} e^{6} + 8 \, A b^{15} d^{3} {\left | b \right |} e^{6} + 42 \, B a^{2} b^{13} d^{2} {\left | b \right |} e^{7} - 24 \, A a b^{14} d^{2} {\left | b \right |} e^{7} - 36 \, B a^{3} b^{12} d {\left | b \right |} e^{8} + 24 \, A a^{2} b^{13} d {\left | b \right |} e^{8} + 11 \, B a^{4} b^{11} {\left | b \right |} e^{9} - 8 \, A a^{3} b^{12} {\left | b \right |} e^{9}\right )}}{b^{7} d^{5} e^{5} - 5 \, a b^{6} d^{4} e^{6} + 10 \, a^{2} b^{5} d^{3} e^{7} - 10 \, a^{3} b^{4} d^{2} e^{8} + 5 \, a^{4} b^{3} d e^{9} - a^{5} b^{2} e^{10}}\right )} {\left (b x + a\right )} - \frac {1155 \, {\left (B a b^{15} d^{4} {\left | b \right |} e^{5} - A b^{16} d^{4} {\left | b \right |} e^{5} - 4 \, B a^{2} b^{14} d^{3} {\left | b \right |} e^{6} + 4 \, A a b^{15} d^{3} {\left | b \right |} e^{6} + 6 \, B a^{3} b^{13} d^{2} {\left | b \right |} e^{7} - 6 \, A a^{2} b^{14} d^{2} {\left | b \right |} e^{7} - 4 \, B a^{4} b^{12} d {\left | b \right |} e^{8} + 4 \, A a^{3} b^{13} d {\left | b \right |} e^{8} + B a^{5} b^{11} {\left | b \right |} e^{9} - A a^{4} b^{12} {\left | b \right |} e^{9}\right )}}{b^{7} d^{5} e^{5} - 5 \, a b^{6} d^{4} e^{6} + 10 \, a^{2} b^{5} d^{3} e^{7} - 10 \, a^{3} b^{4} d^{2} e^{8} + 5 \, a^{4} b^{3} d e^{9} - a^{5} b^{2} e^{10}}\right )} {\left (b x + a\right )}^{\frac {3}{2}}}{3465 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {11}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b*x+a)^(1/2)/(e*x+d)^(13/2),x, algorithm="giac")

[Out]

2/3465*((2*(4*(b*x + a)*(2*(3*B*b^12*d*abs(b)*e^8 - 11*B*a*b^11*abs(b)*e^9 + 8*A*b^12*abs(b)*e^9)*(b*x + a)/(b
^7*d^5*e^5 - 5*a*b^6*d^4*e^6 + 10*a^2*b^5*d^3*e^7 - 10*a^3*b^4*d^2*e^8 + 5*a^4*b^3*d*e^9 - a^5*b^2*e^10) + 11*
(3*B*b^13*d^2*abs(b)*e^7 - 14*B*a*b^12*d*abs(b)*e^8 + 8*A*b^13*d*abs(b)*e^8 + 11*B*a^2*b^11*abs(b)*e^9 - 8*A*a
*b^12*abs(b)*e^9)/(b^7*d^5*e^5 - 5*a*b^6*d^4*e^6 + 10*a^2*b^5*d^3*e^7 - 10*a^3*b^4*d^2*e^8 + 5*a^4*b^3*d*e^9 -
a^5*b^2*e^10)) + 99*(3*B*b^14*d^3*abs(b)*e^6 - 17*B*a*b^13*d^2*abs(b)*e^7 + 8*A*b^14*d^2*abs(b)*e^7 + 25*B*a^
2*b^12*d*abs(b)*e^8 - 16*A*a*b^13*d*abs(b)*e^8 - 11*B*a^3*b^11*abs(b)*e^9 + 8*A*a^2*b^12*abs(b)*e^9)/(b^7*d^5*
e^5 - 5*a*b^6*d^4*e^6 + 10*a^2*b^5*d^3*e^7 - 10*a^3*b^4*d^2*e^8 + 5*a^4*b^3*d*e^9 - a^5*b^2*e^10))*(b*x + a) +
231*(3*B*b^15*d^4*abs(b)*e^5 - 20*B*a*b^14*d^3*abs(b)*e^6 + 8*A*b^15*d^3*abs(b)*e^6 + 42*B*a^2*b^13*d^2*abs(b
)*e^7 - 24*A*a*b^14*d^2*abs(b)*e^7 - 36*B*a^3*b^12*d*abs(b)*e^8 + 24*A*a^2*b^13*d*abs(b)*e^8 + 11*B*a^4*b^11*a
bs(b)*e^9 - 8*A*a^3*b^12*abs(b)*e^9)/(b^7*d^5*e^5 - 5*a*b^6*d^4*e^6 + 10*a^2*b^5*d^3*e^7 - 10*a^3*b^4*d^2*e^8
+ 5*a^4*b^3*d*e^9 - a^5*b^2*e^10))*(b*x + a) - 1155*(B*a*b^15*d^4*abs(b)*e^5 - A*b^16*d^4*abs(b)*e^5 - 4*B*a^2
*b^14*d^3*abs(b)*e^6 + 4*A*a*b^15*d^3*abs(b)*e^6 + 6*B*a^3*b^13*d^2*abs(b)*e^7 - 6*A*a^2*b^14*d^2*abs(b)*e^7 -
4*B*a^4*b^12*d*abs(b)*e^8 + 4*A*a^3*b^13*d*abs(b)*e^8 + B*a^5*b^11*abs(b)*e^9 - A*a^4*b^12*abs(b)*e^9)/(b^7*d
^5*e^5 - 5*a*b^6*d^4*e^6 + 10*a^2*b^5*d^3*e^7 - 10*a^3*b^4*d^2*e^8 + 5*a^4*b^3*d*e^9 - a^5*b^2*e^10))*(b*x + a
)^(3/2)/(b^2*d + (b*x + a)*b*e - a*b*e)^(11/2)

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maple [B]  time = 0.01, size = 505, normalized size = 1.98 \begin {gather*} -\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (128 A \,b^{4} e^{4} x^{4}-176 B a \,b^{3} e^{4} x^{4}+48 B \,b^{4} d \,e^{3} x^{4}-192 A a \,b^{3} e^{4} x^{3}+704 A \,b^{4} d \,e^{3} x^{3}+264 B \,a^{2} b^{2} e^{4} x^{3}-1040 B a \,b^{3} d \,e^{3} x^{3}+264 B \,b^{4} d^{2} e^{2} x^{3}+240 A \,a^{2} b^{2} e^{4} x^{2}-1056 A a \,b^{3} d \,e^{3} x^{2}+1584 A \,b^{4} d^{2} e^{2} x^{2}-330 B \,a^{3} b \,e^{4} x^{2}+1542 B \,a^{2} b^{2} d \,e^{3} x^{2}-2574 B a \,b^{3} d^{2} e^{2} x^{2}+594 B \,b^{4} d^{3} e \,x^{2}-280 A \,a^{3} b \,e^{4} x +1320 A \,a^{2} b^{2} d \,e^{3} x -2376 A a \,b^{3} d^{2} e^{2} x +1848 A \,b^{4} d^{3} e x +385 B \,a^{4} e^{4} x -1920 B \,a^{3} b d \,e^{3} x +3762 B \,a^{2} b^{2} d^{2} e^{2} x -3432 B a \,b^{3} d^{3} e x +693 B \,b^{4} d^{4} x +315 A \,a^{4} e^{4}-1540 A \,a^{3} b d \,e^{3}+2970 A \,a^{2} b^{2} d^{2} e^{2}-2772 A a \,b^{3} d^{3} e +1155 A \,b^{4} d^{4}+70 B \,a^{4} d \,e^{3}-330 B \,a^{3} b \,d^{2} e^{2}+594 B \,a^{2} b^{2} d^{3} e -462 B a \,b^{3} d^{4}\right )}{3465 \left (e x +d \right )^{\frac {11}{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)^(1/2)/(e*x+d)^(13/2),x)

[Out]

-2/3465*(b*x+a)^(3/2)*(128*A*b^4*e^4*x^4-176*B*a*b^3*e^4*x^4+48*B*b^4*d*e^3*x^4-192*A*a*b^3*e^4*x^3+704*A*b^4*
d*e^3*x^3+264*B*a^2*b^2*e^4*x^3-1040*B*a*b^3*d*e^3*x^3+264*B*b^4*d^2*e^2*x^3+240*A*a^2*b^2*e^4*x^2-1056*A*a*b^
3*d*e^3*x^2+1584*A*b^4*d^2*e^2*x^2-330*B*a^3*b*e^4*x^2+1542*B*a^2*b^2*d*e^3*x^2-2574*B*a*b^3*d^2*e^2*x^2+594*B
*b^4*d^3*e*x^2-280*A*a^3*b*e^4*x+1320*A*a^2*b^2*d*e^3*x-2376*A*a*b^3*d^2*e^2*x+1848*A*b^4*d^3*e*x+385*B*a^4*e^
4*x-1920*B*a^3*b*d*e^3*x+3762*B*a^2*b^2*d^2*e^2*x-3432*B*a*b^3*d^3*e*x+693*B*b^4*d^4*x+315*A*a^4*e^4-1540*A*a^
3*b*d*e^3+2970*A*a^2*b^2*d^2*e^2-2772*A*a*b^3*d^3*e+1155*A*b^4*d^4+70*B*a^4*d*e^3-330*B*a^3*b*d^2*e^2+594*B*a^
2*b^2*d^3*e-462*B*a*b^3*d^4)/(e*x+d)^(11/2)/(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)^(1/2)/(e*x+d)^(13/2),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 = 2.65, size = 585, normalized size = 2.29 \begin {gather*} -\frac {\sqrt {d+e\,x}\,\left (\frac {\sqrt {a+b\,x}\,\left (140\,B\,a^5\,d\,e^3+630\,A\,a^5\,e^4-660\,B\,a^4\,b\,d^2\,e^2-3080\,A\,a^4\,b\,d\,e^3+1188\,B\,a^3\,b^2\,d^3\,e+5940\,A\,a^3\,b^2\,d^2\,e^2-924\,B\,a^2\,b^3\,d^4-5544\,A\,a^2\,b^3\,d^3\,e+2310\,A\,a\,b^4\,d^4\right )}{3465\,e^6\,{\left (a\,e-b\,d\right )}^5}+\frac {x\,\sqrt {a+b\,x}\,\left (770\,B\,a^5\,e^4-3700\,B\,a^4\,b\,d\,e^3+70\,A\,a^4\,b\,e^4+6864\,B\,a^3\,b^2\,d^2\,e^2-440\,A\,a^3\,b^2\,d\,e^3-5676\,B\,a^2\,b^3\,d^3\,e+1188\,A\,a^2\,b^3\,d^2\,e^2+462\,B\,a\,b^4\,d^4-1848\,A\,a\,b^4\,d^3\,e+2310\,A\,b^5\,d^4\right )}{3465\,e^6\,{\left (a\,e-b\,d\right )}^5}+\frac {32\,b^4\,x^5\,\sqrt {a+b\,x}\,\left (8\,A\,b\,e-11\,B\,a\,e+3\,B\,b\,d\right )}{3465\,e^3\,{\left (a\,e-b\,d\right )}^5}-\frac {16\,b^3\,x^4\,\left (a\,e-11\,b\,d\right )\,\sqrt {a+b\,x}\,\left (8\,A\,b\,e-11\,B\,a\,e+3\,B\,b\,d\right )}{3465\,e^4\,{\left (a\,e-b\,d\right )}^5}+\frac {4\,b^2\,x^3\,\sqrt {a+b\,x}\,\left (3\,a^2\,e^2-22\,a\,b\,d\,e+99\,b^2\,d^2\right )\,\left (8\,A\,b\,e-11\,B\,a\,e+3\,B\,b\,d\right )}{3465\,e^5\,{\left (a\,e-b\,d\right )}^5}-\frac {2\,b\,x^2\,\sqrt {a+b\,x}\,\left (8\,A\,b\,e-11\,B\,a\,e+3\,B\,b\,d\right )\,\left (5\,a^3\,e^3-33\,a^2\,b\,d\,e^2+99\,a\,b^2\,d^2\,e-231\,b^3\,d^3\right )}{3465\,e^6\,{\left (a\,e-b\,d\right )}^5}\right )}{x^6+\frac {d^6}{e^6}+\frac {6\,d\,x^5}{e}+\frac {6\,d^5\,x}{e^5}+\frac {15\,d^2\,x^4}{e^2}+\frac {20\,d^3\,x^3}{e^3}+\frac {15\,d^4\,x^2}{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)^(13/2),x)

[Out]

-((d + e*x)^(1/2)*(((a + b*x)^(1/2)*(630*A*a^5*e^4 + 2310*A*a*b^4*d^4 + 140*B*a^5*d*e^3 - 924*B*a^2*b^3*d^4 -
5544*A*a^2*b^3*d^3*e + 1188*B*a^3*b^2*d^3*e - 660*B*a^4*b*d^2*e^2 + 5940*A*a^3*b^2*d^2*e^2 - 3080*A*a^4*b*d*e^
3))/(3465*e^6*(a*e - b*d)^5) + (x*(a + b*x)^(1/2)*(2310*A*b^5*d^4 + 770*B*a^5*e^4 + 70*A*a^4*b*e^4 + 462*B*a*b
^4*d^4 - 440*A*a^3*b^2*d*e^3 - 5676*B*a^2*b^3*d^3*e + 1188*A*a^2*b^3*d^2*e^2 + 6864*B*a^3*b^2*d^2*e^2 - 1848*A
*a*b^4*d^3*e - 3700*B*a^4*b*d*e^3))/(3465*e^6*(a*e - b*d)^5) + (32*b^4*x^5*(a + b*x)^(1/2)*(8*A*b*e - 11*B*a*e
+ 3*B*b*d))/(3465*e^3*(a*e - b*d)^5) - (16*b^3*x^4*(a*e - 11*b*d)*(a + b*x)^(1/2)*(8*A*b*e - 11*B*a*e + 3*B*b
*d))/(3465*e^4*(a*e - b*d)^5) + (4*b^2*x^3*(a + b*x)^(1/2)*(3*a^2*e^2 + 99*b^2*d^2 - 22*a*b*d*e)*(8*A*b*e - 11
*B*a*e + 3*B*b*d))/(3465*e^5*(a*e - b*d)^5) - (2*b*x^2*(a + b*x)^(1/2)*(8*A*b*e - 11*B*a*e + 3*B*b*d)*(5*a^3*e
^3 - 231*b^3*d^3 + 99*a*b^2*d^2*e - 33*a^2*b*d*e^2))/(3465*e^6*(a*e - b*d)^5)))/(x^6 + d^6/e^6 + (6*d*x^5)/e +
(6*d^5*x)/e^5 + (15*d^2*x^4)/e^2 + (20*d^3*x^3)/e^3 + (15*d^4*x^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)*(b*x+a)**(1/2)/(e*x+d)**(13/2),x)

[Out]

Timed out

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