3.1491 \(\int \frac{(c+d x)^{5/2}}{(a+b x)^{15/2}} \, dx\)
Optimal. Leaf size=136 \[ \frac{32 d^3 (c+d x)^{7/2}}{3003 (a+b x)^{7/2} (b c-a d)^4}-\frac{16 d^2 (c+d x)^{7/2}}{429 (a+b x)^{9/2} (b c-a d)^3}+\frac{12 d (c+d x)^{7/2}}{143 (a+b x)^{11/2} (b c-a d)^2}-\frac{2 (c+d x)^{7/2}}{13 (a+b x)^{13/2} (b c-a d)} \]
[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))
________________________________________________________________________________________
Rubi [A] time = 0.0278713, antiderivative size = 136, normalized size of antiderivative = 1.,
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} \[ \frac{32 d^3 (c+d x)^{7/2}}{3003 (a+b x)^{7/2} (b c-a d)^4}-\frac{16 d^2 (c+d x)^{7/2}}{429 (a+b x)^{9/2} (b c-a d)^3}+\frac{12 d (c+d x)^{7/2}}{143 (a+b x)^{11/2} (b c-a d)^2}-\frac{2 (c+d x)^{7/2}}{13 (a+b x)^{13/2} (b c-a d)} \]
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 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 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]
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*}
Mathematica [A] time = 0.0786368, size = 118, normalized size = 0.87 \[ \frac{2 (c+d x)^{7/2} \left (143 a^2 b d^2 (2 d x-7 c)+429 a^3 d^3+13 a b^2 d \left (63 c^2-28 c d x+8 d^2 x^2\right )+b^3 \left (126 c^2 d x-231 c^3-56 c d^2 x^2+16 d^3 x^3\right )\right )}{3003 (a+b x)^{13/2} (b c-a d)^4} \]
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))
________________________________________________________________________________________
Maple [A] time = 0.008, size = 171, normalized size = 1.3 \begin{align*}{\frac{32\,{b}^{3}{d}^{3}{x}^{3}+208\,a{b}^{2}{d}^{3}{x}^{2}-112\,{b}^{3}c{d}^{2}{x}^{2}+572\,{a}^{2}b{d}^{3}x-728\,a{b}^{2}c{d}^{2}x+252\,{b}^{3}{c}^{2}dx+858\,{a}^{3}{d}^{3}-2002\,{a}^{2}bc{d}^{2}+1638\,a{b}^{2}{c}^{2}d-462\,{b}^{3}{c}^{3}}{3003\,{d}^{4}{a}^{4}-12012\,b{d}^{3}c{a}^{3}+18018\,{b}^{2}{d}^{2}{c}^{2}{a}^{2}-12012\,{b}^{3}d{c}^{3}a+3003\,{b}^{4}{c}^{4}} \left ( dx+c \right ) ^{{\frac{7}{2}}} \left ( bx+a \right ) ^{-{\frac{13}{2}}}} \end{align*}
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)
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
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
________________________________________________________________________________________
Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
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]
Timed out
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
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
________________________________________________________________________________________
Giac [B] time = 3.24609, size = 3872, normalized size = 28.47 \begin{align*} \text{result too large to display} \end{align*}
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)