3.15.5 \(\int \frac {1}{(a+b x)^{11/2} (c+d x)^{3/2}} \, dx\)

Optimal. Leaf size=206 \[ -\frac {512 d^5 \sqrt {a+b x}}{63 \sqrt {c+d x} (b c-a d)^6}-\frac {256 d^4}{63 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)^5}+\frac {64 d^3}{63 (a+b x)^{3/2} \sqrt {c+d x} (b c-a d)^4}-\frac {32 d^2}{63 (a+b x)^{5/2} \sqrt {c+d x} (b c-a d)^3}+\frac {20 d}{63 (a+b x)^{7/2} \sqrt {c+d x} (b c-a d)^2}-\frac {2}{9 (a+b x)^{9/2} \sqrt {c+d x} (b c-a d)} \]

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Rubi [A]  time = 0.06, antiderivative size = 206, 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} \begin {gather*} -\frac {512 d^5 \sqrt {a+b x}}{63 \sqrt {c+d x} (b c-a d)^6}-\frac {256 d^4}{63 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)^5}+\frac {64 d^3}{63 (a+b x)^{3/2} \sqrt {c+d x} (b c-a d)^4}-\frac {32 d^2}{63 (a+b x)^{5/2} \sqrt {c+d x} (b c-a d)^3}+\frac {20 d}{63 (a+b x)^{7/2} \sqrt {c+d x} (b c-a d)^2}-\frac {2}{9 (a+b x)^{9/2} \sqrt {c+d x} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)^(11/2)*(c + d*x)^(3/2)),x]

[Out]

-2/(9*(b*c - a*d)*(a + b*x)^(9/2)*Sqrt[c + d*x]) + (20*d)/(63*(b*c - a*d)^2*(a + b*x)^(7/2)*Sqrt[c + d*x]) - (
32*d^2)/(63*(b*c - a*d)^3*(a + b*x)^(5/2)*Sqrt[c + d*x]) + (64*d^3)/(63*(b*c - a*d)^4*(a + b*x)^(3/2)*Sqrt[c +
d*x]) - (256*d^4)/(63*(b*c - a*d)^5*Sqrt[a + b*x]*Sqrt[c + d*x]) - (512*d^5*Sqrt[a + b*x])/(63*(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)^{11/2} (c+d x)^{3/2}} \, dx &=-\frac {2}{9 (b c-a d) (a+b x)^{9/2} \sqrt {c+d x}}-\frac {(10 d) \int \frac {1}{(a+b x)^{9/2} (c+d x)^{3/2}} \, dx}{9 (b c-a d)}\\ &=-\frac {2}{9 (b c-a d) (a+b x)^{9/2} \sqrt {c+d x}}+\frac {20 d}{63 (b c-a d)^2 (a+b x)^{7/2} \sqrt {c+d x}}+\frac {\left (80 d^2\right ) \int \frac {1}{(a+b x)^{7/2} (c+d x)^{3/2}} \, dx}{63 (b c-a d)^2}\\ &=-\frac {2}{9 (b c-a d) (a+b x)^{9/2} \sqrt {c+d x}}+\frac {20 d}{63 (b c-a d)^2 (a+b x)^{7/2} \sqrt {c+d x}}-\frac {32 d^2}{63 (b c-a d)^3 (a+b x)^{5/2} \sqrt {c+d x}}-\frac {\left (32 d^3\right ) \int \frac {1}{(a+b x)^{5/2} (c+d x)^{3/2}} \, dx}{21 (b c-a d)^3}\\ &=-\frac {2}{9 (b c-a d) (a+b x)^{9/2} \sqrt {c+d x}}+\frac {20 d}{63 (b c-a d)^2 (a+b x)^{7/2} \sqrt {c+d x}}-\frac {32 d^2}{63 (b c-a d)^3 (a+b x)^{5/2} \sqrt {c+d x}}+\frac {64 d^3}{63 (b c-a d)^4 (a+b x)^{3/2} \sqrt {c+d x}}+\frac {\left (128 d^4\right ) \int \frac {1}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx}{63 (b c-a d)^4}\\ &=-\frac {2}{9 (b c-a d) (a+b x)^{9/2} \sqrt {c+d x}}+\frac {20 d}{63 (b c-a d)^2 (a+b x)^{7/2} \sqrt {c+d x}}-\frac {32 d^2}{63 (b c-a d)^3 (a+b x)^{5/2} \sqrt {c+d x}}+\frac {64 d^3}{63 (b c-a d)^4 (a+b x)^{3/2} \sqrt {c+d x}}-\frac {256 d^4}{63 (b c-a d)^5 \sqrt {a+b x} \sqrt {c+d x}}-\frac {\left (256 d^5\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx}{63 (b c-a d)^5}\\ &=-\frac {2}{9 (b c-a d) (a+b x)^{9/2} \sqrt {c+d x}}+\frac {20 d}{63 (b c-a d)^2 (a+b x)^{7/2} \sqrt {c+d x}}-\frac {32 d^2}{63 (b c-a d)^3 (a+b x)^{5/2} \sqrt {c+d x}}+\frac {64 d^3}{63 (b c-a d)^4 (a+b x)^{3/2} \sqrt {c+d x}}-\frac {256 d^4}{63 (b c-a d)^5 \sqrt {a+b x} \sqrt {c+d x}}-\frac {512 d^5 \sqrt {a+b x}}{63 (b c-a d)^6 \sqrt {c+d x}}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 226, normalized size = 1.10 \begin {gather*} \frac {512 d^5 \sqrt {a+b x}}{63 \sqrt {c+d x} (b c-a d)^5 (a d-b c)}+\frac {256 d^4}{63 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)^4 (a d-b c)}+\frac {64 d^3}{63 (a+b x)^{3/2} \sqrt {c+d x} (b c-a d)^4}-\frac {32 d^2}{63 (a+b x)^{5/2} \sqrt {c+d x} (b c-a d)^3}+\frac {20 d}{63 (a+b x)^{7/2} \sqrt {c+d x} (b c-a d)^2}-\frac {2}{9 (a+b x)^{9/2} \sqrt {c+d x} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x)^(11/2)*(c + d*x)^(3/2)),x]

[Out]

-2/(9*(b*c - a*d)*(a + b*x)^(9/2)*Sqrt[c + d*x]) + (20*d)/(63*(b*c - a*d)^2*(a + b*x)^(7/2)*Sqrt[c + d*x]) - (
32*d^2)/(63*(b*c - a*d)^3*(a + b*x)^(5/2)*Sqrt[c + d*x]) + (64*d^3)/(63*(b*c - a*d)^4*(a + b*x)^(3/2)*Sqrt[c +
d*x]) + (256*d^4)/(63*(b*c - a*d)^4*(-(b*c) + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x]) + (512*d^5*Sqrt[a + b*x])/(63
*(b*c - a*d)^5*(-(b*c) + a*d)*Sqrt[c + d*x])

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IntegrateAlgebraic [A]  time = 0.15, size = 139, normalized size = 0.67 \begin {gather*} -\frac {2 (c+d x)^{9/2} \left (-\frac {45 b^4 d (a+b x)}{c+d x}+\frac {126 b^3 d^2 (a+b x)^2}{(c+d x)^2}-\frac {210 b^2 d^3 (a+b x)^3}{(c+d x)^3}+\frac {63 d^5 (a+b x)^5}{(c+d x)^5}+\frac {315 b d^4 (a+b x)^4}{(c+d x)^4}+7 b^5\right )}{63 (a+b x)^{9/2} (b c-a d)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((a + b*x)^(11/2)*(c + d*x)^(3/2)),x]

[Out]

(-2*(c + d*x)^(9/2)*(7*b^5 + (63*d^5*(a + b*x)^5)/(c + d*x)^5 + (315*b*d^4*(a + b*x)^4)/(c + d*x)^4 - (210*b^2
*d^3*(a + b*x)^3)/(c + d*x)^3 + (126*b^3*d^2*(a + b*x)^2)/(c + d*x)^2 - (45*b^4*d*(a + b*x))/(c + d*x)))/(63*(
b*c - a*d)^6*(a + b*x)^(9/2))

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fricas [B]  time = 15.84, size = 955, normalized size = 4.64 \begin {gather*} -\frac {2 \, {\left (256 \, b^{5} d^{5} x^{5} + 7 \, b^{5} c^{5} - 45 \, a b^{4} c^{4} d + 126 \, a^{2} b^{3} c^{3} d^{2} - 210 \, a^{3} b^{2} c^{2} d^{3} + 315 \, a^{4} b c d^{4} + 63 \, a^{5} d^{5} + 128 \, {\left (b^{5} c d^{4} + 9 \, a b^{4} d^{5}\right )} x^{4} - 32 \, {\left (b^{5} c^{2} d^{3} - 18 \, a b^{4} c d^{4} - 63 \, a^{2} b^{3} d^{5}\right )} x^{3} + 16 \, {\left (b^{5} c^{3} d^{2} - 9 \, a b^{4} c^{2} d^{3} + 63 \, a^{2} b^{3} c d^{4} + 105 \, a^{3} b^{2} d^{5}\right )} x^{2} - 2 \, {\left (5 \, b^{5} c^{4} d - 36 \, a b^{4} c^{3} d^{2} + 126 \, a^{2} b^{3} c^{2} d^{3} - 420 \, a^{3} b^{2} c d^{4} - 315 \, a^{4} b d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{63 \, {\left (a^{5} b^{6} c^{7} - 6 \, a^{6} b^{5} c^{6} d + 15 \, a^{7} b^{4} c^{5} d^{2} - 20 \, a^{8} b^{3} c^{4} d^{3} + 15 \, a^{9} b^{2} c^{3} d^{4} - 6 \, a^{10} b c^{2} d^{5} + a^{11} c d^{6} + {\left (b^{11} c^{6} d - 6 \, a b^{10} c^{5} d^{2} + 15 \, a^{2} b^{9} c^{4} d^{3} - 20 \, a^{3} b^{8} c^{3} d^{4} + 15 \, a^{4} b^{7} c^{2} d^{5} - 6 \, a^{5} b^{6} c d^{6} + a^{6} b^{5} d^{7}\right )} x^{6} + {\left (b^{11} c^{7} - a b^{10} c^{6} d - 15 \, a^{2} b^{9} c^{5} d^{2} + 55 \, a^{3} b^{8} c^{4} d^{3} - 85 \, a^{4} b^{7} c^{3} d^{4} + 69 \, a^{5} b^{6} c^{2} d^{5} - 29 \, a^{6} b^{5} c d^{6} + 5 \, a^{7} b^{4} d^{7}\right )} x^{5} + 5 \, {\left (a b^{10} c^{7} - 4 \, a^{2} b^{9} c^{6} d + 3 \, a^{3} b^{8} c^{5} d^{2} + 10 \, a^{4} b^{7} c^{4} d^{3} - 25 \, a^{5} b^{6} c^{3} d^{4} + 24 \, a^{6} b^{5} c^{2} d^{5} - 11 \, a^{7} b^{4} c d^{6} + 2 \, a^{8} b^{3} d^{7}\right )} x^{4} + 10 \, {\left (a^{2} b^{9} c^{7} - 5 \, a^{3} b^{8} c^{6} d + 9 \, a^{4} b^{7} c^{5} d^{2} - 5 \, a^{5} b^{6} c^{4} d^{3} - 5 \, a^{6} b^{5} c^{3} d^{4} + 9 \, a^{7} b^{4} c^{2} d^{5} - 5 \, a^{8} b^{3} c d^{6} + a^{9} b^{2} d^{7}\right )} x^{3} + 5 \, {\left (2 \, a^{3} b^{8} c^{7} - 11 \, a^{4} b^{7} c^{6} d + 24 \, a^{5} b^{6} c^{5} d^{2} - 25 \, a^{6} b^{5} c^{4} d^{3} + 10 \, a^{7} b^{4} c^{3} d^{4} + 3 \, a^{8} b^{3} c^{2} d^{5} - 4 \, a^{9} b^{2} c d^{6} + a^{10} b d^{7}\right )} x^{2} + {\left (5 \, a^{4} b^{7} c^{7} - 29 \, a^{5} b^{6} c^{6} d + 69 \, a^{6} b^{5} c^{5} d^{2} - 85 \, a^{7} b^{4} c^{4} d^{3} + 55 \, a^{8} b^{3} c^{3} d^{4} - 15 \, a^{9} b^{2} c^{2} d^{5} - a^{10} b c d^{6} + a^{11} d^{7}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(11/2)/(d*x+c)^(3/2),x, algorithm="fricas")

[Out]

-2/63*(256*b^5*d^5*x^5 + 7*b^5*c^5 - 45*a*b^4*c^4*d + 126*a^2*b^3*c^3*d^2 - 210*a^3*b^2*c^2*d^3 + 315*a^4*b*c*
d^4 + 63*a^5*d^5 + 128*(b^5*c*d^4 + 9*a*b^4*d^5)*x^4 - 32*(b^5*c^2*d^3 - 18*a*b^4*c*d^4 - 63*a^2*b^3*d^5)*x^3
+ 16*(b^5*c^3*d^2 - 9*a*b^4*c^2*d^3 + 63*a^2*b^3*c*d^4 + 105*a^3*b^2*d^5)*x^2 - 2*(5*b^5*c^4*d - 36*a*b^4*c^3*
d^2 + 126*a^2*b^3*c^2*d^3 - 420*a^3*b^2*c*d^4 - 315*a^4*b*d^5)*x)*sqrt(b*x + a)*sqrt(d*x + c)/(a^5*b^6*c^7 - 6
*a^6*b^5*c^6*d + 15*a^7*b^4*c^5*d^2 - 20*a^8*b^3*c^4*d^3 + 15*a^9*b^2*c^3*d^4 - 6*a^10*b*c^2*d^5 + a^11*c*d^6
+ (b^11*c^6*d - 6*a*b^10*c^5*d^2 + 15*a^2*b^9*c^4*d^3 - 20*a^3*b^8*c^3*d^4 + 15*a^4*b^7*c^2*d^5 - 6*a^5*b^6*c*
d^6 + a^6*b^5*d^7)*x^6 + (b^11*c^7 - a*b^10*c^6*d - 15*a^2*b^9*c^5*d^2 + 55*a^3*b^8*c^4*d^3 - 85*a^4*b^7*c^3*d
^4 + 69*a^5*b^6*c^2*d^5 - 29*a^6*b^5*c*d^6 + 5*a^7*b^4*d^7)*x^5 + 5*(a*b^10*c^7 - 4*a^2*b^9*c^6*d + 3*a^3*b^8*
c^5*d^2 + 10*a^4*b^7*c^4*d^3 - 25*a^5*b^6*c^3*d^4 + 24*a^6*b^5*c^2*d^5 - 11*a^7*b^4*c*d^6 + 2*a^8*b^3*d^7)*x^4
+ 10*(a^2*b^9*c^7 - 5*a^3*b^8*c^6*d + 9*a^4*b^7*c^5*d^2 - 5*a^5*b^6*c^4*d^3 - 5*a^6*b^5*c^3*d^4 + 9*a^7*b^4*c
^2*d^5 - 5*a^8*b^3*c*d^6 + a^9*b^2*d^7)*x^3 + 5*(2*a^3*b^8*c^7 - 11*a^4*b^7*c^6*d + 24*a^5*b^6*c^5*d^2 - 25*a^
6*b^5*c^4*d^3 + 10*a^7*b^4*c^3*d^4 + 3*a^8*b^3*c^2*d^5 - 4*a^9*b^2*c*d^6 + a^10*b*d^7)*x^2 + (5*a^4*b^7*c^7 -
29*a^5*b^6*c^6*d + 69*a^6*b^5*c^5*d^2 - 85*a^7*b^4*c^4*d^3 + 55*a^8*b^3*c^3*d^4 - 15*a^9*b^2*c^2*d^5 - a^10*b*
c*d^6 + a^11*d^7)*x)

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giac [B]  time = 8.71, size = 2438, normalized size = 11.83

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(11/2)/(d*x+c)^(3/2),x, algorithm="giac")

[Out]

-2*sqrt(b*x + a)*b^2*d^5/((b^6*c^6*abs(b) - 6*a*b^5*c^5*d*abs(b) + 15*a^2*b^4*c^4*d^2*abs(b) - 20*a^3*b^3*c^3*
d^3*abs(b) + 15*a^4*b^2*c^2*d^4*abs(b) - 6*a^5*b*c*d^5*abs(b) + a^6*d^6*abs(b))*sqrt(b^2*c + (b*x + a)*b*d - a
*b*d)) - 4/63*(193*sqrt(b*d)*b^18*c^8*d^4 - 1544*sqrt(b*d)*a*b^17*c^7*d^5 + 5404*sqrt(b*d)*a^2*b^16*c^6*d^6 -
10808*sqrt(b*d)*a^3*b^15*c^5*d^7 + 13510*sqrt(b*d)*a^4*b^14*c^4*d^8 - 10808*sqrt(b*d)*a^5*b^13*c^3*d^9 + 5404*
sqrt(b*d)*a^6*b^12*c^2*d^10 - 1544*sqrt(b*d)*a^7*b^11*c*d^11 + 193*sqrt(b*d)*a^8*b^10*d^12 - 1674*sqrt(b*d)*(s
qrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^16*c^7*d^4 + 11718*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^15*c^6*d^5 - 35154*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^14*c^5*d^6 + 58590*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^13*c^4*d^7 - 58590*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^12*c^3*d^8 + 35154*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^11*c^2*d^9 - 11718*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^10*c*d^10 + 1674*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^7*b^9
*d^11 + 6318*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^14*c^6*d^4 - 37908*
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^13*c^5*d^5 + 94770*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^12*c^4*d^6 - 126360*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^11*c^3*d^7 + 94770*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^10*c^2*d^8 - 37908*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^9*c*d^9 + 6318*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c +
(b*x + a)*b*d - a*b*d))^4*a^6*b^8*d^10 - 13314*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d
- a*b*d))^6*b^12*c^5*d^4 + 66570*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^11*c^4*d^5 - 133140*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^10*c
^3*d^6 + 133140*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^9*c^2*d^7 -
66570*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^8*c*d^8 + 13314*sqrt(b
*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^5*b^7*d^9 + 16128*sqrt(b*d)*(sqrt(b*d)
*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*b^10*c^4*d^4 - 64512*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^9*c^3*d^5 + 96768*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^8*c^2*d^6 - 64512*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^7*c*d^7 + 16128*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a
*b*d))^8*a^4*b^6*d^8 - 8190*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*b^8*c
^3*d^4 + 24570*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^7*c^2*d^5 - 24
570*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^6*c*d^6 + 8190*sqrt(b*d
)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^3*b^5*d^7 + 2898*sqrt(b*d)*(sqrt(b*d)*s
qrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*b^6*c^2*d^4 - 5796*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^5*c*d^5 + 2898*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c +
(b*x + a)*b*d - a*b*d))^12*a^2*b^4*d^6 - 630*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d -
a*b*d))^14*b^4*c*d^4 + 630*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*a*b^3
*d^5 + 63*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^16*b^2*d^4)/((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 - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)^9)

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maple [B]  time = 0.01, size = 356, normalized size = 1.73 \begin {gather*} -\frac {2 \left (256 b^{5} x^{5} d^{5}+1152 a \,b^{4} d^{5} x^{4}+128 b^{5} c \,d^{4} x^{4}+2016 a^{2} b^{3} d^{5} x^{3}+576 a \,b^{4} c \,d^{4} x^{3}-32 b^{5} c^{2} d^{3} x^{3}+1680 a^{3} b^{2} d^{5} x^{2}+1008 a^{2} b^{3} c \,d^{4} x^{2}-144 a \,b^{4} c^{2} d^{3} x^{2}+16 b^{5} c^{3} d^{2} x^{2}+630 a^{4} b \,d^{5} x +840 a^{3} b^{2} c \,d^{4} x -252 a^{2} b^{3} c^{2} d^{3} x +72 a \,b^{4} c^{3} d^{2} x -10 b^{5} c^{4} d x +63 a^{5} d^{5}+315 a^{4} b c \,d^{4}-210 a^{3} b^{2} c^{2} d^{3}+126 a^{2} b^{3} c^{3} d^{2}-45 a \,b^{4} c^{4} d +7 b^{5} c^{5}\right )}{63 \left (b x +a \right )^{\frac {9}{2}} \sqrt {d x +c}\, \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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)^(11/2)/(d*x+c)^(3/2),x)

[Out]

-2/63*(256*b^5*d^5*x^5+1152*a*b^4*d^5*x^4+128*b^5*c*d^4*x^4+2016*a^2*b^3*d^5*x^3+576*a*b^4*c*d^4*x^3-32*b^5*c^
2*d^3*x^3+1680*a^3*b^2*d^5*x^2+1008*a^2*b^3*c*d^4*x^2-144*a*b^4*c^2*d^3*x^2+16*b^5*c^3*d^2*x^2+630*a^4*b*d^5*x
+840*a^3*b^2*c*d^4*x-252*a^2*b^3*c^2*d^3*x+72*a*b^4*c^3*d^2*x-10*b^5*c^4*d*x+63*a^5*d^5+315*a^4*b*c*d^4-210*a^
3*b^2*c^2*d^3+126*a^2*b^3*c^3*d^2-45*a*b^4*c^4*d+7*b^5*c^5)/(b*x+a)^(9/2)/(d*x+c)^(1/2)/(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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(11/2)/(d*x+c)^(3/2),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 = 1.96, size = 454, normalized size = 2.20 \begin {gather*} -\frac {\sqrt {c+d\,x}\,\left (\frac {126\,a^5\,d^5+630\,a^4\,b\,c\,d^4-420\,a^3\,b^2\,c^2\,d^3+252\,a^2\,b^3\,c^3\,d^2-90\,a\,b^4\,c^4\,d+14\,b^5\,c^5}{63\,b^4\,d\,{\left (a\,d-b\,c\right )}^6}+\frac {512\,b\,d^4\,x^5}{63\,{\left (a\,d-b\,c\right )}^6}+\frac {256\,d^3\,x^4\,\left (9\,a\,d+b\,c\right )}{63\,{\left (a\,d-b\,c\right )}^6}+\frac {x\,\left (1260\,a^4\,b\,d^5+1680\,a^3\,b^2\,c\,d^4-504\,a^2\,b^3\,c^2\,d^3+144\,a\,b^4\,c^3\,d^2-20\,b^5\,c^4\,d\right )}{63\,b^4\,d\,{\left (a\,d-b\,c\right )}^6}+\frac {64\,d^2\,x^3\,\left (63\,a^2\,d^2+18\,a\,b\,c\,d-b^2\,c^2\right )}{63\,b\,{\left (a\,d-b\,c\right )}^6}+\frac {32\,d\,x^2\,\left (105\,a^3\,d^3+63\,a^2\,b\,c\,d^2-9\,a\,b^2\,c^2\,d+b^3\,c^3\right )}{63\,b^2\,{\left (a\,d-b\,c\right )}^6}\right )}{x^5\,\sqrt {a+b\,x}+\frac {a^4\,c\,\sqrt {a+b\,x}}{b^4\,d}+\frac {x^4\,\left (4\,a\,d+b\,c\right )\,\sqrt {a+b\,x}}{b\,d}+\frac {2\,a\,x^3\,\left (3\,a\,d+2\,b\,c\right )\,\sqrt {a+b\,x}}{b^2\,d}+\frac {a^3\,x\,\left (a\,d+4\,b\,c\right )\,\sqrt {a+b\,x}}{b^4\,d}+\frac {2\,a^2\,x^2\,\left (2\,a\,d+3\,b\,c\right )\,\sqrt {a+b\,x}}{b^3\,d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*x)^(11/2)*(c + d*x)^(3/2)),x)

[Out]

-((c + d*x)^(1/2)*((126*a^5*d^5 + 14*b^5*c^5 + 252*a^2*b^3*c^3*d^2 - 420*a^3*b^2*c^2*d^3 - 90*a*b^4*c^4*d + 63
0*a^4*b*c*d^4)/(63*b^4*d*(a*d - b*c)^6) + (512*b*d^4*x^5)/(63*(a*d - b*c)^6) + (256*d^3*x^4*(9*a*d + b*c))/(63
*(a*d - b*c)^6) + (x*(1260*a^4*b*d^5 - 20*b^5*c^4*d + 144*a*b^4*c^3*d^2 + 1680*a^3*b^2*c*d^4 - 504*a^2*b^3*c^2
*d^3))/(63*b^4*d*(a*d - b*c)^6) + (64*d^2*x^3*(63*a^2*d^2 - b^2*c^2 + 18*a*b*c*d))/(63*b*(a*d - b*c)^6) + (32*
d*x^2*(105*a^3*d^3 + b^3*c^3 - 9*a*b^2*c^2*d + 63*a^2*b*c*d^2))/(63*b^2*(a*d - b*c)^6)))/(x^5*(a + b*x)^(1/2)
+ (a^4*c*(a + b*x)^(1/2))/(b^4*d) + (x^4*(4*a*d + b*c)*(a + b*x)^(1/2))/(b*d) + (2*a*x^3*(3*a*d + 2*b*c)*(a +
b*x)^(1/2))/(b^2*d) + (a^3*x*(a*d + 4*b*c)*(a + b*x)^(1/2))/(b^4*d) + (2*a^2*x^2*(2*a*d + 3*b*c)*(a + b*x)^(1/
2))/(b^3*d))

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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(1/(b*x+a)**(11/2)/(d*x+c)**(3/2),x)

[Out]

Timed out

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