3.15.69 \(\int \frac {\sqrt {c+d x}}{(a+b x)^{11/2}} \, dx\) [1469]

Optimal. Leaf size=136 \[ -\frac {2 (c+d x)^{3/2}}{9 (b c-a d) (a+b x)^{9/2}}+\frac {4 d (c+d x)^{3/2}}{21 (b c-a d)^2 (a+b x)^{7/2}}-\frac {16 d^2 (c+d x)^{3/2}}{105 (b c-a d)^3 (a+b x)^{5/2}}+\frac {32 d^3 (c+d x)^{3/2}}{315 (b c-a d)^4 (a+b x)^{3/2}} \]

[Out]

-2/9*(d*x+c)^(3/2)/(-a*d+b*c)/(b*x+a)^(9/2)+4/21*d*(d*x+c)^(3/2)/(-a*d+b*c)^2/(b*x+a)^(7/2)-16/105*d^2*(d*x+c)
^(3/2)/(-a*d+b*c)^3/(b*x+a)^(5/2)+32/315*d^3*(d*x+c)^(3/2)/(-a*d+b*c)^4/(b*x+a)^(3/2)

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Rubi [A]
time = 0.02, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 4, 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 {32 d^3 (c+d x)^{3/2}}{315 (a+b x)^{3/2} (b c-a d)^4}-\frac {16 d^2 (c+d x)^{3/2}}{105 (a+b x)^{5/2} (b c-a d)^3}+\frac {4 d (c+d x)^{3/2}}{21 (a+b x)^{7/2} (b c-a d)^2}-\frac {2 (c+d x)^{3/2}}{9 (a+b x)^{9/2} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[c + d*x]/(a + b*x)^(11/2),x]

[Out]

(-2*(c + d*x)^(3/2))/(9*(b*c - a*d)*(a + b*x)^(9/2)) + (4*d*(c + d*x)^(3/2))/(21*(b*c - a*d)^2*(a + b*x)^(7/2)
) - (16*d^2*(c + d*x)^(3/2))/(105*(b*c - a*d)^3*(a + b*x)^(5/2)) + (32*d^3*(c + d*x)^(3/2))/(315*(b*c - a*d)^4
*(a + b*x)^(3/2))

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 {\sqrt {c+d x}}{(a+b x)^{11/2}} \, dx &=-\frac {2 (c+d x)^{3/2}}{9 (b c-a d) (a+b x)^{9/2}}-\frac {(2 d) \int \frac {\sqrt {c+d x}}{(a+b x)^{9/2}} \, dx}{3 (b c-a d)}\\ &=-\frac {2 (c+d x)^{3/2}}{9 (b c-a d) (a+b x)^{9/2}}+\frac {4 d (c+d x)^{3/2}}{21 (b c-a d)^2 (a+b x)^{7/2}}+\frac {\left (8 d^2\right ) \int \frac {\sqrt {c+d x}}{(a+b x)^{7/2}} \, dx}{21 (b c-a d)^2}\\ &=-\frac {2 (c+d x)^{3/2}}{9 (b c-a d) (a+b x)^{9/2}}+\frac {4 d (c+d x)^{3/2}}{21 (b c-a d)^2 (a+b x)^{7/2}}-\frac {16 d^2 (c+d x)^{3/2}}{105 (b c-a d)^3 (a+b x)^{5/2}}-\frac {\left (16 d^3\right ) \int \frac {\sqrt {c+d x}}{(a+b x)^{5/2}} \, dx}{105 (b c-a d)^3}\\ &=-\frac {2 (c+d x)^{3/2}}{9 (b c-a d) (a+b x)^{9/2}}+\frac {4 d (c+d x)^{3/2}}{21 (b c-a d)^2 (a+b x)^{7/2}}-\frac {16 d^2 (c+d x)^{3/2}}{105 (b c-a d)^3 (a+b x)^{5/2}}+\frac {32 d^3 (c+d x)^{3/2}}{315 (b c-a d)^4 (a+b x)^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 0.14, size = 93, normalized size = 0.68 \begin {gather*} -\frac {2 (c+d x)^{3/2} \left (-105 d^3 (a+b x)^3+189 b d^2 (a+b x)^2 (c+d x)-135 b^2 d (a+b x) (c+d x)^2+35 b^3 (c+d x)^3\right )}{315 (b c-a d)^4 (a+b x)^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(c + d*x)^(3/2)*(-105*d^3*(a + b*x)^3 + 189*b*d^2*(a + b*x)^2*(c + d*x) - 135*b^2*d*(a + b*x)*(c + d*x)^2
+ 35*b^3*(c + d*x)^3))/(315*(b*c - a*d)^4*(a + b*x)^(9/2))

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Maple [A]
time = 0.16, size = 208, normalized size = 1.53

method	result	size
gosper	\(\frac {2 \left (d x +c \right )^{\frac {3}{2}} \left (16 b^{3} x^{3} d^{3}+72 d^{3} a \,x^{2} b^{2}-24 b^{3} c \,d^{2} x^{2}+126 a^{2} b \,d^{3} x -108 a \,b^{2} c \,d^{2} x +30 b^{3} c^{2} d x +105 a^{3} d^{3}-189 a^{2} b c \,d^{2}+135 a \,b^{2} c^{2} d -35 b^{3} c^{3}\right )}{315 \left (b x +a \right )^{\frac {9}{2}} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\)	\(171\)
default	\(-\frac {\sqrt {d x +c}}{4 b \left (b x +a \right )^{\frac {9}{2}}}+\frac {\left (a d -b c \right ) \left (-\frac {2 \sqrt {d x +c}}{9 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {9}{2}}}-\frac {8 d \left (-\frac {2 \sqrt {d x +c}}{7 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {7}{2}}}-\frac {6 d \left (-\frac {2 \sqrt {d x +c}}{5 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {5}{2}}}-\frac {4 d \left (-\frac {2 \sqrt {d x +c}}{3 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {3}{2}}}+\frac {4 d \sqrt {d x +c}}{3 \left (-a d +b c \right )^{2} \sqrt {b x +a}}\right )}{5 \left (-a d +b c \right )}\right )}{7 \left (-a d +b c \right )}\right )}{9 \left (-a d +b c \right )}\right )}{8 b}\)	\(208\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(1/2)/(b*x+a)^(11/2),x,method=_RETURNVERBOSE)

[Out]

-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)^(1/2)/(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. 532 vs. \(2 (112) = 224\).
time = 4.56, size = 532, normalized size = 3.91 \begin {gather*} \frac {2 \, {\left (16 \, b^{3} d^{4} x^{4} - 35 \, b^{3} c^{4} + 135 \, a b^{2} c^{3} d - 189 \, a^{2} b c^{2} d^{2} + 105 \, a^{3} c d^{3} - 8 \, {\left (b^{3} c d^{3} - 9 \, a b^{2} d^{4}\right )} x^{3} + 6 \, {\left (b^{3} c^{2} d^{2} - 6 \, a b^{2} c d^{3} + 21 \, a^{2} b d^{4}\right )} x^{2} - {\left (5 \, b^{3} c^{3} d - 27 \, a b^{2} c^{2} d^{2} + 63 \, a^{2} b c d^{3} - 105 \, a^{3} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{315 \, {\left (a^{5} b^{4} c^{4} - 4 \, a^{6} b^{3} c^{3} d + 6 \, a^{7} b^{2} c^{2} d^{2} - 4 \, a^{8} b c d^{3} + a^{9} d^{4} + {\left (b^{9} c^{4} - 4 \, a b^{8} c^{3} d + 6 \, a^{2} b^{7} c^{2} d^{2} - 4 \, a^{3} b^{6} c d^{3} + a^{4} b^{5} d^{4}\right )} x^{5} + 5 \, {\left (a b^{8} c^{4} - 4 \, a^{2} b^{7} c^{3} d + 6 \, a^{3} b^{6} c^{2} d^{2} - 4 \, a^{4} b^{5} c d^{3} + a^{5} b^{4} d^{4}\right )} x^{4} + 10 \, {\left (a^{2} b^{7} c^{4} - 4 \, a^{3} b^{6} c^{3} d + 6 \, a^{4} b^{5} c^{2} d^{2} - 4 \, a^{5} b^{4} c d^{3} + a^{6} b^{3} d^{4}\right )} x^{3} + 10 \, {\left (a^{3} b^{6} c^{4} - 4 \, a^{4} b^{5} c^{3} d + 6 \, a^{5} b^{4} c^{2} d^{2} - 4 \, a^{6} b^{3} c d^{3} + a^{7} b^{2} d^{4}\right )} x^{2} + 5 \, {\left (a^{4} b^{5} c^{4} - 4 \, a^{5} b^{4} c^{3} d + 6 \, a^{6} b^{3} c^{2} d^{2} - 4 \, a^{7} b^{2} c d^{3} + a^{8} b d^{4}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2)/(b*x+a)^(11/2),x, algorithm="fricas")

[Out]

2/315*(16*b^3*d^4*x^4 - 35*b^3*c^4 + 135*a*b^2*c^3*d - 189*a^2*b*c^2*d^2 + 105*a^3*c*d^3 - 8*(b^3*c*d^3 - 9*a*
b^2*d^4)*x^3 + 6*(b^3*c^2*d^2 - 6*a*b^2*c*d^3 + 21*a^2*b*d^4)*x^2 - (5*b^3*c^3*d - 27*a*b^2*c^2*d^2 + 63*a^2*b
*c*d^3 - 105*a^3*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c)/(a^5*b^4*c^4 - 4*a^6*b^3*c^3*d + 6*a^7*b^2*c^2*d^2 - 4*a^
8*b*c*d^3 + a^9*d^4 + (b^9*c^4 - 4*a*b^8*c^3*d + 6*a^2*b^7*c^2*d^2 - 4*a^3*b^6*c*d^3 + a^4*b^5*d^4)*x^5 + 5*(a
*b^8*c^4 - 4*a^2*b^7*c^3*d + 6*a^3*b^6*c^2*d^2 - 4*a^4*b^5*c*d^3 + a^5*b^4*d^4)*x^4 + 10*(a^2*b^7*c^4 - 4*a^3*
b^6*c^3*d + 6*a^4*b^5*c^2*d^2 - 4*a^5*b^4*c*d^3 + a^6*b^3*d^4)*x^3 + 10*(a^3*b^6*c^4 - 4*a^4*b^5*c^3*d + 6*a^5
*b^4*c^2*d^2 - 4*a^6*b^3*c*d^3 + a^7*b^2*d^4)*x^2 + 5*(a^4*b^5*c^4 - 4*a^5*b^4*c^3*d + 6*a^6*b^3*c^2*d^2 - 4*a
^7*b^2*c*d^3 + a^8*b*d^4)*x)

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Sympy [F(-1)] Timed out
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)**(1/2)/(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. 989 vs. \(2 (112) = 224\).
time = 1.93, size = 989, normalized size = 7.27 \begin {gather*} \frac {64 \, {\left (\sqrt {b d} b^{13} c^{5} d^{4} - 5 \, \sqrt {b d} a b^{12} c^{4} d^{5} + 10 \, \sqrt {b d} a^{2} b^{11} c^{3} d^{6} - 10 \, \sqrt {b d} a^{3} b^{10} c^{2} d^{7} + 5 \, \sqrt {b d} a^{4} b^{9} c d^{8} - \sqrt {b d} a^{5} b^{8} d^{9} - 9 \, \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^{11} c^{4} d^{4} + 36 \, \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^{10} c^{3} d^{5} - 54 \, \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^{9} c^{2} d^{6} + 36 \, \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^{8} c d^{7} - 9 \, \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^{7} d^{8} + 36 \, \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^{9} c^{3} d^{4} - 108 \, \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^{8} c^{2} d^{5} + 108 \, \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^{7} c d^{6} - 36 \, \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^{6} d^{7} - 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 )}^{6} b^{7} c^{2} d^{4} + 168 \, \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^{6} c d^{5} - 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 )}^{6} a^{2} b^{5} d^{6} - 189 \, \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^{5} c d^{4} + 189 \, \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^{4} d^{5} - 315 \, \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} b^{3} d^{4}\right )} {\left | b \right |}}{315 \, {\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 )}^{9} b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2)/(b*x+a)^(11/2),x, algorithm="giac")

[Out]

64/315*(sqrt(b*d)*b^13*c^5*d^4 - 5*sqrt(b*d)*a*b^12*c^4*d^5 + 10*sqrt(b*d)*a^2*b^11*c^3*d^6 - 10*sqrt(b*d)*a^3
*b^10*c^2*d^7 + 5*sqrt(b*d)*a^4*b^9*c*d^8 - sqrt(b*d)*a^5*b^8*d^9 - 9*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqr
t(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^11*c^4*d^4 + 36*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x +
a)*b*d - a*b*d))^2*a*b^10*c^3*d^5 - 54*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^9*c^2*d^6 + 36*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^8
*c*d^7 - 9*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^7*d^8 + 36*sqrt(b
*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^9*c^3*d^4 - 108*sqrt(b*d)*(sqrt(b*d)*s
qrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a*b^8*c^2*d^5 + 108*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^7*c*d^6 - 36*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^6*d^7 - 84*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*
b*d))^6*b^7*c^2*d^4 + 168*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^6*c*
d^5 - 84*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^5*d^6 - 189*sqrt(b*
d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*b^5*c*d^4 + 189*sqrt(b*d)*(sqrt(b*d)*sqrt
(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a*b^4*d^5 - 315*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b
^2*c + (b*x + a)*b*d - a*b*d))^10*b^3*d^4)*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)^9*b^2)

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Mupad [B]
time = 1.18, size = 292, normalized size = 2.15 \begin {gather*} \frac {\sqrt {c+d\,x}\,\left (\frac {32\,d^4\,x^4}{315\,b\,{\left (a\,d-b\,c\right )}^4}-\frac {-210\,a^3\,c\,d^3+378\,a^2\,b\,c^2\,d^2-270\,a\,b^2\,c^3\,d+70\,b^3\,c^4}{315\,b^4\,{\left (a\,d-b\,c\right )}^4}+\frac {x\,\left (210\,a^3\,d^4-126\,a^2\,b\,c\,d^3+54\,a\,b^2\,c^2\,d^2-10\,b^3\,c^3\,d\right )}{315\,b^4\,{\left (a\,d-b\,c\right )}^4}+\frac {16\,d^3\,x^3\,\left (9\,a\,d-b\,c\right )}{315\,b^2\,{\left (a\,d-b\,c\right )}^4}+\frac {4\,d^2\,x^2\,\left (21\,a^2\,d^2-6\,a\,b\,c\,d+b^2\,c^2\right )}{105\,b^3\,{\left (a\,d-b\,c\right )}^4}\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)^(1/2)/(a + b*x)^(11/2),x)

[Out]

((c + d*x)^(1/2)*((32*d^4*x^4)/(315*b*(a*d - b*c)^4) - (70*b^3*c^4 - 210*a^3*c*d^3 + 378*a^2*b*c^2*d^2 - 270*a
*b^2*c^3*d)/(315*b^4*(a*d - b*c)^4) + (x*(210*a^3*d^4 - 10*b^3*c^3*d + 54*a*b^2*c^2*d^2 - 126*a^2*b*c*d^3))/(3
15*b^4*(a*d - b*c)^4) + (16*d^3*x^3*(9*a*d - b*c))/(315*b^2*(a*d - b*c)^4) + (4*d^2*x^2*(21*a^2*d^2 + b^2*c^2
- 6*a*b*c*d))/(105*b^3*(a*d - b*c)^4)))/(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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