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

Optimal. Leaf size=66 \[ -\frac {2 (c+d x)^{5/2}}{7 (b c-a d) (a+b x)^{7/2}}+\frac {4 d (c+d x)^{5/2}}{35 (b c-a d)^2 (a+b x)^{5/2}} \]

[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]
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 = {47, 37} \begin {gather*} \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)} \end {gather*}

Antiderivative was successfully verified.

[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*} \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 {(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]
time = 0.09, size = 46, normalized size = 0.70 \begin {gather*} \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}} \end {gather*}

Antiderivative was successfully verified.

[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))

________________________________________________________________________________________

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)^(3/2)/(a + b*x)^(9/2),x]')

[Out]

Timed out

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(200\) vs. \(2(54)=108\).
time = 0.16, size = 201, normalized size = 3.05

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	\(-\frac {\left (d x +c \right )^{\frac {3}{2}}}{2 b \left (b x +a \right )^{\frac {7}{2}}}+\frac {3 \left (a d -b c \right ) \left (-\frac {\sqrt {d x +c}}{3 b \left (b x +a \right )^{\frac {7}{2}}}+\frac {\left (a d -b c \right ) \left (-\frac {2 \sqrt {d x +c}}{7 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {7}{2}}}-\frac {6 d \left (-\frac {2 \sqrt {d x +c}}{5 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {5}{2}}}-\frac {4 d \left (-\frac {2 \sqrt {d x +c}}{3 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {3}{2}}}+\frac {4 d \sqrt {d x +c}}{3 \left (-a d +b c \right )^{2} \sqrt {b x +a}}\right )}{5 \left (-a d +b c \right )}\right )}{7 \left (-a d +b c \right )}\right )}{6 b}\right )}{4 b}\)	\(201\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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)))))

________________________________________________________________________________________

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)^(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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 235 vs. \(2 (54) = 108\).
time = 0.82, size = 235, normalized size = 3.56 \begin {gather*} \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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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(-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)**(3/2)/(b*x+a)**(9/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 191 vs. \(2 (54) = 108\).
time = 0.12, size = 276, normalized size = 4.18 \begin {gather*} \frac {2 \left (-\frac {\left (210 b^{5} d^{8} c-210 b^{4} d^{9} a\right ) \sqrt {c+d x} \sqrt {c+d x}}{-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 |}-\frac {-735 b^{5} d^{8} c^{2}+1470 b^{4} d^{9} a c-735 b^{3} d^{10} a^{2}}{-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 ) \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 (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)^(3/2)/(b*x+a)^(9/2),x)

[Out]

2/35*(d*x + c)^(5/2)*(2*(b^5*c*d^8 - a*b^4*d^9)*(d*x + c)/(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)) - 7*(b^5*c^2*d^8 - 2*a*b^4*c*d^9 + a^2*b^3*d^10)/(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)

________________________________________________________________________________________

Mupad [B]
time = 0.93, size = 178, normalized size = 2.70 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________