3.15.89 \(\int \frac {(c+d x)^{5/2}}{(a+b x)^{11/2}} \, dx\) [1489]

Optimal. Leaf size=66 \[ -\frac {2 (c+d x)^{7/2}}{9 (b c-a d) (a+b x)^{9/2}}+\frac {4 d (c+d x)^{7/2}}{63 (b c-a d)^2 (a+b x)^{7/2}} \]

[Out]

-2/9*(d*x+c)^(7/2)/(-a*d+b*c)/(b*x+a)^(9/2)+4/63*d*(d*x+c)^(7/2)/(-a*d+b*c)^2/(b*x+a)^(7/2)

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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)^{7/2}}{63 (a+b x)^{7/2} (b c-a d)^2}-\frac {2 (c+d x)^{7/2}}{9 (a+b x)^{9/2} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^(5/2)/(a + b*x)^(11/2),x]

[Out]

(-2*(c + d*x)^(7/2))/(9*(b*c - a*d)*(a + b*x)^(9/2)) + (4*d*(c + d*x)^(7/2))/(63*(b*c - a*d)^2*(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]

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)^{5/2}}{(a+b x)^{11/2}} \, dx &=-\frac {2 (c+d x)^{7/2}}{9 (b c-a d) (a+b x)^{9/2}}-\frac {(2 d) \int \frac {(c+d x)^{5/2}}{(a+b x)^{9/2}} \, dx}{9 (b c-a d)}\\ &=-\frac {2 (c+d x)^{7/2}}{9 (b c-a d) (a+b x)^{9/2}}+\frac {4 d (c+d x)^{7/2}}{63 (b c-a d)^2 (a+b x)^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 46, normalized size = 0.70 \begin {gather*} \frac {2 (c+d x)^{7/2} (-7 b c+9 a d+2 b d x)}{63 (b c-a d)^2 (a+b x)^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^(5/2)/(a + b*x)^(11/2),x]

[Out]

(2*(c + d*x)^(7/2)*(-7*b*c + 9*a*d + 2*b*d*x))/(63*(b*c - a*d)^2*(a + b*x)^(9/2))

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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)^(11/2),x]')

[Out]

Timed out

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(273\) vs. \(2(54)=108\).
time = 0.16, size = 274, normalized size = 4.15

method	result	size
gosper	\(\frac {2 \left (d x +c \right )^{\frac {7}{2}} \left (2 b d x +9 a d -7 b c \right )}{63 \left (b x +a \right )^{\frac {9}{2}} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\)	\(54\)
default	\(-\frac {\left (d x +c \right )^{\frac {5}{2}}}{2 b \left (b x +a \right )^{\frac {9}{2}}}+\frac {5 \left (a d -b c \right ) \left (-\frac {\left (d x +c \right )^{\frac {3}{2}}}{3 b \left (b x +a \right )^{\frac {9}{2}}}+\frac {\left (a d -b c \right ) \left (-\frac {\sqrt {d x +c}}{4 b \left (b x +a \right )^{\frac {9}{2}}}+\frac {\left (a d -b c \right ) \left (-\frac {2 \sqrt {d x +c}}{9 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {9}{2}}}-\frac {8 d \left (-\frac {2 \sqrt {d x +c}}{7 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {7}{2}}}-\frac {6 d \left (-\frac {2 \sqrt {d x +c}}{5 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {5}{2}}}-\frac {4 d \left (-\frac {2 \sqrt {d x +c}}{3 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {3}{2}}}+\frac {4 d \sqrt {d x +c}}{3 \left (-a d +b c \right )^{2} \sqrt {b x +a}}\right )}{5 \left (-a d +b c \right )}\right )}{7 \left (-a d +b c \right )}\right )}{9 \left (-a d +b c \right )}\right )}{8 b}\right )}{2 b}\right )}{4 b}\)	\(274\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(5/2)/(b*x+a)^(11/2),x,method=_RETURNVERBOSE)

[Out]

-1/2/b*(d*x+c)^(5/2)/(b*x+a)^(9/2)+5/4*(a*d-b*c)/b*(-1/3/b*(d*x+c)^(3/2)/(b*x+a)^(9/2)+1/2*(a*d-b*c)/b*(-1/4/b
*(d*x+c)^(1/2)/(b*x+a)^(9/2)+1/8*(a*d-b*c)/b*(-2/9*(d*x+c)^(1/2)/(-a*d+b*c)/(b*x+a)^(9/2)-8/9*d/(-a*d+b*c)*(-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)^(11/2),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. 295 vs. \(2 (54) = 108\).
time = 2.70, size = 295, normalized size = 4.47 \begin {gather*} \frac {2 \, {\left (2 \, b d^{4} x^{4} - 7 \, b c^{4} + 9 \, a c^{3} d - {\left (b c d^{3} - 9 \, a d^{4}\right )} x^{3} - 3 \, {\left (5 \, b c^{2} d^{2} - 9 \, a c d^{3}\right )} x^{2} - {\left (19 \, b c^{3} d - 27 \, a c^{2} d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{63 \, {\left (a^{5} b^{2} c^{2} - 2 \, a^{6} b c d + a^{7} d^{2} + {\left (b^{7} c^{2} - 2 \, a b^{6} c d + a^{2} b^{5} d^{2}\right )} x^{5} + 5 \, {\left (a b^{6} c^{2} - 2 \, a^{2} b^{5} c d + a^{3} b^{4} d^{2}\right )} x^{4} + 10 \, {\left (a^{2} b^{5} c^{2} - 2 \, a^{3} b^{4} c d + a^{4} b^{3} d^{2}\right )} x^{3} + 10 \, {\left (a^{3} b^{4} c^{2} - 2 \, a^{4} b^{3} c d + a^{5} b^{2} d^{2}\right )} x^{2} + 5 \, {\left (a^{4} b^{3} c^{2} - 2 \, a^{5} b^{2} c d + a^{6} b d^{2}\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)^(11/2),x, algorithm="fricas")

[Out]

2/63*(2*b*d^4*x^4 - 7*b*c^4 + 9*a*c^3*d - (b*c*d^3 - 9*a*d^4)*x^3 - 3*(5*b*c^2*d^2 - 9*a*c*d^3)*x^2 - (19*b*c^
3*d - 27*a*c^2*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c)/(a^5*b^2*c^2 - 2*a^6*b*c*d + a^7*d^2 + (b^7*c^2 - 2*a*b^6*c
*d + a^2*b^5*d^2)*x^5 + 5*(a*b^6*c^2 - 2*a^2*b^5*c*d + a^3*b^4*d^2)*x^4 + 10*(a^2*b^5*c^2 - 2*a^3*b^4*c*d + a^
4*b^3*d^2)*x^3 + 10*(a^3*b^4*c^2 - 2*a^4*b^3*c*d + a^5*b^2*d^2)*x^2 + 5*(a^4*b^3*c^2 - 2*a^5*b^2*c*d + a^6*b*d
^2)*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)**(11/2),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 249 vs. \(2 (54) = 108\).
time = 0.24, size = 355, normalized size = 5.38 \begin {gather*} \frac {2 \left (-\frac {\left (-28350 b^{7} d^{10} c^{2}+56700 b^{6} d^{11} a c-28350 b^{5} d^{12} a^{2}\right ) \sqrt {c+d x} \sqrt {c+d x}}{893025 b^{8} c^{4} \left |d\right |-3572100 b^{7} d a c^{3} \left |d\right |+5358150 b^{6} d^{2} a^{2} c^{2} \left |d\right |-3572100 b^{5} d^{3} a^{3} c \left |d\right |+893025 b^{4} d^{4} a^{4} \left |d\right |}-\frac {127575 b^{7} d^{10} c^{3}-382725 b^{6} d^{11} a c^{2}+382725 b^{5} d^{12} a^{2} c-127575 b^{4} d^{13} a^{3}}{893025 b^{8} c^{4} \left |d\right |-3572100 b^{7} d a c^{3} \left |d\right |+5358150 b^{6} d^{2} a^{2} c^{2} \left |d\right |-3572100 b^{5} d^{3} a^{3} c \left |d\right |+893025 b^{4} d^{4} a^{4} \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 {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 )^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(5/2)/(b*x+a)^(11/2),x)

[Out]

2/63*(d*x + c)^(7/2)*(2*(b^7*c^2*d^10 - 2*a*b^6*c*d^11 + a^2*b^5*d^12)*(d*x + c)/(b^8*c^4*abs(d) - 4*a*b^7*c^3
*d*abs(d) + 6*a^2*b^6*c^2*d^2*abs(d) - 4*a^3*b^5*c*d^3*abs(d) + a^4*b^4*d^4*abs(d)) - 9*(b^7*c^3*d^10 - 3*a*b^
6*c^2*d^11 + 3*a^2*b^5*c*d^12 - a^3*b^4*d^13)/(b^8*c^4*abs(d) - 4*a*b^7*c^3*d*abs(d) + 6*a^2*b^6*c^2*d^2*abs(d
) - 4*a^3*b^5*c*d^3*abs(d) + a^4*b^4*d^4*abs(d)))/((d*x + c)*b*d - b*c*d + a*d^2)^(9/2)

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Mupad [B]
time = 1.14, size = 229, normalized size = 3.47 \begin {gather*} \frac {\sqrt {c+d\,x}\,\left (\frac {4\,d^4\,x^4}{63\,b^3\,{\left (a\,d-b\,c\right )}^2}-\frac {14\,b\,c^4-18\,a\,c^3\,d}{63\,b^4\,{\left (a\,d-b\,c\right )}^2}+\frac {x^3\,\left (18\,a\,d^4-2\,b\,c\,d^3\right )}{63\,b^4\,{\left (a\,d-b\,c\right )}^2}+\frac {2\,c^2\,d\,x\,\left (27\,a\,d-19\,b\,c\right )}{63\,b^4\,{\left (a\,d-b\,c\right )}^2}+\frac {2\,c\,d^2\,x^2\,\left (9\,a\,d-5\,b\,c\right )}{21\,b^4\,{\left (a\,d-b\,c\right )}^2}\right )}{x^4\,\sqrt {a+b\,x}+\frac {a^4\,\sqrt {a+b\,x}}{b^4}+\frac {6\,a^2\,x^2\,\sqrt {a+b\,x}}{b^2}+\frac {4\,a\,x^3\,\sqrt {a+b\,x}}{b}+\frac {4\,a^3\,x\,\sqrt {a+b\,x}}{b^3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^(5/2)/(a + b*x)^(11/2),x)

[Out]

((c + d*x)^(1/2)*((4*d^4*x^4)/(63*b^3*(a*d - b*c)^2) - (14*b*c^4 - 18*a*c^3*d)/(63*b^4*(a*d - b*c)^2) + (x^3*(
18*a*d^4 - 2*b*c*d^3))/(63*b^4*(a*d - b*c)^2) + (2*c^2*d*x*(27*a*d - 19*b*c))/(63*b^4*(a*d - b*c)^2) + (2*c*d^
2*x^2*(9*a*d - 5*b*c))/(21*b^4*(a*d - b*c)^2)))/(x^4*(a + b*x)^(1/2) + (a^4*(a + b*x)^(1/2))/b^4 + (6*a^2*x^2*
(a + b*x)^(1/2))/b^2 + (4*a*x^3*(a + b*x)^(1/2))/b + (4*a^3*x*(a + b*x)^(1/2))/b^3)

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