3.14.82 \(\int \frac {(c+d x)^{5/2}}{(a+b x)^{9/2}} \, dx\)

Optimal. Leaf size=32 \[ -\frac {2 (c+d x)^{7/2}}{7 (a+b x)^{7/2} (b c-a d)} \]

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Rubi [A]  time = 0.00, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {37} \begin {gather*} -\frac {2 (c+d x)^{7/2}}{7 (a+b x)^{7/2} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^(5/2)/(a + b*x)^(9/2),x]

[Out]

(-2*(c + d*x)^(7/2))/(7*(b*c - a*d)*(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]

Rubi steps

\begin {align*} \int \frac {(c+d x)^{5/2}}{(a+b x)^{9/2}} \, dx &=-\frac {2 (c+d x)^{7/2}}{7 (b c-a d) (a+b x)^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 32, normalized size = 1.00 \begin {gather*} -\frac {2 (c+d x)^{7/2}}{7 (a+b x)^{7/2} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^(5/2)/(a + b*x)^(9/2),x]

[Out]

(-2*(c + d*x)^(7/2))/(7*(b*c - a*d)*(a + b*x)^(7/2))

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IntegrateAlgebraic [A]  time = 0.06, size = 32, normalized size = 1.00 \begin {gather*} -\frac {2 (c+d x)^{7/2}}{7 (a+b x)^{7/2} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(c + d*x)^(5/2)/(a + b*x)^(9/2),x]

[Out]

(-2*(c + d*x)^(7/2))/(7*(b*c - a*d)*(a + b*x)^(7/2))

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fricas [B]  time = 3.85, size = 138, normalized size = 4.31 \begin {gather*} -\frac {2 \, {\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )} \sqrt {b x + a} \sqrt {d x + c}}{7 \, {\left (a^{4} b c - a^{5} d + {\left (b^{5} c - a b^{4} d\right )} x^{4} + 4 \, {\left (a b^{4} c - a^{2} b^{3} d\right )} x^{3} + 6 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} x^{2} + 4 \, {\left (a^{3} b^{2} c - a^{4} b d\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)^(9/2),x, algorithm="fricas")

[Out]

-2/7*(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)*sqrt(b*x + a)*sqrt(d*x + c)/(a^4*b*c - a^5*d + (b^5*c - a*b^4*d
)*x^4 + 4*(a*b^4*c - a^2*b^3*d)*x^3 + 6*(a^2*b^3*c - a^3*b^2*d)*x^2 + 4*(a^3*b^2*c - a^4*b*d)*x)

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giac [B]  time = 2.54, size = 706, normalized size = 22.06 \begin {gather*} -\frac {4 \, {\left (\sqrt {b d} b^{12} c^{6} d^{3} {\left | b \right |} - 6 \, \sqrt {b d} a b^{11} c^{5} d^{4} {\left | b \right |} + 15 \, \sqrt {b d} a^{2} b^{10} c^{4} d^{5} {\left | b \right |} - 20 \, \sqrt {b d} a^{3} b^{9} c^{3} d^{6} {\left | b \right |} + 15 \, \sqrt {b d} a^{4} b^{8} c^{2} d^{7} {\left | b \right |} - 6 \, \sqrt {b d} a^{5} b^{7} c d^{8} {\left | b \right |} + \sqrt {b d} a^{6} b^{6} d^{9} {\left | b \right |} + 21 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} b^{8} c^{4} d^{3} {\left | b \right |} - 84 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a b^{7} c^{3} d^{4} {\left | b \right |} + 126 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{2} b^{6} c^{2} d^{5} {\left | b \right |} - 84 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{3} b^{5} c d^{6} {\left | b \right |} + 21 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{4} b^{4} d^{7} {\left | b \right |} + 35 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{8} b^{4} c^{2} d^{3} {\left | b \right |} - 70 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{8} a b^{3} c d^{4} {\left | b \right |} + 35 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{8} a^{2} b^{2} d^{5} {\left | b \right |} + 7 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{12} d^{3} {\left | b \right |}\right )}}{7 \, {\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{7} b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(5/2)/(b*x+a)^(9/2),x, algorithm="giac")

[Out]

-4/7*(sqrt(b*d)*b^12*c^6*d^3*abs(b) - 6*sqrt(b*d)*a*b^11*c^5*d^4*abs(b) + 15*sqrt(b*d)*a^2*b^10*c^4*d^5*abs(b)
- 20*sqrt(b*d)*a^3*b^9*c^3*d^6*abs(b) + 15*sqrt(b*d)*a^4*b^8*c^2*d^7*abs(b) - 6*sqrt(b*d)*a^5*b^7*c*d^8*abs(b
) + sqrt(b*d)*a^6*b^6*d^9*abs(b) + 21*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^3*abs(b) - 84*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^4*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))^4*a^2*b^6*c^2*d
^5*abs(b) - 84*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^6*abs(b
) + 21*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^7*abs(b) + 35*sqr
t(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^3*abs(b) - 70*sqrt(b*d)*(sq
rt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a*b^3*c*d^4*abs(b) + 35*sqrt(b*d)*(sqrt(b*d)*sq
rt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a^2*b^2*d^5*abs(b) + 7*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a)
- sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*d^3*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)^7*b^4)

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maple [A]  time = 0.00, size = 27, normalized size = 0.84 \begin {gather*} \frac {2 \left (d x +c \right )^{\frac {7}{2}}}{7 \left (b x +a \right )^{\frac {7}{2}} \left (a d -b c \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(5/2)/(b*x+a)^(9/2),x)

[Out]

2/7/(b*x+a)^(7/2)*(d*x+c)^(7/2)/(a*d-b*c)

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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)^(9/2),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 = 0.97, size = 27, normalized size = 0.84 \begin {gather*} \frac {2\,{\left (c+d\,x\right )}^{7/2}}{\left (7\,a\,d-7\,b\,c\right )\,{\left (a+b\,x\right )}^{7/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^(5/2)/(a + b*x)^(9/2),x)

[Out]

(2*(c + d*x)^(7/2))/((7*a*d - 7*b*c)*(a + b*x)^(7/2))

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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)**(9/2),x)

[Out]

Timed out

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