3.1490 \(\int \frac{(c+d x)^{5/2}}{(a+b x)^{13/2}} \, dx\)

Optimal. Leaf size=101 \[ -\frac{16 d^2 (c+d x)^{7/2}}{693 (a+b x)^{7/2} (b c-a d)^3}+\frac{8 d (c+d x)^{7/2}}{99 (a+b x)^{9/2} (b c-a d)^2}-\frac{2 (c+d x)^{7/2}}{11 (a+b x)^{11/2} (b c-a d)} \]

[Out]

(-2*(c + d*x)^(7/2))/(11*(b*c - a*d)*(a + b*x)^(11/2)) + (8*d*(c + d*x)^(7/2))/(99*(b*c - a*d)^2*(a + b*x)^(9/
2)) - (16*d^2*(c + d*x)^(7/2))/(693*(b*c - a*d)^3*(a + b*x)^(7/2))

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Rubi [A]  time = 0.0162974, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {45, 37} \[ -\frac{16 d^2 (c+d x)^{7/2}}{693 (a+b x)^{7/2} (b c-a d)^3}+\frac{8 d (c+d x)^{7/2}}{99 (a+b x)^{9/2} (b c-a d)^2}-\frac{2 (c+d x)^{7/2}}{11 (a+b x)^{11/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^(5/2)/(a + b*x)^(13/2),x]

[Out]

(-2*(c + d*x)^(7/2))/(11*(b*c - a*d)*(a + b*x)^(11/2)) + (8*d*(c + d*x)^(7/2))/(99*(b*c - a*d)^2*(a + b*x)^(9/
2)) - (16*d^2*(c + d*x)^(7/2))/(693*(b*c - a*d)^3*(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)^{13/2}} \, dx &=-\frac{2 (c+d x)^{7/2}}{11 (b c-a d) (a+b x)^{11/2}}-\frac{(4 d) \int \frac{(c+d x)^{5/2}}{(a+b x)^{11/2}} \, dx}{11 (b c-a d)}\\ &=-\frac{2 (c+d x)^{7/2}}{11 (b c-a d) (a+b x)^{11/2}}+\frac{8 d (c+d x)^{7/2}}{99 (b c-a d)^2 (a+b x)^{9/2}}+\frac{\left (8 d^2\right ) \int \frac{(c+d x)^{5/2}}{(a+b x)^{9/2}} \, dx}{99 (b c-a d)^2}\\ &=-\frac{2 (c+d x)^{7/2}}{11 (b c-a d) (a+b x)^{11/2}}+\frac{8 d (c+d x)^{7/2}}{99 (b c-a d)^2 (a+b x)^{9/2}}-\frac{16 d^2 (c+d x)^{7/2}}{693 (b c-a d)^3 (a+b x)^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.0613316, size = 77, normalized size = 0.76 \[ -\frac{2 (c+d x)^{7/2} \left (99 a^2 d^2+22 a b d (2 d x-7 c)+b^2 \left (63 c^2-28 c d x+8 d^2 x^2\right )\right )}{693 (a+b x)^{11/2} (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^(5/2)/(a + b*x)^(13/2),x]

[Out]

(-2*(c + d*x)^(7/2)*(99*a^2*d^2 + 22*a*b*d*(-7*c + 2*d*x) + b^2*(63*c^2 - 28*c*d*x + 8*d^2*x^2)))/(693*(b*c -
a*d)^3*(a + b*x)^(11/2))

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Maple [A]  time = 0.007, size = 105, normalized size = 1. \begin{align*}{\frac{16\,{b}^{2}{d}^{2}{x}^{2}+88\,ab{d}^{2}x-56\,{b}^{2}cdx+198\,{a}^{2}{d}^{2}-308\,abcd+126\,{b}^{2}{c}^{2}}{693\,{a}^{3}{d}^{3}-2079\,{a}^{2}bc{d}^{2}+2079\,a{b}^{2}{c}^{2}d-693\,{b}^{3}{c}^{3}} \left ( dx+c \right ) ^{{\frac{7}{2}}} \left ( bx+a \right ) ^{-{\frac{11}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(5/2)/(b*x+a)^(13/2),x)

[Out]

2/693*(d*x+c)^(7/2)*(8*b^2*d^2*x^2+44*a*b*d^2*x-28*b^2*c*d*x+99*a^2*d^2-154*a*b*c*d+63*b^2*c^2)/(b*x+a)^(11/2)
/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)

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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)^(13/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 112.612, size = 1045, normalized size = 10.35 \begin{align*} -\frac{2 \,{\left (8 \, b^{2} d^{5} x^{5} + 63 \, b^{2} c^{5} - 154 \, a b c^{4} d + 99 \, a^{2} c^{3} d^{2} - 4 \,{\left (b^{2} c d^{4} - 11 \, a b d^{5}\right )} x^{4} +{\left (3 \, b^{2} c^{2} d^{3} - 22 \, a b c d^{4} + 99 \, a^{2} d^{5}\right )} x^{3} +{\left (113 \, b^{2} c^{3} d^{2} - 330 \, a b c^{2} d^{3} + 297 \, a^{2} c d^{4}\right )} x^{2} +{\left (161 \, b^{2} c^{4} d - 418 \, a b c^{3} d^{2} + 297 \, a^{2} c^{2} d^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{693 \,{\left (a^{6} b^{3} c^{3} - 3 \, a^{7} b^{2} c^{2} d + 3 \, a^{8} b c d^{2} - a^{9} d^{3} +{\left (b^{9} c^{3} - 3 \, a b^{8} c^{2} d + 3 \, a^{2} b^{7} c d^{2} - a^{3} b^{6} d^{3}\right )} x^{6} + 6 \,{\left (a b^{8} c^{3} - 3 \, a^{2} b^{7} c^{2} d + 3 \, a^{3} b^{6} c d^{2} - a^{4} b^{5} d^{3}\right )} x^{5} + 15 \,{\left (a^{2} b^{7} c^{3} - 3 \, a^{3} b^{6} c^{2} d + 3 \, a^{4} b^{5} c d^{2} - a^{5} b^{4} d^{3}\right )} x^{4} + 20 \,{\left (a^{3} b^{6} c^{3} - 3 \, a^{4} b^{5} c^{2} d + 3 \, a^{5} b^{4} c d^{2} - a^{6} b^{3} d^{3}\right )} x^{3} + 15 \,{\left (a^{4} b^{5} c^{3} - 3 \, a^{5} b^{4} c^{2} d + 3 \, a^{6} b^{3} c d^{2} - a^{7} b^{2} d^{3}\right )} x^{2} + 6 \,{\left (a^{5} b^{4} c^{3} - 3 \, a^{6} b^{3} c^{2} d + 3 \, a^{7} b^{2} c d^{2} - a^{8} b d^{3}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(5/2)/(b*x+a)^(13/2),x, algorithm="fricas")

[Out]

-2/693*(8*b^2*d^5*x^5 + 63*b^2*c^5 - 154*a*b*c^4*d + 99*a^2*c^3*d^2 - 4*(b^2*c*d^4 - 11*a*b*d^5)*x^4 + (3*b^2*
c^2*d^3 - 22*a*b*c*d^4 + 99*a^2*d^5)*x^3 + (113*b^2*c^3*d^2 - 330*a*b*c^2*d^3 + 297*a^2*c*d^4)*x^2 + (161*b^2*
c^4*d - 418*a*b*c^3*d^2 + 297*a^2*c^2*d^3)*x)*sqrt(b*x + a)*sqrt(d*x + c)/(a^6*b^3*c^3 - 3*a^7*b^2*c^2*d + 3*a
^8*b*c*d^2 - a^9*d^3 + (b^9*c^3 - 3*a*b^8*c^2*d + 3*a^2*b^7*c*d^2 - a^3*b^6*d^3)*x^6 + 6*(a*b^8*c^3 - 3*a^2*b^
7*c^2*d + 3*a^3*b^6*c*d^2 - a^4*b^5*d^3)*x^5 + 15*(a^2*b^7*c^3 - 3*a^3*b^6*c^2*d + 3*a^4*b^5*c*d^2 - a^5*b^4*d
^3)*x^4 + 20*(a^3*b^6*c^3 - 3*a^4*b^5*c^2*d + 3*a^5*b^4*c*d^2 - a^6*b^3*d^3)*x^3 + 15*(a^4*b^5*c^3 - 3*a^5*b^4
*c^2*d + 3*a^6*b^3*c*d^2 - a^7*b^2*d^3)*x^2 + 6*(a^5*b^4*c^3 - 3*a^6*b^3*c^2*d + 3*a^7*b^2*c*d^2 - a^8*b*d^3)*
x)

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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)**(13/2),x)

[Out]

Timed out

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Giac [B]  time = 2.65298, size = 3127, normalized size = 30.96 \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)^(13/2),x, algorithm="giac")

[Out]

-32/693*(sqrt(b*d)*b^16*c^8*d^5*abs(b) - 8*sqrt(b*d)*a*b^15*c^7*d^6*abs(b) + 28*sqrt(b*d)*a^2*b^14*c^6*d^7*abs
(b) - 56*sqrt(b*d)*a^3*b^13*c^5*d^8*abs(b) + 70*sqrt(b*d)*a^4*b^12*c^4*d^9*abs(b) - 56*sqrt(b*d)*a^5*b^11*c^3*
d^10*abs(b) + 28*sqrt(b*d)*a^6*b^10*c^2*d^11*abs(b) - 8*sqrt(b*d)*a^7*b^9*c*d^12*abs(b) + sqrt(b*d)*a^8*b^8*d^
13*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^14*c^7*d^5*abs(b)
+ 77*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^13*c^6*d^6*abs(b) - 231*
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^12*c^5*d^7*abs(b) + 385*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^11*c^4*d^8*abs(b) - 385*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^10*c^3*d^9*abs(b) + 231*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^5*b^9*c^2*d^10*abs(b) - 77*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^8*c*d^11*abs(b) + 11*sqrt(b*d)*(sqrt(b*d)*sqr
t(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^7*b^7*d^12*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^12*c^6*d^5*abs(b) - 330*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^11*c^5*d^6*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))^4*a^2*b^10*c^4*d^7*abs(b) - 1100*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^9*c^3*d^8*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))^4*a^4*b^8*c^2*d^9*abs(b) - 330*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^7*c*d^10*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^6*b^6*d^11*abs(b) + 297*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6
*b^10*c^5*d^5*abs(b) - 1485*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^9*
c^4*d^6*abs(b) + 2970*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^8*c^3*
d^7*abs(b) - 2970*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^7*c^2*d^8*
abs(b) + 1485*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^6*c*d^9*abs(b)
- 297*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^5*d^10*abs(b) + 1485*
sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*b^8*c^4*d^5*abs(b) - 5940*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^7*c^3*d^6*abs(b) + 8910*sqrt(b*d)*(sqr
t(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a^2*b^6*c^2*d^7*abs(b) - 5940*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^5*c*d^8*abs(b) + 1485*sqrt(b*d)*(sqrt(b*d)*sqr
t(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a^4*b^4*d^9*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))^10*b^6*c^3*d^5*abs(b) - 6237*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sq
rt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a*b^5*c^2*d^6*abs(b) + 6237*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^4*c*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))^10*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))^12*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))^12*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))^12*a^2*b^2*d^7*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*
b^2*c*d^5*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*a*b*d^6*a
bs(b) + 462*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^16*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)^11*b^2)