3.1521 \(\int \frac {1}{(a+b x)^{9/2} (c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=207 \[ \frac {512 b d^4 \sqrt {a+b x}}{21 \sqrt {c+d x} (b c-a d)^6}+\frac {256 d^4 \sqrt {a+b x}}{21 (c+d x)^{3/2} (b c-a d)^5}+\frac {64 d^3}{7 \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)^4}-\frac {32 d^2}{21 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)^3}+\frac {4 d}{7 (a+b x)^{5/2} (c+d x)^{3/2} (b c-a d)^2}-\frac {2}{7 (a+b x)^{7/2} (c+d x)^{3/2} (b c-a d)} \]

[Out]

-2/7/(-a*d+b*c)/(b*x+a)^(7/2)/(d*x+c)^(3/2)+4/7*d/(-a*d+b*c)^2/(b*x+a)^(5/2)/(d*x+c)^(3/2)-32/21*d^2/(-a*d+b*c
)^3/(b*x+a)^(3/2)/(d*x+c)^(3/2)+64/7*d^3/(-a*d+b*c)^4/(d*x+c)^(3/2)/(b*x+a)^(1/2)+256/21*d^4*(b*x+a)^(1/2)/(-a
*d+b*c)^5/(d*x+c)^(3/2)+512/21*b*d^4*(b*x+a)^(1/2)/(-a*d+b*c)^6/(d*x+c)^(1/2)

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Rubi [A]  time = 0.06, antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {45, 37} \[ \frac {512 b d^4 \sqrt {a+b x}}{21 \sqrt {c+d x} (b c-a d)^6}+\frac {256 d^4 \sqrt {a+b x}}{21 (c+d x)^{3/2} (b c-a d)^5}+\frac {64 d^3}{7 \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)^4}-\frac {32 d^2}{21 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)^3}+\frac {4 d}{7 (a+b x)^{5/2} (c+d x)^{3/2} (b c-a d)^2}-\frac {2}{7 (a+b x)^{7/2} (c+d x)^{3/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)^(9/2)*(c + d*x)^(5/2)),x]

[Out]

-2/(7*(b*c - a*d)*(a + b*x)^(7/2)*(c + d*x)^(3/2)) + (4*d)/(7*(b*c - a*d)^2*(a + b*x)^(5/2)*(c + d*x)^(3/2)) -
(32*d^2)/(21*(b*c - a*d)^3*(a + b*x)^(3/2)*(c + d*x)^(3/2)) + (64*d^3)/(7*(b*c - a*d)^4*Sqrt[a + b*x]*(c + d*
x)^(3/2)) + (256*d^4*Sqrt[a + b*x])/(21*(b*c - a*d)^5*(c + d*x)^(3/2)) + (512*b*d^4*Sqrt[a + b*x])/(21*(b*c -
a*d)^6*Sqrt[c + d*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])

Rubi steps

\begin {align*} \int \frac {1}{(a+b x)^{9/2} (c+d x)^{5/2}} \, dx &=-\frac {2}{7 (b c-a d) (a+b x)^{7/2} (c+d x)^{3/2}}-\frac {(10 d) \int \frac {1}{(a+b x)^{7/2} (c+d x)^{5/2}} \, dx}{7 (b c-a d)}\\ &=-\frac {2}{7 (b c-a d) (a+b x)^{7/2} (c+d x)^{3/2}}+\frac {4 d}{7 (b c-a d)^2 (a+b x)^{5/2} (c+d x)^{3/2}}+\frac {\left (16 d^2\right ) \int \frac {1}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx}{7 (b c-a d)^2}\\ &=-\frac {2}{7 (b c-a d) (a+b x)^{7/2} (c+d x)^{3/2}}+\frac {4 d}{7 (b c-a d)^2 (a+b x)^{5/2} (c+d x)^{3/2}}-\frac {32 d^2}{21 (b c-a d)^3 (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {\left (32 d^3\right ) \int \frac {1}{(a+b x)^{3/2} (c+d x)^{5/2}} \, dx}{7 (b c-a d)^3}\\ &=-\frac {2}{7 (b c-a d) (a+b x)^{7/2} (c+d x)^{3/2}}+\frac {4 d}{7 (b c-a d)^2 (a+b x)^{5/2} (c+d x)^{3/2}}-\frac {32 d^2}{21 (b c-a d)^3 (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {64 d^3}{7 (b c-a d)^4 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {\left (128 d^4\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx}{7 (b c-a d)^4}\\ &=-\frac {2}{7 (b c-a d) (a+b x)^{7/2} (c+d x)^{3/2}}+\frac {4 d}{7 (b c-a d)^2 (a+b x)^{5/2} (c+d x)^{3/2}}-\frac {32 d^2}{21 (b c-a d)^3 (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {64 d^3}{7 (b c-a d)^4 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {256 d^4 \sqrt {a+b x}}{21 (b c-a d)^5 (c+d x)^{3/2}}+\frac {\left (256 b d^4\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx}{21 (b c-a d)^5}\\ &=-\frac {2}{7 (b c-a d) (a+b x)^{7/2} (c+d x)^{3/2}}+\frac {4 d}{7 (b c-a d)^2 (a+b x)^{5/2} (c+d x)^{3/2}}-\frac {32 d^2}{21 (b c-a d)^3 (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {64 d^3}{7 (b c-a d)^4 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {256 d^4 \sqrt {a+b x}}{21 (b c-a d)^5 (c+d x)^{3/2}}+\frac {512 b d^4 \sqrt {a+b x}}{21 (b c-a d)^6 \sqrt {c+d x}}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 233, normalized size = 1.13 \[ \frac {2 \left (-7 a^5 d^5+35 a^4 b d^4 (3 c+2 d x)+70 a^3 b^2 d^3 \left (3 c^2+12 c d x+8 d^2 x^2\right )+70 a^2 b^3 d^2 \left (-c^3+6 c^2 d x+24 c d^2 x^2+16 d^3 x^3\right )+7 a b^4 d \left (3 c^4-8 c^3 d x+48 c^2 d^2 x^2+192 c d^3 x^3+128 d^4 x^4\right )+b^5 \left (-3 c^5+6 c^4 d x-16 c^3 d^2 x^2+96 c^2 d^3 x^3+384 c d^4 x^4+256 d^5 x^5\right )\right )}{21 (a+b x)^{7/2} (c+d x)^{3/2} (b c-a d)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x)^(9/2)*(c + d*x)^(5/2)),x]

[Out]

(2*(-7*a^5*d^5 + 35*a^4*b*d^4*(3*c + 2*d*x) + 70*a^3*b^2*d^3*(3*c^2 + 12*c*d*x + 8*d^2*x^2) + 70*a^2*b^3*d^2*(
-c^3 + 6*c^2*d*x + 24*c*d^2*x^2 + 16*d^3*x^3) + 7*a*b^4*d*(3*c^4 - 8*c^3*d*x + 48*c^2*d^2*x^2 + 192*c*d^3*x^3
+ 128*d^4*x^4) + b^5*(-3*c^5 + 6*c^4*d*x - 16*c^3*d^2*x^2 + 96*c^2*d^3*x^3 + 384*c*d^4*x^4 + 256*d^5*x^5)))/(2
1*(b*c - a*d)^6*(a + b*x)^(7/2)*(c + d*x)^(3/2))

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fricas [B]  time = 7.60, size = 999, normalized size = 4.83 \[ \frac {2 \, {\left (256 \, b^{5} d^{5} x^{5} - 3 \, b^{5} c^{5} + 21 \, a b^{4} c^{4} d - 70 \, a^{2} b^{3} c^{3} d^{2} + 210 \, a^{3} b^{2} c^{2} d^{3} + 105 \, a^{4} b c d^{4} - 7 \, a^{5} d^{5} + 128 \, {\left (3 \, b^{5} c d^{4} + 7 \, a b^{4} d^{5}\right )} x^{4} + 32 \, {\left (3 \, b^{5} c^{2} d^{3} + 42 \, a b^{4} c d^{4} + 35 \, a^{2} b^{3} d^{5}\right )} x^{3} - 16 \, {\left (b^{5} c^{3} d^{2} - 21 \, a b^{4} c^{2} d^{3} - 105 \, a^{2} b^{3} c d^{4} - 35 \, a^{3} b^{2} d^{5}\right )} x^{2} + 2 \, {\left (3 \, b^{5} c^{4} d - 28 \, a b^{4} c^{3} d^{2} + 210 \, a^{2} b^{3} c^{2} d^{3} + 420 \, a^{3} b^{2} c d^{4} + 35 \, a^{4} b d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{21 \, {\left (a^{4} b^{6} c^{8} - 6 \, a^{5} b^{5} c^{7} d + 15 \, a^{6} b^{4} c^{6} d^{2} - 20 \, a^{7} b^{3} c^{5} d^{3} + 15 \, a^{8} b^{2} c^{4} d^{4} - 6 \, a^{9} b c^{3} d^{5} + a^{10} c^{2} d^{6} + {\left (b^{10} c^{6} d^{2} - 6 \, a b^{9} c^{5} d^{3} + 15 \, a^{2} b^{8} c^{4} d^{4} - 20 \, a^{3} b^{7} c^{3} d^{5} + 15 \, a^{4} b^{6} c^{2} d^{6} - 6 \, a^{5} b^{5} c d^{7} + a^{6} b^{4} d^{8}\right )} x^{6} + 2 \, {\left (b^{10} c^{7} d - 4 \, a b^{9} c^{6} d^{2} + 3 \, a^{2} b^{8} c^{5} d^{3} + 10 \, a^{3} b^{7} c^{4} d^{4} - 25 \, a^{4} b^{6} c^{3} d^{5} + 24 \, a^{5} b^{5} c^{2} d^{6} - 11 \, a^{6} b^{4} c d^{7} + 2 \, a^{7} b^{3} d^{8}\right )} x^{5} + {\left (b^{10} c^{8} + 2 \, a b^{9} c^{7} d - 27 \, a^{2} b^{8} c^{6} d^{2} + 64 \, a^{3} b^{7} c^{5} d^{3} - 55 \, a^{4} b^{6} c^{4} d^{4} - 6 \, a^{5} b^{5} c^{3} d^{5} + 43 \, a^{6} b^{4} c^{2} d^{6} - 28 \, a^{7} b^{3} c d^{7} + 6 \, a^{8} b^{2} d^{8}\right )} x^{4} + 4 \, {\left (a b^{9} c^{8} - 3 \, a^{2} b^{8} c^{7} d - 2 \, a^{3} b^{7} c^{6} d^{2} + 19 \, a^{4} b^{6} c^{5} d^{3} - 30 \, a^{5} b^{5} c^{4} d^{4} + 19 \, a^{6} b^{4} c^{3} d^{5} - 2 \, a^{7} b^{3} c^{2} d^{6} - 3 \, a^{8} b^{2} c d^{7} + a^{9} b d^{8}\right )} x^{3} + {\left (6 \, a^{2} b^{8} c^{8} - 28 \, a^{3} b^{7} c^{7} d + 43 \, a^{4} b^{6} c^{6} d^{2} - 6 \, a^{5} b^{5} c^{5} d^{3} - 55 \, a^{6} b^{4} c^{4} d^{4} + 64 \, a^{7} b^{3} c^{3} d^{5} - 27 \, a^{8} b^{2} c^{2} d^{6} + 2 \, a^{9} b c d^{7} + a^{10} d^{8}\right )} x^{2} + 2 \, {\left (2 \, a^{3} b^{7} c^{8} - 11 \, a^{4} b^{6} c^{7} d + 24 \, a^{5} b^{5} c^{6} d^{2} - 25 \, a^{6} b^{4} c^{5} d^{3} + 10 \, a^{7} b^{3} c^{4} d^{4} + 3 \, a^{8} b^{2} c^{3} d^{5} - 4 \, a^{9} b c^{2} d^{6} + a^{10} c d^{7}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(9/2)/(d*x+c)^(5/2),x, algorithm="fricas")

[Out]

2/21*(256*b^5*d^5*x^5 - 3*b^5*c^5 + 21*a*b^4*c^4*d - 70*a^2*b^3*c^3*d^2 + 210*a^3*b^2*c^2*d^3 + 105*a^4*b*c*d^
4 - 7*a^5*d^5 + 128*(3*b^5*c*d^4 + 7*a*b^4*d^5)*x^4 + 32*(3*b^5*c^2*d^3 + 42*a*b^4*c*d^4 + 35*a^2*b^3*d^5)*x^3
- 16*(b^5*c^3*d^2 - 21*a*b^4*c^2*d^3 - 105*a^2*b^3*c*d^4 - 35*a^3*b^2*d^5)*x^2 + 2*(3*b^5*c^4*d - 28*a*b^4*c^
3*d^2 + 210*a^2*b^3*c^2*d^3 + 420*a^3*b^2*c*d^4 + 35*a^4*b*d^5)*x)*sqrt(b*x + a)*sqrt(d*x + c)/(a^4*b^6*c^8 -
6*a^5*b^5*c^7*d + 15*a^6*b^4*c^6*d^2 - 20*a^7*b^3*c^5*d^3 + 15*a^8*b^2*c^4*d^4 - 6*a^9*b*c^3*d^5 + a^10*c^2*d^
6 + (b^10*c^6*d^2 - 6*a*b^9*c^5*d^3 + 15*a^2*b^8*c^4*d^4 - 20*a^3*b^7*c^3*d^5 + 15*a^4*b^6*c^2*d^6 - 6*a^5*b^5
*c*d^7 + a^6*b^4*d^8)*x^6 + 2*(b^10*c^7*d - 4*a*b^9*c^6*d^2 + 3*a^2*b^8*c^5*d^3 + 10*a^3*b^7*c^4*d^4 - 25*a^4*
b^6*c^3*d^5 + 24*a^5*b^5*c^2*d^6 - 11*a^6*b^4*c*d^7 + 2*a^7*b^3*d^8)*x^5 + (b^10*c^8 + 2*a*b^9*c^7*d - 27*a^2*
b^8*c^6*d^2 + 64*a^3*b^7*c^5*d^3 - 55*a^4*b^6*c^4*d^4 - 6*a^5*b^5*c^3*d^5 + 43*a^6*b^4*c^2*d^6 - 28*a^7*b^3*c*
d^7 + 6*a^8*b^2*d^8)*x^4 + 4*(a*b^9*c^8 - 3*a^2*b^8*c^7*d - 2*a^3*b^7*c^6*d^2 + 19*a^4*b^6*c^5*d^3 - 30*a^5*b^
5*c^4*d^4 + 19*a^6*b^4*c^3*d^5 - 2*a^7*b^3*c^2*d^6 - 3*a^8*b^2*c*d^7 + a^9*b*d^8)*x^3 + (6*a^2*b^8*c^8 - 28*a^
3*b^7*c^7*d + 43*a^4*b^6*c^6*d^2 - 6*a^5*b^5*c^5*d^3 - 55*a^6*b^4*c^4*d^4 + 64*a^7*b^3*c^3*d^5 - 27*a^8*b^2*c^
2*d^6 + 2*a^9*b*c*d^7 + a^10*d^8)*x^2 + 2*(2*a^3*b^7*c^8 - 11*a^4*b^6*c^7*d + 24*a^5*b^5*c^6*d^2 - 25*a^6*b^4*
c^5*d^3 + 10*a^7*b^3*c^4*d^4 + 3*a^8*b^2*c^3*d^5 - 4*a^9*b*c^2*d^6 + a^10*c*d^7)*x)

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giac [B]  time = 7.76, size = 1964, normalized size = 9.49 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(9/2)/(d*x+c)^(5/2),x, algorithm="giac")

[Out]

2/3*sqrt(b*x + a)*(14*(b^9*c^5*d^6*abs(b) - 5*a*b^8*c^4*d^7*abs(b) + 10*a^2*b^7*c^3*d^8*abs(b) - 10*a^3*b^6*c^
2*d^9*abs(b) + 5*a^4*b^5*c*d^10*abs(b) - a^5*b^4*d^11*abs(b))*(b*x + a)/(b^13*c^11*d - 11*a*b^12*c^10*d^2 + 55
*a^2*b^11*c^9*d^3 - 165*a^3*b^10*c^8*d^4 + 330*a^4*b^9*c^7*d^5 - 462*a^5*b^8*c^6*d^6 + 462*a^6*b^7*c^5*d^7 - 3
30*a^7*b^6*c^4*d^8 + 165*a^8*b^5*c^3*d^9 - 55*a^9*b^4*c^2*d^10 + 11*a^10*b^3*c*d^11 - a^11*b^2*d^12) + 15*(b^1
0*c^6*d^5*abs(b) - 6*a*b^9*c^5*d^6*abs(b) + 15*a^2*b^8*c^4*d^7*abs(b) - 20*a^3*b^7*c^3*d^8*abs(b) + 15*a^4*b^6
*c^2*d^9*abs(b) - 6*a^5*b^5*c*d^10*abs(b) + a^6*b^4*d^11*abs(b))/(b^13*c^11*d - 11*a*b^12*c^10*d^2 + 55*a^2*b^
11*c^9*d^3 - 165*a^3*b^10*c^8*d^4 + 330*a^4*b^9*c^7*d^5 - 462*a^5*b^8*c^6*d^6 + 462*a^6*b^7*c^5*d^7 - 330*a^7*
b^6*c^4*d^8 + 165*a^8*b^5*c^3*d^9 - 55*a^9*b^4*c^2*d^10 + 11*a^10*b^3*c*d^11 - a^11*b^2*d^12))/(b^2*c + (b*x +
a)*b*d - a*b*d)^(3/2) + 8/21*(79*sqrt(b*d)*b^15*c^6*d^3 - 474*sqrt(b*d)*a*b^14*c^5*d^4 + 1185*sqrt(b*d)*a^2*b
^13*c^4*d^5 - 1580*sqrt(b*d)*a^3*b^12*c^3*d^6 + 1185*sqrt(b*d)*a^4*b^11*c^2*d^7 - 474*sqrt(b*d)*a^5*b^10*c*d^8
+ 79*sqrt(b*d)*a^6*b^9*d^9 - 511*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*
b^13*c^5*d^3 + 2555*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^12*c^4*d^4
- 5110*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^11*c^3*d^5 + 5110*sq
rt(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^10*c^2*d^6 - 2555*sqrt(b*d)*(s
qrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^4*b^9*c*d^7 + 511*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^8*d^8 + 1344*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt
(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^11*c^4*d^3 - 5376*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^10*c^3*d^4 + 8064*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^9*c^2*d^5 - 5376*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^8*c*d^6 + 1344*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^7*d^7 - 1
750*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*b^9*c^3*d^3 + 5250*sqrt(b*d)*(
sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a*b^8*c^2*d^4 - 5250*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^2*b^7*c*d^5 + 1750*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) -
sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^3*b^6*d^6 + 1015*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b
*x + a)*b*d - a*b*d))^8*b^7*c^2*d^3 - 2030*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^6*c*d^4 + 1015*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^
5*d^5 - 315*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*b^5*c*d^3 + 315*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^4*d^4 + 42*sqrt(b*d)*(sqrt(b*d)*sq
rt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*b^3*d^3)/((b^5*c^5*abs(b) - 5*a*b^4*c^4*d*abs(b) + 10*a^
2*b^3*c^3*d^2*abs(b) - 10*a^3*b^2*c^2*d^3*abs(b) + 5*a^4*b*c*d^4*abs(b) - a^5*d^5*abs(b))*(b^2*c - a*b*d - (sq
rt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)^7)

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maple [B]  time = 0.02, size = 356, normalized size = 1.72 \[ -\frac {2 \left (-256 b^{5} x^{5} d^{5}-896 a \,b^{4} d^{5} x^{4}-384 b^{5} c \,d^{4} x^{4}-1120 a^{2} b^{3} d^{5} x^{3}-1344 a \,b^{4} c \,d^{4} x^{3}-96 b^{5} c^{2} d^{3} x^{3}-560 a^{3} b^{2} d^{5} x^{2}-1680 a^{2} b^{3} c \,d^{4} x^{2}-336 a \,b^{4} c^{2} d^{3} x^{2}+16 b^{5} c^{3} d^{2} x^{2}-70 a^{4} b \,d^{5} x -840 a^{3} b^{2} c \,d^{4} x -420 a^{2} b^{3} c^{2} d^{3} x +56 a \,b^{4} c^{3} d^{2} x -6 b^{5} c^{4} d x +7 a^{5} d^{5}-105 a^{4} b c \,d^{4}-210 a^{3} b^{2} c^{2} d^{3}+70 a^{2} b^{3} c^{3} d^{2}-21 a \,b^{4} c^{4} d +3 b^{5} c^{5}\right )}{21 \left (b x +a \right )^{\frac {7}{2}} \left (d x +c \right )^{\frac {3}{2}} \left (d^{6} a^{6}-6 b \,d^{5} c \,a^{5}+15 b^{2} d^{4} c^{2} a^{4}-20 b^{3} d^{3} c^{3} a^{3}+15 b^{4} d^{2} c^{4} a^{2}-6 b^{5} d \,c^{5} a +b^{6} c^{6}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)^(9/2)/(d*x+c)^(5/2),x)

[Out]

-2/21*(-256*b^5*d^5*x^5-896*a*b^4*d^5*x^4-384*b^5*c*d^4*x^4-1120*a^2*b^3*d^5*x^3-1344*a*b^4*c*d^4*x^3-96*b^5*c
^2*d^3*x^3-560*a^3*b^2*d^5*x^2-1680*a^2*b^3*c*d^4*x^2-336*a*b^4*c^2*d^3*x^2+16*b^5*c^3*d^2*x^2-70*a^4*b*d^5*x-
840*a^3*b^2*c*d^4*x-420*a^2*b^3*c^2*d^3*x+56*a*b^4*c^3*d^2*x-6*b^5*c^4*d*x+7*a^5*d^5-105*a^4*b*c*d^4-210*a^3*b
^2*c^2*d^3+70*a^2*b^3*c^3*d^2-21*a*b^4*c^4*d+3*b^5*c^5)/(b*x+a)^(7/2)/(d*x+c)^(3/2)/(a^6*d^6-6*a^5*b*c*d^5+15*
a^4*b^2*c^2*d^4-20*a^3*b^3*c^3*d^3+15*a^2*b^4*c^4*d^2-6*a*b^5*c^5*d+b^6*c^6)

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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(1/(b*x+a)^(9/2)/(d*x+c)^(5/2),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 = 1.91, size = 478, normalized size = 2.31 \[ \frac {\sqrt {c+d\,x}\,\left (\frac {32\,x^2\,\left (35\,a^3\,d^3+105\,a^2\,b\,c\,d^2+21\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{21\,b\,{\left (a\,d-b\,c\right )}^6}-\frac {14\,a^5\,d^5-210\,a^4\,b\,c\,d^4-420\,a^3\,b^2\,c^2\,d^3+140\,a^2\,b^3\,c^3\,d^2-42\,a\,b^4\,c^4\,d+6\,b^5\,c^5}{21\,b^3\,d^2\,{\left (a\,d-b\,c\right )}^6}+\frac {64\,d\,x^3\,\left (35\,a^2\,d^2+42\,a\,b\,c\,d+3\,b^2\,c^2\right )}{21\,{\left (a\,d-b\,c\right )}^6}+\frac {512\,b^2\,d^3\,x^5}{21\,{\left (a\,d-b\,c\right )}^6}+\frac {256\,b\,d^2\,x^4\,\left (7\,a\,d+3\,b\,c\right )}{21\,{\left (a\,d-b\,c\right )}^6}+\frac {x\,\left (140\,a^4\,b\,d^5+1680\,a^3\,b^2\,c\,d^4+840\,a^2\,b^3\,c^2\,d^3-112\,a\,b^4\,c^3\,d^2+12\,b^5\,c^4\,d\right )}{21\,b^3\,d^2\,{\left (a\,d-b\,c\right )}^6}\right )}{x^5\,\sqrt {a+b\,x}+\frac {x^3\,\sqrt {a+b\,x}\,\left (3\,a^2\,d^2+6\,a\,b\,c\,d+b^2\,c^2\right )}{b^2\,d^2}+\frac {x^4\,\left (3\,a\,d+2\,b\,c\right )\,\sqrt {a+b\,x}}{b\,d}+\frac {a^3\,c^2\,\sqrt {a+b\,x}}{b^3\,d^2}+\frac {a\,x^2\,\sqrt {a+b\,x}\,\left (a^2\,d^2+6\,a\,b\,c\,d+3\,b^2\,c^2\right )}{b^3\,d^2}+\frac {a^2\,c\,x\,\left (2\,a\,d+3\,b\,c\right )\,\sqrt {a+b\,x}}{b^3\,d^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*x)^(9/2)*(c + d*x)^(5/2)),x)

[Out]

((c + d*x)^(1/2)*((32*x^2*(35*a^3*d^3 - b^3*c^3 + 21*a*b^2*c^2*d + 105*a^2*b*c*d^2))/(21*b*(a*d - b*c)^6) - (1
4*a^5*d^5 + 6*b^5*c^5 + 140*a^2*b^3*c^3*d^2 - 420*a^3*b^2*c^2*d^3 - 42*a*b^4*c^4*d - 210*a^4*b*c*d^4)/(21*b^3*
d^2*(a*d - b*c)^6) + (64*d*x^3*(35*a^2*d^2 + 3*b^2*c^2 + 42*a*b*c*d))/(21*(a*d - b*c)^6) + (512*b^2*d^3*x^5)/(
21*(a*d - b*c)^6) + (256*b*d^2*x^4*(7*a*d + 3*b*c))/(21*(a*d - b*c)^6) + (x*(140*a^4*b*d^5 + 12*b^5*c^4*d - 11
2*a*b^4*c^3*d^2 + 1680*a^3*b^2*c*d^4 + 840*a^2*b^3*c^2*d^3))/(21*b^3*d^2*(a*d - b*c)^6)))/(x^5*(a + b*x)^(1/2)
+ (x^3*(a + b*x)^(1/2)*(3*a^2*d^2 + b^2*c^2 + 6*a*b*c*d))/(b^2*d^2) + (x^4*(3*a*d + 2*b*c)*(a + b*x)^(1/2))/(
b*d) + (a^3*c^2*(a + b*x)^(1/2))/(b^3*d^2) + (a*x^2*(a + b*x)^(1/2)*(a^2*d^2 + 3*b^2*c^2 + 6*a*b*c*d))/(b^3*d^
2) + (a^2*c*x*(2*a*d + 3*b*c)*(a + b*x)^(1/2))/(b^3*d^2))

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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(1/(b*x+a)**(9/2)/(d*x+c)**(5/2),x)

[Out]

Timed out

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