3.14.85 \(\int \frac {(c+d x)^{5/2}}{(a+b x)^{15/2}} \, dx\)
Optimal. Leaf size=136 \[ \frac {32 d^3 (c+d x)^{7/2}}{3003 (a+b x)^{7/2} (b c-a d)^4}-\frac {16 d^2 (c+d x)^{7/2}}{429 (a+b x)^{9/2} (b c-a d)^3}+\frac {12 d (c+d x)^{7/2}}{143 (a+b x)^{11/2} (b c-a d)^2}-\frac {2 (c+d x)^{7/2}}{13 (a+b x)^{13/2} (b c-a d)} \]
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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} \begin {gather*} \frac {32 d^3 (c+d x)^{7/2}}{3003 (a+b x)^{7/2} (b c-a d)^4}-\frac {16 d^2 (c+d x)^{7/2}}{429 (a+b x)^{9/2} (b c-a d)^3}+\frac {12 d (c+d x)^{7/2}}{143 (a+b x)^{11/2} (b c-a d)^2}-\frac {2 (c+d x)^{7/2}}{13 (a+b x)^{13/2} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(c + d*x)^(5/2)/(a + b*x)^(15/2),x]
[Out]
(-2*(c + d*x)^(7/2))/(13*(b*c - a*d)*(a + b*x)^(13/2)) + (12*d*(c + d*x)^(7/2))/(143*(b*c - a*d)^2*(a + b*x)^(
11/2)) - (16*d^2*(c + d*x)^(7/2))/(429*(b*c - a*d)^3*(a + b*x)^(9/2)) + (32*d^3*(c + d*x)^(7/2))/(3003*(b*c -
a*d)^4*(a + b*x)^(7/2))
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)^{5/2}}{(a+b x)^{15/2}} \, dx &=-\frac {2 (c+d x)^{7/2}}{13 (b c-a d) (a+b x)^{13/2}}-\frac {(6 d) \int \frac {(c+d x)^{5/2}}{(a+b x)^{13/2}} \, dx}{13 (b c-a d)}\\ &=-\frac {2 (c+d x)^{7/2}}{13 (b c-a d) (a+b x)^{13/2}}+\frac {12 d (c+d x)^{7/2}}{143 (b c-a d)^2 (a+b x)^{11/2}}+\frac {\left (24 d^2\right ) \int \frac {(c+d x)^{5/2}}{(a+b x)^{11/2}} \, dx}{143 (b c-a d)^2}\\ &=-\frac {2 (c+d x)^{7/2}}{13 (b c-a d) (a+b x)^{13/2}}+\frac {12 d (c+d x)^{7/2}}{143 (b c-a d)^2 (a+b x)^{11/2}}-\frac {16 d^2 (c+d x)^{7/2}}{429 (b c-a d)^3 (a+b x)^{9/2}}-\frac {\left (16 d^3\right ) \int \frac {(c+d x)^{5/2}}{(a+b x)^{9/2}} \, dx}{429 (b c-a d)^3}\\ &=-\frac {2 (c+d x)^{7/2}}{13 (b c-a d) (a+b x)^{13/2}}+\frac {12 d (c+d x)^{7/2}}{143 (b c-a d)^2 (a+b x)^{11/2}}-\frac {16 d^2 (c+d x)^{7/2}}{429 (b c-a d)^3 (a+b x)^{9/2}}+\frac {32 d^3 (c+d x)^{7/2}}{3003 (b c-a d)^4 (a+b x)^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 118, normalized size = 0.87 \begin {gather*} \frac {2 (c+d x)^{7/2} \left (429 a^3 d^3+143 a^2 b d^2 (2 d x-7 c)+13 a b^2 d \left (63 c^2-28 c d x+8 d^2 x^2\right )+b^3 \left (-231 c^3+126 c^2 d x-56 c d^2 x^2+16 d^3 x^3\right )\right )}{3003 (a+b x)^{13/2} (b c-a d)^4} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(c + d*x)^(5/2)/(a + b*x)^(15/2),x]
[Out]
(2*(c + d*x)^(7/2)*(429*a^3*d^3 + 143*a^2*b*d^2*(-7*c + 2*d*x) + 13*a*b^2*d*(63*c^2 - 28*c*d*x + 8*d^2*x^2) +
b^3*(-231*c^3 + 126*c^2*d*x - 56*c*d^2*x^2 + 16*d^3*x^3)))/(3003*(b*c - a*d)^4*(a + b*x)^(13/2))
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IntegrateAlgebraic [A] time = 0.15, size = 95, normalized size = 0.70 \begin {gather*} -\frac {2 (c+d x)^{7/2} \left (\frac {231 b^3 (c+d x)^3}{(a+b x)^3}-\frac {819 b^2 d (c+d x)^2}{(a+b x)^2}+\frac {1001 b d^2 (c+d x)}{a+b x}-429 d^3\right )}{3003 (a+b x)^{7/2} (b c-a d)^4} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(c + d*x)^(5/2)/(a + b*x)^(15/2),x]
[Out]
(-2*(c + d*x)^(7/2)*(-429*d^3 + (1001*b*d^2*(c + d*x))/(a + b*x) - (819*b^2*d*(c + d*x)^2)/(a + b*x)^2 + (231*
b^3*(c + d*x)^3)/(a + b*x)^3))/(3003*(b*c - a*d)^4*(a + b*x)^(7/2))
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fricas [B] time = 55.76, size = 765, normalized size = 5.62 \begin {gather*} \frac {2 \, {\left (16 \, b^{3} d^{6} x^{6} - 231 \, b^{3} c^{6} + 819 \, a b^{2} c^{5} d - 1001 \, a^{2} b c^{4} d^{2} + 429 \, a^{3} c^{3} d^{3} - 8 \, {\left (b^{3} c d^{5} - 13 \, a b^{2} d^{6}\right )} x^{5} + 2 \, {\left (3 \, b^{3} c^{2} d^{4} - 26 \, a b^{2} c d^{5} + 143 \, a^{2} b d^{6}\right )} x^{4} - {\left (5 \, b^{3} c^{3} d^{3} - 39 \, a b^{2} c^{2} d^{4} + 143 \, a^{2} b c d^{5} - 429 \, a^{3} d^{6}\right )} x^{3} - {\left (371 \, b^{3} c^{4} d^{2} - 1469 \, a b^{2} c^{3} d^{3} + 2145 \, a^{2} b c^{2} d^{4} - 1287 \, a^{3} c d^{5}\right )} x^{2} - {\left (567 \, b^{3} c^{5} d - 2093 \, a b^{2} c^{4} d^{2} + 2717 \, a^{2} b c^{3} d^{3} - 1287 \, a^{3} c^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3003 \, {\left (a^{7} b^{4} c^{4} - 4 \, a^{8} b^{3} c^{3} d + 6 \, a^{9} b^{2} c^{2} d^{2} - 4 \, a^{10} b c d^{3} + a^{11} d^{4} + {\left (b^{11} c^{4} - 4 \, a b^{10} c^{3} d + 6 \, a^{2} b^{9} c^{2} d^{2} - 4 \, a^{3} b^{8} c d^{3} + a^{4} b^{7} d^{4}\right )} x^{7} + 7 \, {\left (a b^{10} c^{4} - 4 \, a^{2} b^{9} c^{3} d + 6 \, a^{3} b^{8} c^{2} d^{2} - 4 \, a^{4} b^{7} c d^{3} + a^{5} b^{6} d^{4}\right )} x^{6} + 21 \, {\left (a^{2} b^{9} c^{4} - 4 \, a^{3} b^{8} c^{3} d + 6 \, a^{4} b^{7} c^{2} d^{2} - 4 \, a^{5} b^{6} c d^{3} + a^{6} b^{5} d^{4}\right )} x^{5} + 35 \, {\left (a^{3} b^{8} c^{4} - 4 \, a^{4} b^{7} c^{3} d + 6 \, a^{5} b^{6} c^{2} d^{2} - 4 \, a^{6} b^{5} c d^{3} + a^{7} b^{4} d^{4}\right )} x^{4} + 35 \, {\left (a^{4} b^{7} c^{4} - 4 \, a^{5} b^{6} c^{3} d + 6 \, a^{6} b^{5} c^{2} d^{2} - 4 \, a^{7} b^{4} c d^{3} + a^{8} b^{3} d^{4}\right )} x^{3} + 21 \, {\left (a^{5} b^{6} c^{4} - 4 \, a^{6} b^{5} c^{3} d + 6 \, a^{7} b^{4} c^{2} d^{2} - 4 \, a^{8} b^{3} c d^{3} + a^{9} b^{2} d^{4}\right )} x^{2} + 7 \, {\left (a^{6} b^{5} c^{4} - 4 \, a^{7} b^{4} c^{3} d + 6 \, a^{8} b^{3} c^{2} d^{2} - 4 \, a^{9} b^{2} c d^{3} + a^{10} b d^{4}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)^(5/2)/(b*x+a)^(15/2),x, algorithm="fricas")
[Out]
2/3003*(16*b^3*d^6*x^6 - 231*b^3*c^6 + 819*a*b^2*c^5*d - 1001*a^2*b*c^4*d^2 + 429*a^3*c^3*d^3 - 8*(b^3*c*d^5 -
13*a*b^2*d^6)*x^5 + 2*(3*b^3*c^2*d^4 - 26*a*b^2*c*d^5 + 143*a^2*b*d^6)*x^4 - (5*b^3*c^3*d^3 - 39*a*b^2*c^2*d^
4 + 143*a^2*b*c*d^5 - 429*a^3*d^6)*x^3 - (371*b^3*c^4*d^2 - 1469*a*b^2*c^3*d^3 + 2145*a^2*b*c^2*d^4 - 1287*a^3
*c*d^5)*x^2 - (567*b^3*c^5*d - 2093*a*b^2*c^4*d^2 + 2717*a^2*b*c^3*d^3 - 1287*a^3*c^2*d^4)*x)*sqrt(b*x + a)*sq
rt(d*x + c)/(a^7*b^4*c^4 - 4*a^8*b^3*c^3*d + 6*a^9*b^2*c^2*d^2 - 4*a^10*b*c*d^3 + a^11*d^4 + (b^11*c^4 - 4*a*b
^10*c^3*d + 6*a^2*b^9*c^2*d^2 - 4*a^3*b^8*c*d^3 + a^4*b^7*d^4)*x^7 + 7*(a*b^10*c^4 - 4*a^2*b^9*c^3*d + 6*a^3*b
^8*c^2*d^2 - 4*a^4*b^7*c*d^3 + a^5*b^6*d^4)*x^6 + 21*(a^2*b^9*c^4 - 4*a^3*b^8*c^3*d + 6*a^4*b^7*c^2*d^2 - 4*a^
5*b^6*c*d^3 + a^6*b^5*d^4)*x^5 + 35*(a^3*b^8*c^4 - 4*a^4*b^7*c^3*d + 6*a^5*b^6*c^2*d^2 - 4*a^6*b^5*c*d^3 + a^7
*b^4*d^4)*x^4 + 35*(a^4*b^7*c^4 - 4*a^5*b^6*c^3*d + 6*a^6*b^5*c^2*d^2 - 4*a^7*b^4*c*d^3 + a^8*b^3*d^4)*x^3 + 2
1*(a^5*b^6*c^4 - 4*a^6*b^5*c^3*d + 6*a^7*b^4*c^2*d^2 - 4*a^8*b^3*c*d^3 + a^9*b^2*d^4)*x^2 + 7*(a^6*b^5*c^4 - 4
*a^7*b^4*c^3*d + 6*a^8*b^3*c^2*d^2 - 4*a^9*b^2*c*d^3 + a^10*b*d^4)*x)
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giac [B] time = 6.09, size = 2868, normalized size = 21.09
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)^(5/2)/(b*x+a)^(15/2),x, algorithm="giac")
[Out]
64/3003*(sqrt(b*d)*b^18*c^9*d^6*abs(b) - 9*sqrt(b*d)*a*b^17*c^8*d^7*abs(b) + 36*sqrt(b*d)*a^2*b^16*c^7*d^8*abs
(b) - 84*sqrt(b*d)*a^3*b^15*c^6*d^9*abs(b) + 126*sqrt(b*d)*a^4*b^14*c^5*d^10*abs(b) - 126*sqrt(b*d)*a^5*b^13*c
^4*d^11*abs(b) + 84*sqrt(b*d)*a^6*b^12*c^3*d^12*abs(b) - 36*sqrt(b*d)*a^7*b^11*c^2*d^13*abs(b) + 9*sqrt(b*d)*a
^8*b^10*c*d^14*abs(b) - sqrt(b*d)*a^9*b^9*d^15*abs(b) - 13*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (
b*x + a)*b*d - a*b*d))^2*b^16*c^8*d^6*abs(b) + 104*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^7*d^7*abs(b) - 364*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^6*d^8*abs(b) + 728*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^5*d^9*abs(b) - 910*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^4*d^10*abs(b) + 728*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^3*d^11*abs(b) - 364*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^2*d^12*abs(b) + 104*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*c*d^13*abs(b) - 13*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^8*b^8*d^14*
abs(b) + 78*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^7*d^6*abs(b) -
546*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^6*d^7*abs(b) + 1638*s
qrt(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^5*d^8*abs(b) - 2730*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^4*d^9*abs(b) + 2730*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^3*d^10*abs(b) - 1638*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^2*d^11*abs(b) + 546*sqrt(b*d)*(sq
rt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^6*b^8*c*d^12*abs(b) - 78*sqrt(b*d)*(sqrt(b*d)
*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^7*b^7*d^13*abs(b) - 286*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^6*d^6*abs(b) + 1716*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^5*d^7*abs(b) - 4290*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqr
t(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^2*b^10*c^4*d^8*abs(b) + 5720*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^3*d^9*abs(b) - 4290*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^2*d^10*abs(b) + 1716*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*c*d^11*abs(b) - 286*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x +
a)*b*d - a*b*d))^6*a^6*b^6*d^12*abs(b) - 2288*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^5*d^6*abs(b) + 11440*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^4*d^7*abs(b) - 22880*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^3*d^8*abs(b) + 22880*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^2*d^9*abs(b) - 11440*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*c*d^10*abs(b) + 2288*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a^5*b^5*d
^11*abs(b) - 10296*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^4*d^6*ab
s(b) + 41184*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^3*d^7*abs(b)
- 61776*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^2*d^8*abs(b) +
41184*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*c*d^9*abs(b) - 102
96*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^4*b^4*d^10*abs(b) - 16302*sq
rt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*b^6*c^3*d^6*abs(b) + 48906*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^2*d^7*abs(b) - 48906*sqrt(b*d)*(s
qrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*a^2*b^4*c*d^8*abs(b) + 16302*sqrt(b*d)*(sqrt(
b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*a^3*b^3*d^9*abs(b) - 18018*sqrt(b*d)*(sqrt(b*d)*s
qrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*b^4*c^2*d^6*abs(b) + 36036*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*c*d^7*abs(b) - 18018*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a)
- sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*a^2*b^2*d^8*abs(b) - 9009*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*c*d^6*abs(b) + 9009*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c +
(b*x + a)*b*d - a*b*d))^16*a*b*d^7*abs(b) - 3003*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b
*d - a*b*d))^18*d^6*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)^13*b)
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maple [A] time = 0.01, size = 171, normalized size = 1.26 \begin {gather*} \frac {2 \left (d x +c \right )^{\frac {7}{2}} \left (16 b^{3} x^{3} d^{3}+104 a \,b^{2} d^{3} x^{2}-56 b^{3} c \,d^{2} x^{2}+286 a^{2} b \,d^{3} x -364 a \,b^{2} c \,d^{2} x +126 b^{3} c^{2} d x +429 a^{3} d^{3}-1001 a^{2} b c \,d^{2}+819 a \,b^{2} c^{2} d -231 b^{3} c^{3}\right )}{3003 \left (b x +a \right )^{\frac {13}{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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d*x+c)^(5/2)/(b*x+a)^(15/2),x)
[Out]
2/3003*(d*x+c)^(7/2)*(16*b^3*d^3*x^3+104*a*b^2*d^3*x^2-56*b^3*c*d^2*x^2+286*a^2*b*d^3*x-364*a*b^2*c*d^2*x+126*
b^3*c^2*d*x+429*a^3*d^3-1001*a^2*b*c*d^2+819*a*b^2*c^2*d-231*b^3*c^3)/(b*x+a)^(13/2)/(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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)^(5/2)/(b*x+a)^(15/2),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 = 1.62, size = 459, normalized size = 3.38 \begin {gather*} \frac {\sqrt {c+d\,x}\,\left (\frac {x^2\,\left (2574\,a^3\,c\,d^5-4290\,a^2\,b\,c^2\,d^4+2938\,a\,b^2\,c^3\,d^3-742\,b^3\,c^4\,d^2\right )}{3003\,b^6\,{\left (a\,d-b\,c\right )}^4}-\frac {-858\,a^3\,c^3\,d^3+2002\,a^2\,b\,c^4\,d^2-1638\,a\,b^2\,c^5\,d+462\,b^3\,c^6}{3003\,b^6\,{\left (a\,d-b\,c\right )}^4}+\frac {x^3\,\left (858\,a^3\,d^6-286\,a^2\,b\,c\,d^5+78\,a\,b^2\,c^2\,d^4-10\,b^3\,c^3\,d^3\right )}{3003\,b^6\,{\left (a\,d-b\,c\right )}^4}+\frac {32\,d^6\,x^6}{3003\,b^3\,{\left (a\,d-b\,c\right )}^4}-\frac {x\,\left (-2574\,a^3\,c^2\,d^4+5434\,a^2\,b\,c^3\,d^3-4186\,a\,b^2\,c^4\,d^2+1134\,b^3\,c^5\,d\right )}{3003\,b^6\,{\left (a\,d-b\,c\right )}^4}+\frac {16\,d^5\,x^5\,\left (13\,a\,d-b\,c\right )}{3003\,b^4\,{\left (a\,d-b\,c\right )}^4}+\frac {4\,d^4\,x^4\,\left (143\,a^2\,d^2-26\,a\,b\,c\,d+3\,b^2\,c^2\right )}{3003\,b^5\,{\left (a\,d-b\,c\right )}^4}\right )}{x^6\,\sqrt {a+b\,x}+\frac {a^6\,\sqrt {a+b\,x}}{b^6}+\frac {15\,a^2\,x^4\,\sqrt {a+b\,x}}{b^2}+\frac {20\,a^3\,x^3\,\sqrt {a+b\,x}}{b^3}+\frac {15\,a^4\,x^2\,\sqrt {a+b\,x}}{b^4}+\frac {6\,a\,x^5\,\sqrt {a+b\,x}}{b}+\frac {6\,a^5\,x\,\sqrt {a+b\,x}}{b^5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c + d*x)^(5/2)/(a + b*x)^(15/2),x)
[Out]
((c + d*x)^(1/2)*((x^2*(2574*a^3*c*d^5 - 742*b^3*c^4*d^2 + 2938*a*b^2*c^3*d^3 - 4290*a^2*b*c^2*d^4))/(3003*b^6
*(a*d - b*c)^4) - (462*b^3*c^6 - 858*a^3*c^3*d^3 + 2002*a^2*b*c^4*d^2 - 1638*a*b^2*c^5*d)/(3003*b^6*(a*d - b*c
)^4) + (x^3*(858*a^3*d^6 - 10*b^3*c^3*d^3 + 78*a*b^2*c^2*d^4 - 286*a^2*b*c*d^5))/(3003*b^6*(a*d - b*c)^4) + (3
2*d^6*x^6)/(3003*b^3*(a*d - b*c)^4) - (x*(1134*b^3*c^5*d - 2574*a^3*c^2*d^4 - 4186*a*b^2*c^4*d^2 + 5434*a^2*b*
c^3*d^3))/(3003*b^6*(a*d - b*c)^4) + (16*d^5*x^5*(13*a*d - b*c))/(3003*b^4*(a*d - b*c)^4) + (4*d^4*x^4*(143*a^
2*d^2 + 3*b^2*c^2 - 26*a*b*c*d))/(3003*b^5*(a*d - b*c)^4)))/(x^6*(a + b*x)^(1/2) + (a^6*(a + b*x)^(1/2))/b^6 +
(15*a^2*x^4*(a + b*x)^(1/2))/b^2 + (20*a^3*x^3*(a + b*x)^(1/2))/b^3 + (15*a^4*x^2*(a + b*x)^(1/2))/b^4 + (6*a
*x^5*(a + b*x)^(1/2))/b + (6*a^5*x*(a + b*x)^(1/2))/b^5)
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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((d*x+c)**(5/2)/(b*x+a)**(15/2),x)
[Out]
Timed out
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