3.14.83 \(\int \frac {(c+d x)^{5/2}}{(a+b x)^{11/2}} \, dx\)
Optimal. Leaf size=66 \[ \frac {4 d (c+d x)^{7/2}}{63 (a+b x)^{7/2} (b c-a d)^2}-\frac {2 (c+d x)^{7/2}}{9 (a+b x)^{9/2} (b c-a d)} \]
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Rubi [A] time = 0.01, antiderivative size = 66, normalized size of antiderivative = 1.00,
number of steps used = 2, 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 {4 d (c+d x)^{7/2}}{63 (a+b x)^{7/2} (b c-a d)^2}-\frac {2 (c+d x)^{7/2}}{9 (a+b x)^{9/2} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(c + d*x)^(5/2)/(a + b*x)^(11/2),x]
[Out]
(-2*(c + d*x)^(7/2))/(9*(b*c - a*d)*(a + b*x)^(9/2)) + (4*d*(c + d*x)^(7/2))/(63*(b*c - a*d)^2*(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)^{11/2}} \, dx &=-\frac {2 (c+d x)^{7/2}}{9 (b c-a d) (a+b x)^{9/2}}-\frac {(2 d) \int \frac {(c+d x)^{5/2}}{(a+b x)^{9/2}} \, dx}{9 (b c-a d)}\\ &=-\frac {2 (c+d x)^{7/2}}{9 (b c-a d) (a+b x)^{9/2}}+\frac {4 d (c+d x)^{7/2}}{63 (b c-a d)^2 (a+b x)^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 46, normalized size = 0.70 \begin {gather*} \frac {2 (c+d x)^{7/2} (9 a d-7 b c+2 b d x)}{63 (a+b x)^{9/2} (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(c + d*x)^(5/2)/(a + b*x)^(11/2),x]
[Out]
(2*(c + d*x)^(7/2)*(-7*b*c + 9*a*d + 2*b*d*x))/(63*(b*c - a*d)^2*(a + b*x)^(9/2))
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IntegrateAlgebraic [A] time = 0.12, size = 57, normalized size = 0.86 \begin {gather*} -\frac {2 \left (\frac {7 b (c+d x)^{9/2}}{(a+b x)^{9/2}}-\frac {9 d (c+d x)^{7/2}}{(a+b x)^{7/2}}\right )}{63 (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(c + d*x)^(5/2)/(a + b*x)^(11/2),x]
[Out]
(-2*((-9*d*(c + d*x)^(7/2))/(a + b*x)^(7/2) + (7*b*(c + d*x)^(9/2))/(a + b*x)^(9/2)))/(63*(b*c - a*d)^2)
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fricas [B] time = 13.62, size = 295, normalized size = 4.47 \begin {gather*} \frac {2 \, {\left (2 \, b d^{4} x^{4} - 7 \, b c^{4} + 9 \, a c^{3} d - {\left (b c d^{3} - 9 \, a d^{4}\right )} x^{3} - 3 \, {\left (5 \, b c^{2} d^{2} - 9 \, a c d^{3}\right )} x^{2} - {\left (19 \, b c^{3} d - 27 \, a c^{2} d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{63 \, {\left (a^{5} b^{2} c^{2} - 2 \, a^{6} b c d + a^{7} d^{2} + {\left (b^{7} c^{2} - 2 \, a b^{6} c d + a^{2} b^{5} d^{2}\right )} x^{5} + 5 \, {\left (a b^{6} c^{2} - 2 \, a^{2} b^{5} c d + a^{3} b^{4} d^{2}\right )} x^{4} + 10 \, {\left (a^{2} b^{5} c^{2} - 2 \, a^{3} b^{4} c d + a^{4} b^{3} d^{2}\right )} x^{3} + 10 \, {\left (a^{3} b^{4} c^{2} - 2 \, a^{4} b^{3} c d + a^{5} b^{2} d^{2}\right )} x^{2} + 5 \, {\left (a^{4} b^{3} c^{2} - 2 \, a^{5} b^{2} c d + a^{6} b d^{2}\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)^(11/2),x, algorithm="fricas")
[Out]
2/63*(2*b*d^4*x^4 - 7*b*c^4 + 9*a*c^3*d - (b*c*d^3 - 9*a*d^4)*x^3 - 3*(5*b*c^2*d^2 - 9*a*c*d^3)*x^2 - (19*b*c^
3*d - 27*a*c^2*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c)/(a^5*b^2*c^2 - 2*a^6*b*c*d + a^7*d^2 + (b^7*c^2 - 2*a*b^6*c
*d + a^2*b^5*d^2)*x^5 + 5*(a*b^6*c^2 - 2*a^2*b^5*c*d + a^3*b^4*d^2)*x^4 + 10*(a^2*b^5*c^2 - 2*a^3*b^4*c*d + a^
4*b^3*d^2)*x^3 + 10*(a^3*b^4*c^2 - 2*a^4*b^3*c*d + a^5*b^2*d^2)*x^2 + 5*(a^4*b^3*c^2 - 2*a^5*b^2*c*d + a^6*b*d
^2)*x)
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giac [B] time = 3.80, size = 1826, normalized size = 27.67
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)^(11/2),x, algorithm="giac")
[Out]
8/63*(sqrt(b*d)*b^14*c^7*d^4*abs(b) - 7*sqrt(b*d)*a*b^13*c^6*d^5*abs(b) + 21*sqrt(b*d)*a^2*b^12*c^5*d^6*abs(b)
- 35*sqrt(b*d)*a^3*b^11*c^4*d^7*abs(b) + 35*sqrt(b*d)*a^4*b^10*c^3*d^8*abs(b) - 21*sqrt(b*d)*a^5*b^9*c^2*d^9*
abs(b) + 7*sqrt(b*d)*a^6*b^8*c*d^10*abs(b) - sqrt(b*d)*a^7*b^7*d^11*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^12*c^6*d^4*abs(b) + 54*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqr
t(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^11*c^5*d^5*abs(b) - 135*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^10*c^4*d^6*abs(b) + 180*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^9*c^3*d^7*abs(b) - 135*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^8*c^2*d^8*abs(b) + 54*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^7*c*d^9*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^6*b^6*d^10*abs(b) - 27*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^1
0*c^5*d^4*abs(b) + 135*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^9*c^4*d
^5*abs(b) - 270*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^8*c^3*d^6*ab
s(b) + 270*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^7*c^2*d^7*abs(b)
- 135*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^6*c*d^8*abs(b) + 27*sq
rt(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^5*d^9*abs(b) - 189*sqrt(b*d)*(
sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*b^8*c^4*d^4*abs(b) + 756*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^7*c^3*d^5*abs(b) - 1134*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^6*c^2*d^6*abs(b) + 756*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^5*c*d^7*abs(b) - 189*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - s
qrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^4*b^4*d^8*abs(b) - 189*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c
+ (b*x + a)*b*d - a*b*d))^8*b^6*c^3*d^4*abs(b) + 567*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^5*c^2*d^5*abs(b) - 567*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^4*c*d^6*abs(b) + 189*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^3*d^7*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^
4*c^2*d^4*abs(b) + 630*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^3*c*d^
5*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^2*b^2*d^6*abs(b)
- 105*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*b^2*c*d^4*abs(b) + 105*sqr
t(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*a*b*d^5*abs(b) - 63*sqrt(b*d)*(sqrt(
b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*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^3)
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maple [A] time = 0.00, size = 54, normalized size = 0.82 \begin {gather*} \frac {2 \left (d x +c \right )^{\frac {7}{2}} \left (2 b d x +9 a d -7 b c \right )}{63 \left (b x +a \right )^{\frac {9}{2}} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d*x+c)^(5/2)/(b*x+a)^(11/2),x)
[Out]
2/63*(d*x+c)^(7/2)*(2*b*d*x+9*a*d-7*b*c)/(b*x+a)^(9/2)/(a^2*d^2-2*a*b*c*d+b^2*c^2)
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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)^(11/2),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 = 1.14, size = 229, normalized size = 3.47 \begin {gather*} \frac {\sqrt {c+d\,x}\,\left (\frac {4\,d^4\,x^4}{63\,b^3\,{\left (a\,d-b\,c\right )}^2}-\frac {14\,b\,c^4-18\,a\,c^3\,d}{63\,b^4\,{\left (a\,d-b\,c\right )}^2}+\frac {x^3\,\left (18\,a\,d^4-2\,b\,c\,d^3\right )}{63\,b^4\,{\left (a\,d-b\,c\right )}^2}+\frac {2\,c^2\,d\,x\,\left (27\,a\,d-19\,b\,c\right )}{63\,b^4\,{\left (a\,d-b\,c\right )}^2}+\frac {2\,c\,d^2\,x^2\,\left (9\,a\,d-5\,b\,c\right )}{21\,b^4\,{\left (a\,d-b\,c\right )}^2}\right )}{x^4\,\sqrt {a+b\,x}+\frac {a^4\,\sqrt {a+b\,x}}{b^4}+\frac {6\,a^2\,x^2\,\sqrt {a+b\,x}}{b^2}+\frac {4\,a\,x^3\,\sqrt {a+b\,x}}{b}+\frac {4\,a^3\,x\,\sqrt {a+b\,x}}{b^3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c + d*x)^(5/2)/(a + b*x)^(11/2),x)
[Out]
((c + d*x)^(1/2)*((4*d^4*x^4)/(63*b^3*(a*d - b*c)^2) - (14*b*c^4 - 18*a*c^3*d)/(63*b^4*(a*d - b*c)^2) + (x^3*(
18*a*d^4 - 2*b*c*d^3))/(63*b^4*(a*d - b*c)^2) + (2*c^2*d*x*(27*a*d - 19*b*c))/(63*b^4*(a*d - b*c)^2) + (2*c*d^
2*x^2*(9*a*d - 5*b*c))/(21*b^4*(a*d - b*c)^2)))/(x^4*(a + b*x)^(1/2) + (a^4*(a + b*x)^(1/2))/b^4 + (6*a^2*x^2*
(a + b*x)^(1/2))/b^2 + (4*a*x^3*(a + b*x)^(1/2))/b + (4*a^3*x*(a + b*x)^(1/2))/b^3)
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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)**(11/2),x)
[Out]
Timed out
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