3.15.88 \(\int \frac {(c+d x)^{5/2}}{(a+b x)^{9/2}} \, dx\) [1488]

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

[Out]

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

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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 = 0.05, size = 32, normalized size = 1.00 \begin {gather*} -\frac {2 (c+d x)^{7/2}}{7 (b c-a d) (a+b x)^{7/2}} \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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Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Timed out

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(233\) vs. \(2(26)=52\).
time = 0.16, size = 234, normalized size = 7.31

method	result	size
gosper	\(\frac {2 \left (d x +c \right )^{\frac {7}{2}}}{7 \left (b x +a \right )^{\frac {7}{2}} \left (a d -b c \right )}\)	\(27\)
default	\(-\frac {\left (d x +c \right )^{\frac {5}{2}}}{b \left (b x +a \right )^{\frac {7}{2}}}+\frac {5 \left (a d -b c \right ) \left (-\frac {\left (d x +c \right )^{\frac {3}{2}}}{2 b \left (b x +a \right )^{\frac {7}{2}}}+\frac {3 \left (a d -b c \right ) \left (-\frac {\sqrt {d x +c}}{3 b \left (b x +a \right )^{\frac {7}{2}}}+\frac {\left (a d -b c \right ) \left (-\frac {2 \sqrt {d x +c}}{7 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {7}{2}}}-\frac {6 d \left (-\frac {2 \sqrt {d x +c}}{5 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {5}{2}}}-\frac {4 d \left (-\frac {2 \sqrt {d x +c}}{3 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {3}{2}}}+\frac {4 d \sqrt {d x +c}}{3 \left (-a d +b c \right )^{2} \sqrt {b x +a}}\right )}{5 \left (-a d +b c \right )}\right )}{7 \left (-a d +b c \right )}\right )}{6 b}\right )}{4 b}\right )}{2 b}\)	\(234\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/b*(d*x+c)^(5/2)/(b*x+a)^(7/2)+5/2*(a*d-b*c)/b*(-1/2/b*(d*x+c)^(3/2)/(b*x+a)^(7/2)+3/4*(a*d-b*c)/b*(-1/3/b*(
d*x+c)^(1/2)/(b*x+a)^(7/2)+1/6*(a*d-b*c)/b*(-2/7*(d*x+c)^(1/2)/(-a*d+b*c)/(b*x+a)^(7/2)-6/7*d/(-a*d+b*c)*(-2/5
*(d*x+c)^(1/2)/(-a*d+b*c)/(b*x+a)^(5/2)-4/5*d/(-a*d+b*c)*(-2/3*(d*x+c)^(1/2)/(-a*d+b*c)/(b*x+a)^(3/2)+4/3*d*(d
*x+c)^(1/2)/(-a*d+b*c)^2/(b*x+a)^(1/2))))))

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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)^(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 detail

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 138 vs. \(2 (26) = 52\).
time = 0.80, 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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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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (26) = 52\).
time = 0.16, size = 199, normalized size = 6.22 \begin {gather*} -\frac {2 \left (-525 b^{5} d^{8} c^{2}+1050 b^{4} d^{9} a c-525 b^{3} d^{10} a^{2}\right ) \sqrt {c+d x} \sqrt {c+d x} \sqrt {c+d x} \sqrt {c+d x} \sqrt {c+d x} \sqrt {c+d x} \sqrt {c+d x} \sqrt {a d^{2}-b c d+b d \left (c+d x\right )}}{\left (-3675 b^{6} c^{3} \left |d\right |+11025 b^{5} d a c^{2} \left |d\right |-11025 b^{4} d^{2} a^{2} c \left |d\right |+3675 b^{3} d^{3} a^{3} \left |d\right |\right ) \left (a d^{2}-b c d+b d \left (c+d x\right )\right )^{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)

[Out]

-2/7*(b^5*c^2*d^8 - 2*a*b^4*c*d^9 + a^2*b^3*d^10)*(d*x + c)^(7/2)/((b^6*c^3*abs(d) - 3*a*b^5*c^2*d*abs(d) + 3*
a^2*b^4*c*d^2*abs(d) - a^3*b^3*d^3*abs(d))*((d*x + c)*b*d - b*c*d + a*d^2)^(7/2))

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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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