3.1480 \(\int \frac {(c+d x)^{3/2}}{(a+b x)^{13/2}} \, dx\)

Optimal. Leaf size=136 \[ \frac {32 d^3 (c+d x)^{5/2}}{1155 (a+b x)^{5/2} (b c-a d)^4}-\frac {16 d^2 (c+d x)^{5/2}}{231 (a+b x)^{7/2} (b c-a d)^3}+\frac {4 d (c+d x)^{5/2}}{33 (a+b x)^{9/2} (b c-a d)^2}-\frac {2 (c+d x)^{5/2}}{11 (a+b x)^{11/2} (b c-a d)} \]

[Out]

-2/11*(d*x+c)^(5/2)/(-a*d+b*c)/(b*x+a)^(11/2)+4/33*d*(d*x+c)^(5/2)/(-a*d+b*c)^2/(b*x+a)^(9/2)-16/231*d^2*(d*x+
c)^(5/2)/(-a*d+b*c)^3/(b*x+a)^(7/2)+32/1155*d^3*(d*x+c)^(5/2)/(-a*d+b*c)^4/(b*x+a)^(5/2)

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Rubi [A]  time = 0.03, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {45, 37} \[ \frac {32 d^3 (c+d x)^{5/2}}{1155 (a+b x)^{5/2} (b c-a d)^4}-\frac {16 d^2 (c+d x)^{5/2}}{231 (a+b x)^{7/2} (b c-a d)^3}+\frac {4 d (c+d x)^{5/2}}{33 (a+b x)^{9/2} (b c-a d)^2}-\frac {2 (c+d x)^{5/2}}{11 (a+b x)^{11/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^(3/2)/(a + b*x)^(13/2),x]

[Out]

(-2*(c + d*x)^(5/2))/(11*(b*c - a*d)*(a + b*x)^(11/2)) + (4*d*(c + d*x)^(5/2))/(33*(b*c - a*d)^2*(a + b*x)^(9/
2)) - (16*d^2*(c + d*x)^(5/2))/(231*(b*c - a*d)^3*(a + b*x)^(7/2)) + (32*d^3*(c + d*x)^(5/2))/(1155*(b*c - a*d
)^4*(a + b*x)^(5/2))

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])

Rubi steps

\begin {align*} \int \frac {(c+d x)^{3/2}}{(a+b x)^{13/2}} \, dx &=-\frac {2 (c+d x)^{5/2}}{11 (b c-a d) (a+b x)^{11/2}}-\frac {(6 d) \int \frac {(c+d x)^{3/2}}{(a+b x)^{11/2}} \, dx}{11 (b c-a d)}\\ &=-\frac {2 (c+d x)^{5/2}}{11 (b c-a d) (a+b x)^{11/2}}+\frac {4 d (c+d x)^{5/2}}{33 (b c-a d)^2 (a+b x)^{9/2}}+\frac {\left (8 d^2\right ) \int \frac {(c+d x)^{3/2}}{(a+b x)^{9/2}} \, dx}{33 (b c-a d)^2}\\ &=-\frac {2 (c+d x)^{5/2}}{11 (b c-a d) (a+b x)^{11/2}}+\frac {4 d (c+d x)^{5/2}}{33 (b c-a d)^2 (a+b x)^{9/2}}-\frac {16 d^2 (c+d x)^{5/2}}{231 (b c-a d)^3 (a+b x)^{7/2}}-\frac {\left (16 d^3\right ) \int \frac {(c+d x)^{3/2}}{(a+b x)^{7/2}} \, dx}{231 (b c-a d)^3}\\ &=-\frac {2 (c+d x)^{5/2}}{11 (b c-a d) (a+b x)^{11/2}}+\frac {4 d (c+d x)^{5/2}}{33 (b c-a d)^2 (a+b x)^{9/2}}-\frac {16 d^2 (c+d x)^{5/2}}{231 (b c-a d)^3 (a+b x)^{7/2}}+\frac {32 d^3 (c+d x)^{5/2}}{1155 (b c-a d)^4 (a+b x)^{5/2}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 118, normalized size = 0.87 \[ \frac {2 (c+d x)^{5/2} \left (231 a^3 d^3+99 a^2 b d^2 (2 d x-5 c)+11 a b^2 d \left (35 c^2-20 c d x+8 d^2 x^2\right )+b^3 \left (-105 c^3+70 c^2 d x-40 c d^2 x^2+16 d^3 x^3\right )\right )}{1155 (a+b x)^{11/2} (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^(3/2)/(a + b*x)^(13/2),x]

[Out]

(2*(c + d*x)^(5/2)*(231*a^3*d^3 + 99*a^2*b*d^2*(-5*c + 2*d*x) + 11*a*b^2*d*(35*c^2 - 20*c*d*x + 8*d^2*x^2) + b
^3*(-105*c^3 + 70*c^2*d*x - 40*c*d^2*x^2 + 16*d^3*x^3)))/(1155*(b*c - a*d)^4*(a + b*x)^(11/2))

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fricas [B]  time = 9.76, size = 649, normalized size = 4.77 \[ \frac {2 \, {\left (16 \, b^{3} d^{5} x^{5} - 105 \, b^{3} c^{5} + 385 \, a b^{2} c^{4} d - 495 \, a^{2} b c^{3} d^{2} + 231 \, a^{3} c^{2} d^{3} - 8 \, {\left (b^{3} c d^{4} - 11 \, a b^{2} d^{5}\right )} x^{4} + 2 \, {\left (3 \, b^{3} c^{2} d^{3} - 22 \, a b^{2} c d^{4} + 99 \, a^{2} b d^{5}\right )} x^{3} - {\left (5 \, b^{3} c^{3} d^{2} - 33 \, a b^{2} c^{2} d^{3} + 99 \, a^{2} b c d^{4} - 231 \, a^{3} d^{5}\right )} x^{2} - 2 \, {\left (70 \, b^{3} c^{4} d - 275 \, a b^{2} c^{3} d^{2} + 396 \, a^{2} b c^{2} d^{3} - 231 \, a^{3} c d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{1155 \, {\left (a^{6} b^{4} c^{4} - 4 \, a^{7} b^{3} c^{3} d + 6 \, a^{8} b^{2} c^{2} d^{2} - 4 \, a^{9} b c d^{3} + a^{10} d^{4} + {\left (b^{10} c^{4} - 4 \, a b^{9} c^{3} d + 6 \, a^{2} b^{8} c^{2} d^{2} - 4 \, a^{3} b^{7} c d^{3} + a^{4} b^{6} d^{4}\right )} x^{6} + 6 \, {\left (a b^{9} c^{4} - 4 \, a^{2} b^{8} c^{3} d + 6 \, a^{3} b^{7} c^{2} d^{2} - 4 \, a^{4} b^{6} c d^{3} + a^{5} b^{5} d^{4}\right )} x^{5} + 15 \, {\left (a^{2} b^{8} c^{4} - 4 \, a^{3} b^{7} c^{3} d + 6 \, a^{4} b^{6} c^{2} d^{2} - 4 \, a^{5} b^{5} c d^{3} + a^{6} b^{4} d^{4}\right )} x^{4} + 20 \, {\left (a^{3} b^{7} c^{4} - 4 \, a^{4} b^{6} c^{3} d + 6 \, a^{5} b^{5} c^{2} d^{2} - 4 \, a^{6} b^{4} c d^{3} + a^{7} b^{3} d^{4}\right )} x^{3} + 15 \, {\left (a^{4} b^{6} c^{4} - 4 \, a^{5} b^{5} c^{3} d + 6 \, a^{6} b^{4} c^{2} d^{2} - 4 \, a^{7} b^{3} c d^{3} + a^{8} b^{2} d^{4}\right )} x^{2} + 6 \, {\left (a^{5} b^{5} c^{4} - 4 \, a^{6} b^{4} c^{3} d + 6 \, a^{7} b^{3} c^{2} d^{2} - 4 \, a^{8} b^{2} c d^{3} + a^{9} b d^{4}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(3/2)/(b*x+a)^(13/2),x, algorithm="fricas")

[Out]

2/1155*(16*b^3*d^5*x^5 - 105*b^3*c^5 + 385*a*b^2*c^4*d - 495*a^2*b*c^3*d^2 + 231*a^3*c^2*d^3 - 8*(b^3*c*d^4 -
11*a*b^2*d^5)*x^4 + 2*(3*b^3*c^2*d^3 - 22*a*b^2*c*d^4 + 99*a^2*b*d^5)*x^3 - (5*b^3*c^3*d^2 - 33*a*b^2*c^2*d^3
+ 99*a^2*b*c*d^4 - 231*a^3*d^5)*x^2 - 2*(70*b^3*c^4*d - 275*a*b^2*c^3*d^2 + 396*a^2*b*c^2*d^3 - 231*a^3*c*d^4)
*x)*sqrt(b*x + a)*sqrt(d*x + c)/(a^6*b^4*c^4 - 4*a^7*b^3*c^3*d + 6*a^8*b^2*c^2*d^2 - 4*a^9*b*c*d^3 + a^10*d^4
+ (b^10*c^4 - 4*a*b^9*c^3*d + 6*a^2*b^8*c^2*d^2 - 4*a^3*b^7*c*d^3 + a^4*b^6*d^4)*x^6 + 6*(a*b^9*c^4 - 4*a^2*b^
8*c^3*d + 6*a^3*b^7*c^2*d^2 - 4*a^4*b^6*c*d^3 + a^5*b^5*d^4)*x^5 + 15*(a^2*b^8*c^4 - 4*a^3*b^7*c^3*d + 6*a^4*b
^6*c^2*d^2 - 4*a^5*b^5*c*d^3 + a^6*b^4*d^4)*x^4 + 20*(a^3*b^7*c^4 - 4*a^4*b^6*c^3*d + 6*a^5*b^5*c^2*d^2 - 4*a^
6*b^4*c*d^3 + a^7*b^3*d^4)*x^3 + 15*(a^4*b^6*c^4 - 4*a^5*b^5*c^3*d + 6*a^6*b^4*c^2*d^2 - 4*a^7*b^3*c*d^3 + a^8
*b^2*d^4)*x^2 + 6*(a^5*b^5*c^4 - 4*a^6*b^4*c^3*d + 6*a^7*b^3*c^2*d^2 - 4*a^8*b^2*c*d^3 + a^9*b*d^4)*x)

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giac [B]  time = 3.28, size = 1823, normalized size = 13.40 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(3/2)/(b*x+a)^(13/2),x, algorithm="giac")

[Out]

64/1155*(sqrt(b*d)*b^14*c^7*d^5*abs(b) - 7*sqrt(b*d)*a*b^13*c^6*d^6*abs(b) + 21*sqrt(b*d)*a^2*b^12*c^5*d^7*abs
(b) - 35*sqrt(b*d)*a^3*b^11*c^4*d^8*abs(b) + 35*sqrt(b*d)*a^4*b^10*c^3*d^9*abs(b) - 21*sqrt(b*d)*a^5*b^9*c^2*d
^10*abs(b) + 7*sqrt(b*d)*a^6*b^8*c*d^11*abs(b) - sqrt(b*d)*a^7*b^7*d^12*abs(b) - 11*sqrt(b*d)*(sqrt(b*d)*sqrt(
b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^12*c^6*d^5*abs(b) + 66*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a)
- sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^11*c^5*d^6*abs(b) - 165*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt
(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2*b^10*c^4*d^7*abs(b) + 220*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2
*c + (b*x + a)*b*d - a*b*d))^2*a^3*b^9*c^3*d^8*abs(b) - 165*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c +
(b*x + a)*b*d - a*b*d))^2*a^4*b^8*c^2*d^9*abs(b) + 66*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x +
a)*b*d - a*b*d))^2*a^5*b^7*c*d^10*abs(b) - 11*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d
- a*b*d))^2*a^6*b^6*d^11*abs(b) + 55*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d)
)^4*b^10*c^5*d^5*abs(b) - 275*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a*b^
9*c^4*d^6*abs(b) + 550*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^2*b^8*c^3
*d^7*abs(b) - 550*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^3*b^7*c^2*d^8*
abs(b) + 275*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^4*b^6*c*d^9*abs(b)
- 55*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^5*b^5*d^10*abs(b) - 165*sqr
t(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*b^8*c^4*d^5*abs(b) + 660*sqrt(b*d)*(s
qrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a*b^7*c^3*d^6*abs(b) - 990*sqrt(b*d)*(sqrt(b*d
)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^2*b^6*c^2*d^7*abs(b) + 660*sqrt(b*d)*(sqrt(b*d)*sqr
t(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^3*b^5*c*d^8*abs(b) - 165*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x +
a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^4*b^4*d^9*abs(b) - 825*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqr
t(b^2*c + (b*x + a)*b*d - a*b*d))^8*b^6*c^3*d^5*abs(b) + 2475*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c
+ (b*x + a)*b*d - a*b*d))^8*a*b^5*c^2*d^6*abs(b) - 2475*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x
+ a)*b*d - a*b*d))^8*a^2*b^4*c*d^7*abs(b) + 825*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b
*d - a*b*d))^8*a^3*b^3*d^8*abs(b) - 2541*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b
*d))^10*b^4*c^2*d^5*abs(b) + 5082*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10
*a*b^3*c*d^6*abs(b) - 2541*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^2*b^
2*d^7*abs(b) - 2079*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*b^2*c*d^5*abs
(b) + 2079*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*a*b*d^6*abs(b) - 1155*
sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*d^5*abs(b))/(b^2*c - a*b*d - (sqr
t(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)^11

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maple [A]  time = 0.01, size = 171, normalized size = 1.26 \[ \frac {2 \left (d x +c \right )^{\frac {5}{2}} \left (16 b^{3} x^{3} d^{3}+88 a \,b^{2} d^{3} x^{2}-40 b^{3} c \,d^{2} x^{2}+198 a^{2} b \,d^{3} x -220 a \,b^{2} c \,d^{2} x +70 b^{3} c^{2} d x +231 a^{3} d^{3}-495 a^{2} b c \,d^{2}+385 a \,b^{2} c^{2} d -105 b^{3} c^{3}\right )}{1155 \left (b x +a \right )^{\frac {11}{2}} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(3/2)/(b*x+a)^(13/2),x)

[Out]

2/1155*(d*x+c)^(5/2)*(16*b^3*d^3*x^3+88*a*b^2*d^3*x^2-40*b^3*c*d^2*x^2+198*a^2*b*d^3*x-220*a*b^2*c*d^2*x+70*b^
3*c^2*d*x+231*a^3*d^3-495*a^2*b*c*d^2+385*a*b^2*c^2*d-105*b^3*c^3)/(b*x+a)^(11/2)/(a^4*d^4-4*a^3*b*c*d^3+6*a^2
*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(3/2)/(b*x+a)^(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*d-b*c>0)', see `assume?` for
more details)Is a*d-b*c zero or nonzero?

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mupad [B]  time = 1.33, size = 376, normalized size = 2.76 \[ \frac {\sqrt {c+d\,x}\,\left (\frac {x^2\,\left (462\,a^3\,d^5-198\,a^2\,b\,c\,d^4+66\,a\,b^2\,c^2\,d^3-10\,b^3\,c^3\,d^2\right )}{1155\,b^5\,{\left (a\,d-b\,c\right )}^4}-\frac {-462\,a^3\,c^2\,d^3+990\,a^2\,b\,c^3\,d^2-770\,a\,b^2\,c^4\,d+210\,b^3\,c^5}{1155\,b^5\,{\left (a\,d-b\,c\right )}^4}+\frac {x\,\left (924\,a^3\,c\,d^4-1584\,a^2\,b\,c^2\,d^3+1100\,a\,b^2\,c^3\,d^2-280\,b^3\,c^4\,d\right )}{1155\,b^5\,{\left (a\,d-b\,c\right )}^4}+\frac {32\,d^5\,x^5}{1155\,b^2\,{\left (a\,d-b\,c\right )}^4}+\frac {16\,d^4\,x^4\,\left (11\,a\,d-b\,c\right )}{1155\,b^3\,{\left (a\,d-b\,c\right )}^4}+\frac {4\,d^3\,x^3\,\left (99\,a^2\,d^2-22\,a\,b\,c\,d+3\,b^2\,c^2\right )}{1155\,b^4\,{\left (a\,d-b\,c\right )}^4}\right )}{x^5\,\sqrt {a+b\,x}+\frac {a^5\,\sqrt {a+b\,x}}{b^5}+\frac {10\,a^2\,x^3\,\sqrt {a+b\,x}}{b^2}+\frac {10\,a^3\,x^2\,\sqrt {a+b\,x}}{b^3}+\frac {5\,a\,x^4\,\sqrt {a+b\,x}}{b}+\frac {5\,a^4\,x\,\sqrt {a+b\,x}}{b^4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^(3/2)/(a + b*x)^(13/2),x)

[Out]

((c + d*x)^(1/2)*((x^2*(462*a^3*d^5 - 10*b^3*c^3*d^2 + 66*a*b^2*c^2*d^3 - 198*a^2*b*c*d^4))/(1155*b^5*(a*d - b
*c)^4) - (210*b^3*c^5 - 462*a^3*c^2*d^3 + 990*a^2*b*c^3*d^2 - 770*a*b^2*c^4*d)/(1155*b^5*(a*d - b*c)^4) + (x*(
924*a^3*c*d^4 - 280*b^3*c^4*d + 1100*a*b^2*c^3*d^2 - 1584*a^2*b*c^2*d^3))/(1155*b^5*(a*d - b*c)^4) + (32*d^5*x
^5)/(1155*b^2*(a*d - b*c)^4) + (16*d^4*x^4*(11*a*d - b*c))/(1155*b^3*(a*d - b*c)^4) + (4*d^3*x^3*(99*a^2*d^2 +
3*b^2*c^2 - 22*a*b*c*d))/(1155*b^4*(a*d - b*c)^4)))/(x^5*(a + b*x)^(1/2) + (a^5*(a + b*x)^(1/2))/b^5 + (10*a^
2*x^3*(a + b*x)^(1/2))/b^2 + (10*a^3*x^2*(a + b*x)^(1/2))/b^3 + (5*a*x^4*(a + b*x)^(1/2))/b + (5*a^4*x*(a + b*
x)^(1/2))/b^4)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**(3/2)/(b*x+a)**(13/2),x)

[Out]

Timed out

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