3.1479 \(\int \frac{(c+d x)^{3/2}}{(a+b x)^{11/2}} \, dx\)
Optimal. Leaf size=101 \[ -\frac{16 d^2 (c+d x)^{5/2}}{315 (a+b x)^{5/2} (b c-a d)^3}+\frac{8 d (c+d x)^{5/2}}{63 (a+b x)^{7/2} (b c-a d)^2}-\frac{2 (c+d x)^{5/2}}{9 (a+b x)^{9/2} (b c-a d)} \]
[Out]
(-2*(c + d*x)^(5/2))/(9*(b*c - a*d)*(a + b*x)^(9/2)) + (8*d*(c + d*x)^(5/2))/(63*(b*c - a*d)^2*(a + b*x)^(7/2)
) - (16*d^2*(c + d*x)^(5/2))/(315*(b*c - a*d)^3*(a + b*x)^(5/2))
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Rubi [A] time = 0.015951, 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} \[ -\frac{16 d^2 (c+d x)^{5/2}}{315 (a+b x)^{5/2} (b c-a d)^3}+\frac{8 d (c+d x)^{5/2}}{63 (a+b x)^{7/2} (b c-a d)^2}-\frac{2 (c+d x)^{5/2}}{9 (a+b x)^{9/2} (b c-a d)} \]
Antiderivative was successfully verified.
[In]
Int[(c + d*x)^(3/2)/(a + b*x)^(11/2),x]
[Out]
(-2*(c + d*x)^(5/2))/(9*(b*c - a*d)*(a + b*x)^(9/2)) + (8*d*(c + d*x)^(5/2))/(63*(b*c - a*d)^2*(a + b*x)^(7/2)
) - (16*d^2*(c + d*x)^(5/2))/(315*(b*c - a*d)^3*(a + b*x)^(5/2))
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)^{3/2}}{(a+b x)^{11/2}} \, dx &=-\frac{2 (c+d x)^{5/2}}{9 (b c-a d) (a+b x)^{9/2}}-\frac{(4 d) \int \frac{(c+d x)^{3/2}}{(a+b x)^{9/2}} \, dx}{9 (b c-a d)}\\ &=-\frac{2 (c+d x)^{5/2}}{9 (b c-a d) (a+b x)^{9/2}}+\frac{8 d (c+d x)^{5/2}}{63 (b c-a d)^2 (a+b x)^{7/2}}+\frac{\left (8 d^2\right ) \int \frac{(c+d x)^{3/2}}{(a+b x)^{7/2}} \, dx}{63 (b c-a d)^2}\\ &=-\frac{2 (c+d x)^{5/2}}{9 (b c-a d) (a+b x)^{9/2}}+\frac{8 d (c+d x)^{5/2}}{63 (b c-a d)^2 (a+b x)^{7/2}}-\frac{16 d^2 (c+d x)^{5/2}}{315 (b c-a d)^3 (a+b x)^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0572788, size = 77, normalized size = 0.76 \[ -\frac{2 (c+d x)^{5/2} \left (63 a^2 d^2+18 a b d (2 d x-5 c)+b^2 \left (35 c^2-20 c d x+8 d^2 x^2\right )\right )}{315 (a+b x)^{9/2} (b c-a d)^3} \]
Antiderivative was successfully verified.
[In]
Integrate[(c + d*x)^(3/2)/(a + b*x)^(11/2),x]
[Out]
(-2*(c + d*x)^(5/2)*(63*a^2*d^2 + 18*a*b*d*(-5*c + 2*d*x) + b^2*(35*c^2 - 20*c*d*x + 8*d^2*x^2)))/(315*(b*c -
a*d)^3*(a + b*x)^(9/2))
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Maple [A] time = 0.006, size = 105, normalized size = 1. \begin{align*}{\frac{16\,{b}^{2}{d}^{2}{x}^{2}+72\,ab{d}^{2}x-40\,{b}^{2}cdx+126\,{a}^{2}{d}^{2}-180\,abcd+70\,{b}^{2}{c}^{2}}{315\,{a}^{3}{d}^{3}-945\,{a}^{2}bc{d}^{2}+945\,a{b}^{2}{c}^{2}d-315\,{b}^{3}{c}^{3}} \left ( dx+c \right ) ^{{\frac{5}{2}}} \left ( bx+a \right ) ^{-{\frac{9}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d*x+c)^(3/2)/(b*x+a)^(11/2),x)
[Out]
2/315*(d*x+c)^(5/2)*(8*b^2*d^2*x^2+36*a*b*d^2*x-20*b^2*c*d*x+63*a^2*d^2-90*a*b*c*d+35*b^2*c^2)/(b*x+a)^(9/2)/(
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)^(3/2)/(b*x+a)^(11/2),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 48.1185, size = 860, normalized size = 8.51 \begin{align*} -\frac{2 \,{\left (8 \, b^{2} d^{4} x^{4} + 35 \, b^{2} c^{4} - 90 \, a b c^{3} d + 63 \, a^{2} c^{2} d^{2} - 4 \,{\left (b^{2} c d^{3} - 9 \, a b d^{4}\right )} x^{3} + 3 \,{\left (b^{2} c^{2} d^{2} - 6 \, a b c d^{3} + 21 \, a^{2} d^{4}\right )} x^{2} + 2 \,{\left (25 \, b^{2} c^{3} d - 72 \, a b c^{2} d^{2} + 63 \, a^{2} c d^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{315 \,{\left (a^{5} b^{3} c^{3} - 3 \, a^{6} b^{2} c^{2} d + 3 \, a^{7} b c d^{2} - a^{8} d^{3} +{\left (b^{8} c^{3} - 3 \, a b^{7} c^{2} d + 3 \, a^{2} b^{6} c d^{2} - a^{3} b^{5} d^{3}\right )} x^{5} + 5 \,{\left (a b^{7} c^{3} - 3 \, a^{2} b^{6} c^{2} d + 3 \, a^{3} b^{5} c d^{2} - a^{4} b^{4} d^{3}\right )} x^{4} + 10 \,{\left (a^{2} b^{6} c^{3} - 3 \, a^{3} b^{5} c^{2} d + 3 \, a^{4} b^{4} c d^{2} - a^{5} b^{3} d^{3}\right )} x^{3} + 10 \,{\left (a^{3} b^{5} c^{3} - 3 \, a^{4} b^{4} c^{2} d + 3 \, a^{5} b^{3} c d^{2} - a^{6} b^{2} d^{3}\right )} x^{2} + 5 \,{\left (a^{4} b^{4} c^{3} - 3 \, a^{5} b^{3} c^{2} d + 3 \, a^{6} b^{2} c d^{2} - a^{7} b d^{3}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)^(3/2)/(b*x+a)^(11/2),x, algorithm="fricas")
[Out]
-2/315*(8*b^2*d^4*x^4 + 35*b^2*c^4 - 90*a*b*c^3*d + 63*a^2*c^2*d^2 - 4*(b^2*c*d^3 - 9*a*b*d^4)*x^3 + 3*(b^2*c^
2*d^2 - 6*a*b*c*d^3 + 21*a^2*d^4)*x^2 + 2*(25*b^2*c^3*d - 72*a*b*c^2*d^2 + 63*a^2*c*d^3)*x)*sqrt(b*x + a)*sqrt
(d*x + c)/(a^5*b^3*c^3 - 3*a^6*b^2*c^2*d + 3*a^7*b*c*d^2 - a^8*d^3 + (b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d^
2 - a^3*b^5*d^3)*x^5 + 5*(a*b^7*c^3 - 3*a^2*b^6*c^2*d + 3*a^3*b^5*c*d^2 - a^4*b^4*d^3)*x^4 + 10*(a^2*b^6*c^3 -
3*a^3*b^5*c^2*d + 3*a^4*b^4*c*d^2 - a^5*b^3*d^3)*x^3 + 10*(a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^5*b^3*c*d^2 -
a^6*b^2*d^3)*x^2 + 5*(a^4*b^4*c^3 - 3*a^5*b^3*c^2*d + 3*a^6*b^2*c*d^2 - a^7*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)**(3/2)/(b*x+a)**(11/2),x)
[Out]
Timed out
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Giac [B] time = 1.7454, size = 1882, normalized size = 18.63 \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)^(3/2)/(b*x+a)^(11/2),x, algorithm="giac")
[Out]
-32/315*(sqrt(b*d)*b^12*c^6*d^4*abs(b) - 6*sqrt(b*d)*a*b^11*c^5*d^5*abs(b) + 15*sqrt(b*d)*a^2*b^10*c^4*d^6*abs
(b) - 20*sqrt(b*d)*a^3*b^9*c^3*d^7*abs(b) + 15*sqrt(b*d)*a^4*b^8*c^2*d^8*abs(b) - 6*sqrt(b*d)*a^5*b^7*c*d^9*ab
s(b) + sqrt(b*d)*a^6*b^6*d^10*abs(b) - 9*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b
*d))^2*b^10*c^5*d^4*abs(b) + 45*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^9*c^4*d^5*abs(b) - 90*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^8*c^
3*d^6*abs(b) + 90*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^7*c^2*d^7*
abs(b) - 45*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^6*c*d^8*abs(b) +
9*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^5*d^9*abs(b) + 36*sqrt(b*
d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^8*c^4*d^4*abs(b) - 144*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^7*c^3*d^5*abs(b) + 216*sqrt(b*d)*(sqrt(b*d)*sq
rt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^2*b^6*c^2*d^6*abs(b) - 144*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^5*c*d^7*abs(b) + 36*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^4*d^8*abs(b) + 126*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2
*c + (b*x + a)*b*d - a*b*d))^6*b^6*c^3*d^4*abs(b) - 378*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^5*c^2*d^5*abs(b) + 378*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^4*c*d^6*abs(b) - 126*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^3*d^7*abs(b) + 441*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*b
^4*c^2*d^4*abs(b) - 882*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^3*c*d^
5*abs(b) + 441*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^2*d^6*abs(b)
+ 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^2*c*d^4*abs(b) - 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*d^5*abs(b) + 210*sqrt(b*d)*(sqrt(
b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*d^4*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*b)