3.15.70 \(\int \frac {\sqrt {c+d x}}{(a+b x)^{13/2}} \, dx\) [1470]

Optimal. Leaf size=171 \[ -\frac {2 (c+d x)^{3/2}}{11 (b c-a d) (a+b x)^{11/2}}+\frac {16 d (c+d x)^{3/2}}{99 (b c-a d)^2 (a+b x)^{9/2}}-\frac {32 d^2 (c+d x)^{3/2}}{231 (b c-a d)^3 (a+b x)^{7/2}}+\frac {128 d^3 (c+d x)^{3/2}}{1155 (b c-a d)^4 (a+b x)^{5/2}}-\frac {256 d^4 (c+d x)^{3/2}}{3465 (b c-a d)^5 (a+b x)^{3/2}} \]

[Out]

-2/11*(d*x+c)^(3/2)/(-a*d+b*c)/(b*x+a)^(11/2)+16/99*d*(d*x+c)^(3/2)/(-a*d+b*c)^2/(b*x+a)^(9/2)-32/231*d^2*(d*x
+c)^(3/2)/(-a*d+b*c)^3/(b*x+a)^(7/2)+128/1155*d^3*(d*x+c)^(3/2)/(-a*d+b*c)^4/(b*x+a)^(5/2)-256/3465*d^4*(d*x+c
)^(3/2)/(-a*d+b*c)^5/(b*x+a)^(3/2)

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Rubi [A]
time = 0.03, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 5, 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 {256 d^4 (c+d x)^{3/2}}{3465 (a+b x)^{3/2} (b c-a d)^5}+\frac {128 d^3 (c+d x)^{3/2}}{1155 (a+b x)^{5/2} (b c-a d)^4}-\frac {32 d^2 (c+d x)^{3/2}}{231 (a+b x)^{7/2} (b c-a d)^3}+\frac {16 d (c+d x)^{3/2}}{99 (a+b x)^{9/2} (b c-a d)^2}-\frac {2 (c+d x)^{3/2}}{11 (a+b x)^{11/2} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[c + d*x]/(a + b*x)^(13/2),x]

[Out]

(-2*(c + d*x)^(3/2))/(11*(b*c - a*d)*(a + b*x)^(11/2)) + (16*d*(c + d*x)^(3/2))/(99*(b*c - a*d)^2*(a + b*x)^(9
/2)) - (32*d^2*(c + d*x)^(3/2))/(231*(b*c - a*d)^3*(a + b*x)^(7/2)) + (128*d^3*(c + d*x)^(3/2))/(1155*(b*c - a
*d)^4*(a + b*x)^(5/2)) - (256*d^4*(c + d*x)^(3/2))/(3465*(b*c - a*d)^5*(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)^{13/2}} \, dx &=-\frac {2 (c+d x)^{3/2}}{11 (b c-a d) (a+b x)^{11/2}}-\frac {(8 d) \int \frac {\sqrt {c+d x}}{(a+b x)^{11/2}} \, dx}{11 (b c-a d)}\\ &=-\frac {2 (c+d x)^{3/2}}{11 (b c-a d) (a+b x)^{11/2}}+\frac {16 d (c+d x)^{3/2}}{99 (b c-a d)^2 (a+b x)^{9/2}}+\frac {\left (16 d^2\right ) \int \frac {\sqrt {c+d x}}{(a+b x)^{9/2}} \, dx}{33 (b c-a d)^2}\\ &=-\frac {2 (c+d x)^{3/2}}{11 (b c-a d) (a+b x)^{11/2}}+\frac {16 d (c+d x)^{3/2}}{99 (b c-a d)^2 (a+b x)^{9/2}}-\frac {32 d^2 (c+d x)^{3/2}}{231 (b c-a d)^3 (a+b x)^{7/2}}-\frac {\left (64 d^3\right ) \int \frac {\sqrt {c+d x}}{(a+b x)^{7/2}} \, dx}{231 (b c-a d)^3}\\ &=-\frac {2 (c+d x)^{3/2}}{11 (b c-a d) (a+b x)^{11/2}}+\frac {16 d (c+d x)^{3/2}}{99 (b c-a d)^2 (a+b x)^{9/2}}-\frac {32 d^2 (c+d x)^{3/2}}{231 (b c-a d)^3 (a+b x)^{7/2}}+\frac {128 d^3 (c+d x)^{3/2}}{1155 (b c-a d)^4 (a+b x)^{5/2}}+\frac {\left (128 d^4\right ) \int \frac {\sqrt {c+d x}}{(a+b x)^{5/2}} \, dx}{1155 (b c-a d)^4}\\ &=-\frac {2 (c+d x)^{3/2}}{11 (b c-a d) (a+b x)^{11/2}}+\frac {16 d (c+d x)^{3/2}}{99 (b c-a d)^2 (a+b x)^{9/2}}-\frac {32 d^2 (c+d x)^{3/2}}{231 (b c-a d)^3 (a+b x)^{7/2}}+\frac {128 d^3 (c+d x)^{3/2}}{1155 (b c-a d)^4 (a+b x)^{5/2}}-\frac {256 d^4 (c+d x)^{3/2}}{3465 (b c-a d)^5 (a+b x)^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 0.14, size = 117, normalized size = 0.68 \begin {gather*} -\frac {2 (c+d x)^{3/2} \left (1155 d^4-\frac {2772 b d^3 (c+d x)}{a+b x}+\frac {2970 b^2 d^2 (c+d x)^2}{(a+b x)^2}-\frac {1540 b^3 d (c+d x)^3}{(a+b x)^3}+\frac {315 b^4 (c+d x)^4}{(a+b x)^4}\right )}{3465 (b c-a d)^5 (a+b x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c + d*x]/(a + b*x)^(13/2),x]

[Out]

(-2*(c + d*x)^(3/2)*(1155*d^4 - (2772*b*d^3*(c + d*x))/(a + b*x) + (2970*b^2*d^2*(c + d*x)^2)/(a + b*x)^2 - (1
540*b^3*d*(c + d*x)^3)/(a + b*x)^3 + (315*b^4*(c + d*x)^4)/(a + b*x)^4))/(3465*(b*c - a*d)^5*(a + b*x)^(3/2))

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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)^(1/2)/(a + b*x)^(13/2),x]')

[Out]

Timed out

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Maple [A]
time = 0.16, size = 248, normalized size = 1.45

method	result	size
default	\(-\frac {\sqrt {d x +c}}{5 b \left (b x +a \right )^{\frac {11}{2}}}+\frac {\left (a d -b c \right ) \left (-\frac {2 \sqrt {d x +c}}{11 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {11}{2}}}-\frac {10 d \left (-\frac {2 \sqrt {d x +c}}{9 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {9}{2}}}-\frac {8 d \left (-\frac {2 \sqrt {d x +c}}{7 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {7}{2}}}-\frac {6 d \left (-\frac {2 \sqrt {d x +c}}{5 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {5}{2}}}-\frac {4 d \left (-\frac {2 \sqrt {d x +c}}{3 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {3}{2}}}+\frac {4 d \sqrt {d x +c}}{3 \left (-a d +b c \right )^{2} \sqrt {b x +a}}\right )}{5 \left (-a d +b c \right )}\right )}{7 \left (-a d +b c \right )}\right )}{9 \left (-a d +b c \right )}\right )}{11 \left (-a d +b c \right )}\right )}{10 b}\)	\(248\)
gosper	\(\frac {2 \left (d x +c \right )^{\frac {3}{2}} \left (128 d^{4} x^{4} b^{4}+704 a \,b^{3} d^{4} x^{3}-192 b^{4} c \,d^{3} x^{3}+1584 a^{2} b^{2} d^{4} x^{2}-1056 a \,b^{3} c \,d^{3} x^{2}+240 b^{4} c^{2} d^{2} x^{2}+1848 a^{3} b \,d^{4} x -2376 a^{2} b^{2} c \,d^{3} x +1320 a \,b^{3} c^{2} d^{2} x -280 b^{4} c^{3} d x +1155 a^{4} d^{4}-2772 a^{3} b c \,d^{3}+2970 a^{2} b^{2} c^{2} d^{2}-1540 a \,b^{3} c^{3} d +315 b^{4} c^{4}\right )}{3465 \left (b x +a \right )^{\frac {11}{2}} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}\)	\(256\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(1/2)/(b*x+a)^(13/2),x,method=_RETURNVERBOSE)

[Out]

-1/5/b*(d*x+c)^(1/2)/(b*x+a)^(11/2)+1/10*(a*d-b*c)/b*(-2/11/(-a*d+b*c)/(b*x+a)^(11/2)*(d*x+c)^(1/2)-10/11*d/(-
a*d+b*c)*(-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)^(13/2),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. 781 vs. \(2 (141) = 282\).
time = 5.20, size = 781, normalized size = 4.57 \begin {gather*} -\frac {2 \, {\left (128 \, b^{4} d^{5} x^{5} + 315 \, b^{4} c^{5} - 1540 \, a b^{3} c^{4} d + 2970 \, a^{2} b^{2} c^{3} d^{2} - 2772 \, a^{3} b c^{2} d^{3} + 1155 \, a^{4} c d^{4} - 64 \, {\left (b^{4} c d^{4} - 11 \, a b^{3} d^{5}\right )} x^{4} + 16 \, {\left (3 \, b^{4} c^{2} d^{3} - 22 \, a b^{3} c d^{4} + 99 \, a^{2} b^{2} d^{5}\right )} x^{3} - 8 \, {\left (5 \, b^{4} c^{3} d^{2} - 33 \, a b^{3} c^{2} d^{3} + 99 \, a^{2} b^{2} c d^{4} - 231 \, a^{3} b d^{5}\right )} x^{2} + {\left (35 \, b^{4} c^{4} d - 220 \, a b^{3} c^{3} d^{2} + 594 \, a^{2} b^{2} c^{2} d^{3} - 924 \, a^{3} b c d^{4} + 1155 \, a^{4} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3465 \, {\left (a^{6} b^{5} c^{5} - 5 \, a^{7} b^{4} c^{4} d + 10 \, a^{8} b^{3} c^{3} d^{2} - 10 \, a^{9} b^{2} c^{2} d^{3} + 5 \, a^{10} b c d^{4} - a^{11} d^{5} + {\left (b^{11} c^{5} - 5 \, a b^{10} c^{4} d + 10 \, a^{2} b^{9} c^{3} d^{2} - 10 \, a^{3} b^{8} c^{2} d^{3} + 5 \, a^{4} b^{7} c d^{4} - a^{5} b^{6} d^{5}\right )} x^{6} + 6 \, {\left (a b^{10} c^{5} - 5 \, a^{2} b^{9} c^{4} d + 10 \, a^{3} b^{8} c^{3} d^{2} - 10 \, a^{4} b^{7} c^{2} d^{3} + 5 \, a^{5} b^{6} c d^{4} - a^{6} b^{5} d^{5}\right )} x^{5} + 15 \, {\left (a^{2} b^{9} c^{5} - 5 \, a^{3} b^{8} c^{4} d + 10 \, a^{4} b^{7} c^{3} d^{2} - 10 \, a^{5} b^{6} c^{2} d^{3} + 5 \, a^{6} b^{5} c d^{4} - a^{7} b^{4} d^{5}\right )} x^{4} + 20 \, {\left (a^{3} b^{8} c^{5} - 5 \, a^{4} b^{7} c^{4} d + 10 \, a^{5} b^{6} c^{3} d^{2} - 10 \, a^{6} b^{5} c^{2} d^{3} + 5 \, a^{7} b^{4} c d^{4} - a^{8} b^{3} d^{5}\right )} x^{3} + 15 \, {\left (a^{4} b^{7} c^{5} - 5 \, a^{5} b^{6} c^{4} d + 10 \, a^{6} b^{5} c^{3} d^{2} - 10 \, a^{7} b^{4} c^{2} d^{3} + 5 \, a^{8} b^{3} c d^{4} - a^{9} b^{2} d^{5}\right )} x^{2} + 6 \, {\left (a^{5} b^{6} c^{5} - 5 \, a^{6} b^{5} c^{4} d + 10 \, a^{7} b^{4} c^{3} d^{2} - 10 \, a^{8} b^{3} c^{2} d^{3} + 5 \, a^{9} b^{2} c d^{4} - a^{10} b d^{5}\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)^(13/2),x, algorithm="fricas")

[Out]

-2/3465*(128*b^4*d^5*x^5 + 315*b^4*c^5 - 1540*a*b^3*c^4*d + 2970*a^2*b^2*c^3*d^2 - 2772*a^3*b*c^2*d^3 + 1155*a
^4*c*d^4 - 64*(b^4*c*d^4 - 11*a*b^3*d^5)*x^4 + 16*(3*b^4*c^2*d^3 - 22*a*b^3*c*d^4 + 99*a^2*b^2*d^5)*x^3 - 8*(5
*b^4*c^3*d^2 - 33*a*b^3*c^2*d^3 + 99*a^2*b^2*c*d^4 - 231*a^3*b*d^5)*x^2 + (35*b^4*c^4*d - 220*a*b^3*c^3*d^2 +
594*a^2*b^2*c^2*d^3 - 924*a^3*b*c*d^4 + 1155*a^4*d^5)*x)*sqrt(b*x + a)*sqrt(d*x + c)/(a^6*b^5*c^5 - 5*a^7*b^4*
c^4*d + 10*a^8*b^3*c^3*d^2 - 10*a^9*b^2*c^2*d^3 + 5*a^10*b*c*d^4 - a^11*d^5 + (b^11*c^5 - 5*a*b^10*c^4*d + 10*
a^2*b^9*c^3*d^2 - 10*a^3*b^8*c^2*d^3 + 5*a^4*b^7*c*d^4 - a^5*b^6*d^5)*x^6 + 6*(a*b^10*c^5 - 5*a^2*b^9*c^4*d +
10*a^3*b^8*c^3*d^2 - 10*a^4*b^7*c^2*d^3 + 5*a^5*b^6*c*d^4 - a^6*b^5*d^5)*x^5 + 15*(a^2*b^9*c^5 - 5*a^3*b^8*c^4
*d + 10*a^4*b^7*c^3*d^2 - 10*a^5*b^6*c^2*d^3 + 5*a^6*b^5*c*d^4 - a^7*b^4*d^5)*x^4 + 20*(a^3*b^8*c^5 - 5*a^4*b^
7*c^4*d + 10*a^5*b^6*c^3*d^2 - 10*a^6*b^5*c^2*d^3 + 5*a^7*b^4*c*d^4 - a^8*b^3*d^5)*x^3 + 15*(a^4*b^7*c^5 - 5*a
^5*b^6*c^4*d + 10*a^6*b^5*c^3*d^2 - 10*a^7*b^4*c^2*d^3 + 5*a^8*b^3*c*d^4 - a^9*b^2*d^5)*x^2 + 6*(a^5*b^6*c^5 -
5*a^6*b^5*c^4*d + 10*a^7*b^4*c^3*d^2 - 10*a^8*b^3*c^2*d^3 + 5*a^9*b^2*c*d^4 - a^10*b*d^5)*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)**(1/2)/(b*x+a)**(13/2),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 645 vs. \(2 (141) = 282\).
time = 0.20, size = 767, normalized size = 4.49 \begin {gather*} \frac {2 \left (\left (\left (\left (\frac {3991680 b^{9} d^{12} \sqrt {c+d x} \sqrt {c+d x}}{-108056025 b^{10} c^{5} \left |d\right |[phone] b^{9} d a c^{4} \left |d\right |-1080560250 b^{8} d^{2} a^{2} c^{3} \left |d\right |[phone] b^{7} d^{3} a^{3} c^{2} \left |d\right |-540280125 b^{6} d^{4} a^{4} c \left |d\right |[phone] b^{5} d^{5} a^{5} \left |d\right |}-\frac {21954240 b^{9} d^{12} c-21954240 b^{8} d^{13} a}{-108056025 b^{10} c^{5} \left |d\right |[phone] b^{9} d a c^{4} \left |d\right |-1080560250 b^{8} d^{2} a^{2} c^{3} \left |d\right |[phone] b^{7} d^{3} a^{3} c^{2} \left |d\right |-540280125 b^{6} d^{4} a^{4} c \left |d\right |[phone] b^{5} d^{5} a^{5} \left |d\right |}\right ) \sqrt {c+d x} \sqrt {c+d x}-\frac {-49397040 b^{9} d^{12} c^{2}[phone] b^{8} d^{13} a c-49397040 b^{7} d^{14} a^{2}}{-108056025 b^{10} c^{5} \left |d\right |[phone] b^{9} d a c^{4} \left |d\right |-1080560250 b^{8} d^{2} a^{2} c^{3} \left |d\right |[phone] b^{7} d^{3} a^{3} c^{2} \left |d\right |-540280125 b^{6} d^{4} a^{4} c \left |d\right |[phone] b^{5} d^{5} a^{5} \left |d\right |}\right ) \sqrt {c+d x} \sqrt {c+d x}-\frac {57629880 b^{9} d^{12} c^{3}-172889640 b^{8} d^{13} a c^{2}[phone] b^{7} d^{14} a^{2} c-57629880 b^{6} d^{15} a^{3}}{-108056025 b^{10} c^{5} \left |d\right |[phone] b^{9} d a c^{4} \left |d\right |-1080560250 b^{8} d^{2} a^{2} c^{3} \left |d\right |[phone] b^{7} d^{3} a^{3} c^{2} \left |d\right |-540280125 b^{6} d^{4} a^{4} c \left |d\right |[phone] b^{5} d^{5} a^{5} \left |d\right |}\right ) \sqrt {c+d x} \sqrt {c+d x}-\frac {-36018675 b^{9} d^{12} c^{4}[phone] b^{8} d^{13} a c^{3}-216112050 b^{7} d^{14} a^{2} c^{2}[phone] b^{6} d^{15} a^{3} c-36018675 b^{5} d^{16} a^{4}}{-108056025 b^{10} c^{5} \left |d\right |[phone] b^{9} d a c^{4} \left |d\right |-1080560250 b^{8} d^{2} a^{2} c^{3} \left |d\right |[phone] b^{7} d^{3} a^{3} c^{2} \left |d\right |-540280125 b^{6} d^{4} a^{4} c \left |d\right |[phone] b^{5} d^{5} a^{5} \left |d\right |}\right ) \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 )^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2)/(b*x+a)^(13/2),x)

[Out]

-2/3465*(8*(2*(4*(2*(d*x + c)*b^9*d^12/(b^10*c^5*abs(d) - 5*a*b^9*c^4*d*abs(d) + 10*a^2*b^8*c^3*d^2*abs(d) - 1
0*a^3*b^7*c^2*d^3*abs(d) + 5*a^4*b^6*c*d^4*abs(d) - a^5*b^5*d^5*abs(d)) - 11*(b^9*c*d^12 - a*b^8*d^13)/(b^10*c
^5*abs(d) - 5*a*b^9*c^4*d*abs(d) + 10*a^2*b^8*c^3*d^2*abs(d) - 10*a^3*b^7*c^2*d^3*abs(d) + 5*a^4*b^6*c*d^4*abs
(d) - a^5*b^5*d^5*abs(d)))*(d*x + c) + 99*(b^9*c^2*d^12 - 2*a*b^8*c*d^13 + a^2*b^7*d^14)/(b^10*c^5*abs(d) - 5*
a*b^9*c^4*d*abs(d) + 10*a^2*b^8*c^3*d^2*abs(d) - 10*a^3*b^7*c^2*d^3*abs(d) + 5*a^4*b^6*c*d^4*abs(d) - a^5*b^5*
d^5*abs(d)))*(d*x + c) - 231*(b^9*c^3*d^12 - 3*a*b^8*c^2*d^13 + 3*a^2*b^7*c*d^14 - a^3*b^6*d^15)/(b^10*c^5*abs
(d) - 5*a*b^9*c^4*d*abs(d) + 10*a^2*b^8*c^3*d^2*abs(d) - 10*a^3*b^7*c^2*d^3*abs(d) + 5*a^4*b^6*c*d^4*abs(d) -
a^5*b^5*d^5*abs(d)))*(d*x + c) + 1155*(b^9*c^4*d^12 - 4*a*b^8*c^3*d^13 + 6*a^2*b^7*c^2*d^14 - 4*a^3*b^6*c*d^15
+ a^4*b^5*d^16)/(b^10*c^5*abs(d) - 5*a*b^9*c^4*d*abs(d) + 10*a^2*b^8*c^3*d^2*abs(d) - 10*a^3*b^7*c^2*d^3*abs(
d) + 5*a^4*b^6*c*d^4*abs(d) - a^5*b^5*d^5*abs(d)))*(d*x + c)^(3/2)/((d*x + c)*b*d - b*c*d + a*d^2)^(11/2)

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Mupad [B]
time = 1.43, size = 397, normalized size = 2.32 \begin {gather*} \frac {\sqrt {c+d\,x}\,\left (\frac {2310\,a^4\,c\,d^4-5544\,a^3\,b\,c^2\,d^3+5940\,a^2\,b^2\,c^3\,d^2-3080\,a\,b^3\,c^4\,d+630\,b^4\,c^5}{3465\,b^5\,{\left (a\,d-b\,c\right )}^5}+\frac {x\,\left (2310\,a^4\,d^5-1848\,a^3\,b\,c\,d^4+1188\,a^2\,b^2\,c^2\,d^3-440\,a\,b^3\,c^3\,d^2+70\,b^4\,c^4\,d\right )}{3465\,b^5\,{\left (a\,d-b\,c\right )}^5}+\frac {256\,d^5\,x^5}{3465\,b\,{\left (a\,d-b\,c\right )}^5}+\frac {16\,d^2\,x^2\,\left (231\,a^3\,d^3-99\,a^2\,b\,c\,d^2+33\,a\,b^2\,c^2\,d-5\,b^3\,c^3\right )}{3465\,b^4\,{\left (a\,d-b\,c\right )}^5}+\frac {128\,d^4\,x^4\,\left (11\,a\,d-b\,c\right )}{3465\,b^2\,{\left (a\,d-b\,c\right )}^5}+\frac {32\,d^3\,x^3\,\left (99\,a^2\,d^2-22\,a\,b\,c\,d+3\,b^2\,c^2\right )}{3465\,b^3\,{\left (a\,d-b\,c\right )}^5}\right )}{x^5\,\sqrt {a+b\,x}+\frac {a^5\,\sqrt {a+b\,x}}{b^5}+\frac {10\,a^2\,x^3\,\sqrt {a+b\,x}}{b^2}+\frac {10\,a^3\,x^2\,\sqrt {a+b\,x}}{b^3}+\frac {5\,a\,x^4\,\sqrt {a+b\,x}}{b}+\frac {5\,a^4\,x\,\sqrt {a+b\,x}}{b^4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^(1/2)/(a + b*x)^(13/2),x)

[Out]

((c + d*x)^(1/2)*((630*b^4*c^5 + 2310*a^4*c*d^4 - 5544*a^3*b*c^2*d^3 + 5940*a^2*b^2*c^3*d^2 - 3080*a*b^3*c^4*d
)/(3465*b^5*(a*d - b*c)^5) + (x*(2310*a^4*d^5 + 70*b^4*c^4*d - 440*a*b^3*c^3*d^2 + 1188*a^2*b^2*c^2*d^3 - 1848
*a^3*b*c*d^4))/(3465*b^5*(a*d - b*c)^5) + (256*d^5*x^5)/(3465*b*(a*d - b*c)^5) + (16*d^2*x^2*(231*a^3*d^3 - 5*
b^3*c^3 + 33*a*b^2*c^2*d - 99*a^2*b*c*d^2))/(3465*b^4*(a*d - b*c)^5) + (128*d^4*x^4*(11*a*d - b*c))/(3465*b^2*
(a*d - b*c)^5) + (32*d^3*x^3*(99*a^2*d^2 + 3*b^2*c^2 - 22*a*b*c*d))/(3465*b^3*(a*d - b*c)^5)))/(x^5*(a + b*x)^
(1/2) + (a^5*(a + b*x)^(1/2))/b^5 + (10*a^2*x^3*(a + b*x)^(1/2))/b^2 + (10*a^3*x^2*(a + b*x)^(1/2))/b^3 + (5*a
*x^4*(a + b*x)^(1/2))/b + (5*a^4*x*(a + b*x)^(1/2))/b^4)

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