\(\int \frac {(c+d x)^{3/2}}{(a+b x)^{9/2}} \, dx\) [1478]

Optimal result

Integrand size = 19, antiderivative size = 66 \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^{9/2}} \, dx=-\frac {2 (c+d x)^{5/2}}{7 (b c-a d) (a+b x)^{7/2}}+\frac {4 d (c+d x)^{5/2}}{35 (b c-a d)^2 (a+b x)^{5/2}} \]

[Out]

-2/7*(d*x+c)^(5/2)/(-a*d+b*c)/(b*x+a)^(7/2)+4/35*d*(d*x+c)^(5/2)/(-a*d+b*c)^2/(b*x+a)^(5/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {47, 37} \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^{9/2}} \, dx=\frac {4 d (c+d x)^{5/2}}{35 (a+b x)^{5/2} (b c-a d)^2}-\frac {2 (c+d x)^{5/2}}{7 (a+b x)^{7/2} (b c-a d)} \]

[In]

Int[(c + d*x)^(3/2)/(a + b*x)^(9/2),x]

[Out]

(-2*(c + d*x)^(5/2))/(7*(b*c - a*d)*(a + b*x)^(7/2)) + (4*d*(c + d*x)^(5/2))/(35*(b*c - a*d)^2*(a + b*x)^(5/2)
)

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 47

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*} \text {integral}& = -\frac {2 (c+d x)^{5/2}}{7 (b c-a d) (a+b x)^{7/2}}-\frac {(2 d) \int \frac {(c+d x)^{3/2}}{(a+b x)^{7/2}} \, dx}{7 (b c-a d)} \\ & = -\frac {2 (c+d x)^{5/2}}{7 (b c-a d) (a+b x)^{7/2}}+\frac {4 d (c+d x)^{5/2}}{35 (b c-a d)^2 (a+b x)^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.70 \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^{9/2}} \, dx=-\frac {2 (c+d x)^{5/2} (5 b c-7 a d-2 b d x)}{35 (b c-a d)^2 (a+b x)^{7/2}} \]

[In]

Integrate[(c + d*x)^(3/2)/(a + b*x)^(9/2),x]

[Out]

(-2*(c + d*x)^(5/2)*(5*b*c - 7*a*d - 2*b*d*x))/(35*(b*c - a*d)^2*(a + b*x)^(7/2))

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.82

method	result	size
gosper	\(\frac {2 \left (d x +c \right )^{\frac {5}{2}} \left (2 b d x +7 a d -5 b c \right )}{35 \left (b x +a \right )^{\frac {7}{2}} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\)	\(54\)
default	\(-\frac {\left (d x +c \right )^{\frac {3}{2}}}{2 b \left (b x +a \right )^{\frac {7}{2}}}+\frac {3 \left (a d -b c \right ) \left (-\frac {\sqrt {d x +c}}{3 b \left (b x +a \right )^{\frac {7}{2}}}+\frac {\left (a d -b c \right ) \left (-\frac {2 \sqrt {d x +c}}{7 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {7}{2}}}-\frac {6 d \left (-\frac {2 \sqrt {d x +c}}{5 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {5}{2}}}-\frac {4 d \left (-\frac {2 \sqrt {d x +c}}{3 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {3}{2}}}+\frac {4 d \sqrt {d x +c}}{3 \left (-a d +b c \right )^{2} \sqrt {b x +a}}\right )}{5 \left (-a d +b c \right )}\right )}{7 \left (-a d +b c \right )}\right )}{6 b}\right )}{4 b}\)	\(201\)

[In]

int((d*x+c)^(3/2)/(b*x+a)^(9/2),x,method=_RETURNVERBOSE)

[Out]

2/35*(d*x+c)^(5/2)*(2*b*d*x+7*a*d-5*b*c)/(b*x+a)^(7/2)/(a^2*d^2-2*a*b*c*d+b^2*c^2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 235 vs. \(2 (54) = 108\).

Time = 0.72 (sec) , antiderivative size = 235, normalized size of antiderivative = 3.56 \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^{9/2}} \, dx=\frac {2 \, {\left (2 \, b d^{3} x^{3} - 5 \, b c^{3} + 7 \, a c^{2} d - {\left (b c d^{2} - 7 \, a d^{3}\right )} x^{2} - 2 \, {\left (4 \, b c^{2} d - 7 \, a c d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{35 \, {\left (a^{4} b^{2} c^{2} - 2 \, a^{5} b c d + a^{6} d^{2} + {\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} x^{4} + 4 \, {\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} x^{3} + 6 \, {\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2}\right )} x^{2} + 4 \, {\left (a^{3} b^{3} c^{2} - 2 \, a^{4} b^{2} c d + a^{5} b d^{2}\right )} x\right )}} \]

[In]

integrate((d*x+c)^(3/2)/(b*x+a)^(9/2),x, algorithm="fricas")

[Out]

2/35*(2*b*d^3*x^3 - 5*b*c^3 + 7*a*c^2*d - (b*c*d^2 - 7*a*d^3)*x^2 - 2*(4*b*c^2*d - 7*a*c*d^2)*x)*sqrt(b*x + a)
*sqrt(d*x + c)/(a^4*b^2*c^2 - 2*a^5*b*c*d + a^6*d^2 + (b^6*c^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*x^4 + 4*(a*b^5*c^2
- 2*a^2*b^4*c*d + a^3*b^3*d^2)*x^3 + 6*(a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2)*x^2 + 4*(a^3*b^3*c^2 - 2*a
^4*b^2*c*d + a^5*b*d^2)*x)

Sympy [F]

\[ \int \frac {(c+d x)^{3/2}}{(a+b x)^{9/2}} \, dx=\int \frac {\left (c + d x\right )^{\frac {3}{2}}}{\left (a + b x\right )^{\frac {9}{2}}}\, dx \]

[In]

integrate((d*x+c)**(3/2)/(b*x+a)**(9/2),x)

[Out]

Integral((c + d*x)**(3/2)/(a + b*x)**(9/2), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^{9/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((d*x+c)^(3/2)/(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. 1024 vs. \(2 (54) = 108\).

Time = 0.49 (sec) , antiderivative size = 1024, normalized size of antiderivative = 15.52 \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^{9/2}} \, dx=\frac {8 \, {\left (\sqrt {b d} b^{10} c^{5} d^{3} {\left | b \right |} - 5 \, \sqrt {b d} a b^{9} c^{4} d^{4} {\left | b \right |} + 10 \, \sqrt {b d} a^{2} b^{8} c^{3} d^{5} {\left | b \right |} - 10 \, \sqrt {b d} a^{3} b^{7} c^{2} d^{6} {\left | b \right |} + 5 \, \sqrt {b d} a^{4} b^{6} c d^{7} {\left | b \right |} - \sqrt {b d} a^{5} b^{5} d^{8} {\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 )}^{2} b^{8} c^{4} d^{3} {\left | b \right |} + 28 \, \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 )}^{2} a b^{7} c^{3} d^{4} {\left | b \right |} - 42 \, \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 )}^{2} a^{2} b^{6} c^{2} d^{5} {\left | b \right |} + 28 \, \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 )}^{2} a^{3} b^{5} c d^{6} {\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 )}^{2} a^{4} b^{4} d^{7} {\left | b \right |} - 14 \, \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^{6} c^{3} d^{3} {\left | b \right |} + 42 \, \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^{5} c^{2} d^{4} {\left | b \right |} - 42 \, \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^{4} c d^{5} {\left | b \right |} + 14 \, \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^{3} d^{6} {\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 )}^{6} b^{4} c^{2} d^{3} {\left | b \right |} + 140 \, \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 )}^{6} a b^{3} c d^{4} {\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 )}^{6} a^{2} b^{2} d^{5} {\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^{2} c d^{3} {\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 b 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 )}^{10} d^{3} {\left | b \right |}\right )}}{35 \, {\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^{2}} \]

[In]

integrate((d*x+c)^(3/2)/(b*x+a)^(9/2),x, algorithm="giac")

[Out]

8/35*(sqrt(b*d)*b^10*c^5*d^3*abs(b) - 5*sqrt(b*d)*a*b^9*c^4*d^4*abs(b) + 10*sqrt(b*d)*a^2*b^8*c^3*d^5*abs(b) -
10*sqrt(b*d)*a^3*b^7*c^2*d^6*abs(b) + 5*sqrt(b*d)*a^4*b^6*c*d^7*abs(b) - sqrt(b*d)*a^5*b^5*d^8*abs(b) - 7*sqr
t(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^8*c^4*d^3*abs(b) + 28*sqrt(b*d)*(sq
rt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^7*c^3*d^4*abs(b) - 42*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^6*c^2*d^5*abs(b) + 28*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^5*c*d^6*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))^2*a^4*b^4*d^7*abs(b) - 14*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*
c + (b*x + a)*b*d - a*b*d))^4*b^6*c^3*d^3*abs(b) + 42*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^5*c^2*d^4*abs(b) - 42*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^4*c*d^5*abs(b) + 14*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^3*d^6*abs(b) - 70*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*b^4*c^
2*d^3*abs(b) + 140*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^3*c*d^4*abs
(b) - 70*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^2*d^5*abs(b) - 35*s
qrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*b^2*c*d^3*abs(b) + 35*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*d^4*abs(b) - 35*sqrt(b*d)*(sqrt(b*d)*sqrt(b
*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*d^3*abs(b))/((b^2*c - a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqr
t(b^2*c + (b*x + a)*b*d - a*b*d))^2)^7*b^2)

Mupad [B] (verification not implemented)

Time = 0.86 (sec) , antiderivative size = 178, normalized size of antiderivative = 2.70 \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^{9/2}} \, dx=\frac {\sqrt {c+d\,x}\,\left (\frac {4\,d^3\,x^3}{35\,b^2\,{\left (a\,d-b\,c\right )}^2}-\frac {10\,b\,c^3-14\,a\,c^2\,d}{35\,b^3\,{\left (a\,d-b\,c\right )}^2}+\frac {x^2\,\left (14\,a\,d^3-2\,b\,c\,d^2\right )}{35\,b^3\,{\left (a\,d-b\,c\right )}^2}+\frac {4\,c\,d\,x\,\left (7\,a\,d-4\,b\,c\right )}{35\,b^3\,{\left (a\,d-b\,c\right )}^2}\right )}{x^3\,\sqrt {a+b\,x}+\frac {a^3\,\sqrt {a+b\,x}}{b^3}+\frac {3\,a\,x^2\,\sqrt {a+b\,x}}{b}+\frac {3\,a^2\,x\,\sqrt {a+b\,x}}{b^2}} \]

[In]

int((c + d*x)^(3/2)/(a + b*x)^(9/2),x)

[Out]

((c + d*x)^(1/2)*((4*d^3*x^3)/(35*b^2*(a*d - b*c)^2) - (10*b*c^3 - 14*a*c^2*d)/(35*b^3*(a*d - b*c)^2) + (x^2*(
14*a*d^3 - 2*b*c*d^2))/(35*b^3*(a*d - b*c)^2) + (4*c*d*x*(7*a*d - 4*b*c))/(35*b^3*(a*d - b*c)^2)))/(x^3*(a + b
*x)^(1/2) + (a^3*(a + b*x)^(1/2))/b^3 + (3*a*x^2*(a + b*x)^(1/2))/b + (3*a^2*x*(a + b*x)^(1/2))/b^2)