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

Optimal result

Integrand size = 19, antiderivative size = 32 \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^{9/2}} \, dx=-\frac {2 (c+d x)^{7/2}}{7 (b c-a d) (a+b x)^{7/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {37} \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^{9/2}} \, dx=-\frac {2 (c+d x)^{7/2}}{7 (a+b x)^{7/2} (b c-a d)} \]

[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*} \text {integral}& = -\frac {2 (c+d x)^{7/2}}{7 (b c-a d) (a+b x)^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^{9/2}} \, dx=-\frac {2 (c+d x)^{7/2}}{7 (b c-a d) (a+b x)^{7/2}} \]

[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))

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84

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

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 138 vs. \(2 (26) = 52\).

Time = 0.69 (sec) , antiderivative size = 138, normalized size of antiderivative = 4.31 \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^{9/2}} \, dx=-\frac {2 \, {\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )} \sqrt {b x + a} \sqrt {d x + c}}{7 \, {\left (a^{4} b c - a^{5} d + {\left (b^{5} c - a b^{4} d\right )} x^{4} + 4 \, {\left (a b^{4} c - a^{2} b^{3} d\right )} x^{3} + 6 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} x^{2} + 4 \, {\left (a^{3} b^{2} c - a^{4} b d\right )} x\right )}} \]

[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)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^{9/2}} \, dx=\text {Exception raised: ValueError} \]

[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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 706 vs. \(2 (26) = 52\).

Time = 0.55 (sec) , antiderivative size = 706, normalized size of antiderivative = 22.06 \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^{9/2}} \, dx=-\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}} \]

[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)

Mupad [B] (verification not implemented)

Time = 1.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^{9/2}} \, dx=\frac {2\,{\left (c+d\,x\right )}^{7/2}}{\left (7\,a\,d-7\,b\,c\right )\,{\left (a+b\,x\right )}^{7/2}} \]

[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))