3.15.91 \(\int \frac {(c+d x)^{5/2}}{(a+b x)^{15/2}} \, dx\) [1491]

Optimal. Leaf size=136 \[ -\frac {2 (c+d x)^{7/2}}{13 (b c-a d) (a+b x)^{13/2}}+\frac {12 d (c+d x)^{7/2}}{143 (b c-a d)^2 (a+b x)^{11/2}}-\frac {16 d^2 (c+d x)^{7/2}}{429 (b c-a d)^3 (a+b x)^{9/2}}+\frac {32 d^3 (c+d x)^{7/2}}{3003 (b c-a d)^4 (a+b x)^{7/2}} \]

[Out]

-2/13*(d*x+c)^(7/2)/(-a*d+b*c)/(b*x+a)^(13/2)+12/143*d*(d*x+c)^(7/2)/(-a*d+b*c)^2/(b*x+a)^(11/2)-16/429*d^2*(d
*x+c)^(7/2)/(-a*d+b*c)^3/(b*x+a)^(9/2)+32/3003*d^3*(d*x+c)^(7/2)/(-a*d+b*c)^4/(b*x+a)^(7/2)

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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)^{7/2}}{3003 (a+b x)^{7/2} (b c-a d)^4}-\frac {16 d^2 (c+d x)^{7/2}}{429 (a+b x)^{9/2} (b c-a d)^3}+\frac {12 d (c+d x)^{7/2}}{143 (a+b x)^{11/2} (b c-a d)^2}-\frac {2 (c+d x)^{7/2}}{13 (a+b x)^{13/2} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^(5/2)/(a + b*x)^(15/2),x]

[Out]

(-2*(c + d*x)^(7/2))/(13*(b*c - a*d)*(a + b*x)^(13/2)) + (12*d*(c + d*x)^(7/2))/(143*(b*c - a*d)^2*(a + b*x)^(
11/2)) - (16*d^2*(c + d*x)^(7/2))/(429*(b*c - a*d)^3*(a + b*x)^(9/2)) + (32*d^3*(c + d*x)^(7/2))/(3003*(b*c -
a*d)^4*(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)^{15/2}} \, dx &=-\frac {2 (c+d x)^{7/2}}{13 (b c-a d) (a+b x)^{13/2}}-\frac {(6 d) \int \frac {(c+d x)^{5/2}}{(a+b x)^{13/2}} \, dx}{13 (b c-a d)}\\ &=-\frac {2 (c+d x)^{7/2}}{13 (b c-a d) (a+b x)^{13/2}}+\frac {12 d (c+d x)^{7/2}}{143 (b c-a d)^2 (a+b x)^{11/2}}+\frac {\left (24 d^2\right ) \int \frac {(c+d x)^{5/2}}{(a+b x)^{11/2}} \, dx}{143 (b c-a d)^2}\\ &=-\frac {2 (c+d x)^{7/2}}{13 (b c-a d) (a+b x)^{13/2}}+\frac {12 d (c+d x)^{7/2}}{143 (b c-a d)^2 (a+b x)^{11/2}}-\frac {16 d^2 (c+d x)^{7/2}}{429 (b c-a d)^3 (a+b x)^{9/2}}-\frac {\left (16 d^3\right ) \int \frac {(c+d x)^{5/2}}{(a+b x)^{9/2}} \, dx}{429 (b c-a d)^3}\\ &=-\frac {2 (c+d x)^{7/2}}{13 (b c-a d) (a+b x)^{13/2}}+\frac {12 d (c+d x)^{7/2}}{143 (b c-a d)^2 (a+b x)^{11/2}}-\frac {16 d^2 (c+d x)^{7/2}}{429 (b c-a d)^3 (a+b x)^{9/2}}+\frac {32 d^3 (c+d x)^{7/2}}{3003 (b c-a d)^4 (a+b x)^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 95, normalized size = 0.70 \begin {gather*} -\frac {2 (c+d x)^{7/2} \left (-429 d^3+\frac {1001 b d^2 (c+d x)}{a+b x}-\frac {819 b^2 d (c+d x)^2}{(a+b x)^2}+\frac {231 b^3 (c+d x)^3}{(a+b x)^3}\right )}{3003 (b c-a d)^4 (a+b x)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^(5/2)/(a + b*x)^(15/2),x]

[Out]

(-2*(c + d*x)^(7/2)*(-429*d^3 + (1001*b*d^2*(c + d*x))/(a + b*x) - (819*b^2*d*(c + d*x)^2)/(a + b*x)^2 + (231*
b^3*(c + d*x)^3)/(a + b*x)^3))/(3003*(b*c - a*d)^4*(a + b*x)^(7/2))

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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[(c + d*x)^(5/2)/(a + b*x)^(15/2),x]')

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4061 deep

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(353\) vs. \(2(112)=224\).
time = 0.17, size = 354, normalized size = 2.60

method	result	size
gosper	\(\frac {2 \left (d x +c \right )^{\frac {7}{2}} \left (16 b^{3} x^{3} d^{3}+104 d^{3} a \,x^{2} b^{2}-56 b^{3} c \,d^{2} x^{2}+286 a^{2} b \,d^{3} x -364 a \,b^{2} c \,d^{2} x +126 b^{3} c^{2} d x +429 a^{3} d^{3}-1001 a^{2} b c \,d^{2}+819 a \,b^{2} c^{2} d -231 b^{3} c^{3}\right )}{3003 \left (b x +a \right )^{\frac {13}{2}} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\)	\(171\)
default	\(-\frac {\left (d x +c \right )^{\frac {5}{2}}}{4 b \left (b x +a \right )^{\frac {13}{2}}}+\frac {5 \left (a d -b c \right ) \left (-\frac {\left (d x +c \right )^{\frac {3}{2}}}{5 b \left (b x +a \right )^{\frac {13}{2}}}+\frac {3 \left (a d -b c \right ) \left (-\frac {\sqrt {d x +c}}{6 b \left (b x +a \right )^{\frac {13}{2}}}+\frac {\left (a d -b c \right ) \left (-\frac {2 \sqrt {d x +c}}{13 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {13}{2}}}-\frac {12 d \left (-\frac {2 \sqrt {d x +c}}{11 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {11}{2}}}-\frac {10 d \left (-\frac {2 \sqrt {d x +c}}{9 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {9}{2}}}-\frac {8 d \left (-\frac {2 \sqrt {d x +c}}{7 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {7}{2}}}-\frac {6 d \left (-\frac {2 \sqrt {d x +c}}{5 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {5}{2}}}-\frac {4 d \left (-\frac {2 \sqrt {d x +c}}{3 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {3}{2}}}+\frac {4 d \sqrt {d x +c}}{3 \left (-a d +b c \right )^{2} \sqrt {b x +a}}\right )}{5 \left (-a d +b c \right )}\right )}{7 \left (-a d +b c \right )}\right )}{9 \left (-a d +b c \right )}\right )}{11 \left (-a d +b c \right )}\right )}{13 \left (-a d +b c \right )}\right )}{12 b}\right )}{10 b}\right )}{8 b}\)	\(354\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(5/2)/(b*x+a)^(15/2),x,method=_RETURNVERBOSE)

[Out]

-1/4/b*(d*x+c)^(5/2)/(b*x+a)^(13/2)+5/8*(a*d-b*c)/b*(-1/5/b*(d*x+c)^(3/2)/(b*x+a)^(13/2)+3/10*(a*d-b*c)/b*(-1/
6/b*(d*x+c)^(1/2)/(b*x+a)^(13/2)+1/12*(a*d-b*c)/b*(-2/13/(-a*d+b*c)/(b*x+a)^(13/2)*(d*x+c)^(1/2)-12/13*d/(-a*d
+b*c)*(-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)^(5/2)/(b*x+a)^(15/2),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. 765 vs. \(2 (112) = 224\).
time = 12.24, size = 765, normalized size = 5.62 \begin {gather*} \frac {2 \, {\left (16 \, b^{3} d^{6} x^{6} - 231 \, b^{3} c^{6} + 819 \, a b^{2} c^{5} d - 1001 \, a^{2} b c^{4} d^{2} + 429 \, a^{3} c^{3} d^{3} - 8 \, {\left (b^{3} c d^{5} - 13 \, a b^{2} d^{6}\right )} x^{5} + 2 \, {\left (3 \, b^{3} c^{2} d^{4} - 26 \, a b^{2} c d^{5} + 143 \, a^{2} b d^{6}\right )} x^{4} - {\left (5 \, b^{3} c^{3} d^{3} - 39 \, a b^{2} c^{2} d^{4} + 143 \, a^{2} b c d^{5} - 429 \, a^{3} d^{6}\right )} x^{3} - {\left (371 \, b^{3} c^{4} d^{2} - 1469 \, a b^{2} c^{3} d^{3} + 2145 \, a^{2} b c^{2} d^{4} - 1287 \, a^{3} c d^{5}\right )} x^{2} - {\left (567 \, b^{3} c^{5} d - 2093 \, a b^{2} c^{4} d^{2} + 2717 \, a^{2} b c^{3} d^{3} - 1287 \, a^{3} c^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3003 \, {\left (a^{7} b^{4} c^{4} - 4 \, a^{8} b^{3} c^{3} d + 6 \, a^{9} b^{2} c^{2} d^{2} - 4 \, a^{10} b c d^{3} + a^{11} d^{4} + {\left (b^{11} c^{4} - 4 \, a b^{10} c^{3} d + 6 \, a^{2} b^{9} c^{2} d^{2} - 4 \, a^{3} b^{8} c d^{3} + a^{4} b^{7} d^{4}\right )} x^{7} + 7 \, {\left (a b^{10} c^{4} - 4 \, a^{2} b^{9} c^{3} d + 6 \, a^{3} b^{8} c^{2} d^{2} - 4 \, a^{4} b^{7} c d^{3} + a^{5} b^{6} d^{4}\right )} x^{6} + 21 \, {\left (a^{2} b^{9} c^{4} - 4 \, a^{3} b^{8} c^{3} d + 6 \, a^{4} b^{7} c^{2} d^{2} - 4 \, a^{5} b^{6} c d^{3} + a^{6} b^{5} d^{4}\right )} x^{5} + 35 \, {\left (a^{3} b^{8} c^{4} - 4 \, a^{4} b^{7} c^{3} d + 6 \, a^{5} b^{6} c^{2} d^{2} - 4 \, a^{6} b^{5} c d^{3} + a^{7} b^{4} d^{4}\right )} x^{4} + 35 \, {\left (a^{4} b^{7} c^{4} - 4 \, a^{5} b^{6} c^{3} d + 6 \, a^{6} b^{5} c^{2} d^{2} - 4 \, a^{7} b^{4} c d^{3} + a^{8} b^{3} d^{4}\right )} x^{3} + 21 \, {\left (a^{5} b^{6} c^{4} - 4 \, a^{6} b^{5} c^{3} d + 6 \, a^{7} b^{4} c^{2} d^{2} - 4 \, a^{8} b^{3} c d^{3} + a^{9} b^{2} d^{4}\right )} x^{2} + 7 \, {\left (a^{6} b^{5} c^{4} - 4 \, a^{7} b^{4} c^{3} d + 6 \, a^{8} b^{3} c^{2} d^{2} - 4 \, a^{9} b^{2} c d^{3} + a^{10} b d^{4}\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)^(15/2),x, algorithm="fricas")

[Out]

2/3003*(16*b^3*d^6*x^6 - 231*b^3*c^6 + 819*a*b^2*c^5*d - 1001*a^2*b*c^4*d^2 + 429*a^3*c^3*d^3 - 8*(b^3*c*d^5 -
13*a*b^2*d^6)*x^5 + 2*(3*b^3*c^2*d^4 - 26*a*b^2*c*d^5 + 143*a^2*b*d^6)*x^4 - (5*b^3*c^3*d^3 - 39*a*b^2*c^2*d^
4 + 143*a^2*b*c*d^5 - 429*a^3*d^6)*x^3 - (371*b^3*c^4*d^2 - 1469*a*b^2*c^3*d^3 + 2145*a^2*b*c^2*d^4 - 1287*a^3
*c*d^5)*x^2 - (567*b^3*c^5*d - 2093*a*b^2*c^4*d^2 + 2717*a^2*b*c^3*d^3 - 1287*a^3*c^2*d^4)*x)*sqrt(b*x + a)*sq
rt(d*x + c)/(a^7*b^4*c^4 - 4*a^8*b^3*c^3*d + 6*a^9*b^2*c^2*d^2 - 4*a^10*b*c*d^3 + a^11*d^4 + (b^11*c^4 - 4*a*b
^10*c^3*d + 6*a^2*b^9*c^2*d^2 - 4*a^3*b^8*c*d^3 + a^4*b^7*d^4)*x^7 + 7*(a*b^10*c^4 - 4*a^2*b^9*c^3*d + 6*a^3*b
^8*c^2*d^2 - 4*a^4*b^7*c*d^3 + a^5*b^6*d^4)*x^6 + 21*(a^2*b^9*c^4 - 4*a^3*b^8*c^3*d + 6*a^4*b^7*c^2*d^2 - 4*a^
5*b^6*c*d^3 + a^6*b^5*d^4)*x^5 + 35*(a^3*b^8*c^4 - 4*a^4*b^7*c^3*d + 6*a^5*b^6*c^2*d^2 - 4*a^6*b^5*c*d^3 + a^7
*b^4*d^4)*x^4 + 35*(a^4*b^7*c^4 - 4*a^5*b^6*c^3*d + 6*a^6*b^5*c^2*d^2 - 4*a^7*b^4*c*d^3 + a^8*b^3*d^4)*x^3 + 2
1*(a^5*b^6*c^4 - 4*a^6*b^5*c^3*d + 6*a^7*b^4*c^2*d^2 - 4*a^8*b^3*c*d^3 + a^9*b^2*d^4)*x^2 + 7*(a^6*b^5*c^4 - 4
*a^7*b^4*c^3*d + 6*a^8*b^3*c^2*d^2 - 4*a^9*b^2*c*d^3 + a^10*b*d^4)*x)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**(5/2)/(b*x+a)**(15/2),x)

[Out]

Exception raised: SystemError

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 662 vs. \(2 (112) = 224\).
time = 0.48, size = 811, normalized size = 5.96 \begin {gather*} \frac {2 \left (\left (\left (-\frac {\left (-2162160 b^{11} d^{14} c^{2}+4324320 b^{10} d^{15} a c-2162160 b^{9} d^{16} a^{2}\right ) \sqrt {c+d x} \sqrt {c+d x}}{405810405 b^{12} c^{6} \left |d\right |-2434862430 b^{11} d a c^{5} \left |d\right |[phone] b^{10} d^{2} a^{2} c^{4} \left |d\right |-8116208100 b^{9} d^{3} a^{3} c^{3} \left |d\right |[phone] b^{8} d^{4} a^{4} c^{2} \left |d\right |-2434862430 b^{7} d^{5} a^{5} c \left |d\right |[phone] b^{6} d^{6} a^{6} \left |d\right |}-\frac {14054040 b^{11} d^{14} c^{3}-42162120 b^{10} d^{15} a c^{2}[phone] b^{9} d^{16} a^{2} c-14054040 b^{8} d^{17} a^{3}}{405810405 b^{12} c^{6} \left |d\right |-2434862430 b^{11} d a c^{5} \left |d\right |[phone] b^{10} d^{2} a^{2} c^{4} \left |d\right |-8116208100 b^{9} d^{3} a^{3} c^{3} \left |d\right |[phone] b^{8} d^{4} a^{4} c^{2} \left |d\right |-2434862430 b^{7} d^{5} a^{5} c \left |d\right |[phone] b^{6} d^{6} a^{6} \left |d\right |}\right ) \sqrt {c+d x} \sqrt {c+d x}-\frac {-38648610 b^{11} d^{14} c^{4}[phone] b^{10} d^{15} a c^{3}-231891660 b^{9} d^{16} a^{2} c^{2}[phone] b^{8} d^{17} a^{3} c-38648610 b^{7} d^{18} a^{4}}{405810405 b^{12} c^{6} \left |d\right |-2434862430 b^{11} d a c^{5} \left |d\right |[phone] b^{10} d^{2} a^{2} c^{4} \left |d\right |-8116208100 b^{9} d^{3} a^{3} c^{3} \left |d\right |[phone] b^{8} d^{4} a^{4} c^{2} \left |d\right |-2434862430 b^{7} d^{5} a^{5} c \left |d\right |[phone] b^{6} d^{6} a^{6} \left |d\right |}\right ) \sqrt {c+d x} \sqrt {c+d x}-\frac {57972915 b^{11} d^{14} c^{5}-289864575 b^{10} d^{15} a c^{4}[phone] b^{9} d^{16} a^{2} c^{3}-579729150 b^{8} d^{17} a^{3} c^{2}[phone] b^{7} d^{18} a^{4} c-57972915 b^{6} d^{19} a^{5}}{405810405 b^{12} c^{6} \left |d\right |-2434862430 b^{11} d a c^{5} \left |d\right |[phone] b^{10} d^{2} a^{2} c^{4} \left |d\right |-8116208100 b^{9} d^{3} a^{3} c^{3} \left |d\right |[phone] b^{8} d^{4} a^{4} c^{2} \left |d\right |-2434862430 b^{7} d^{5} a^{5} c \left |d\right |[phone] b^{6} d^{6} a^{6} \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 )^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(5/2)/(b*x+a)^(15/2),x)

[Out]

2/3003*(2*(4*(d*x + c)*(2*(b^11*c^2*d^14 - 2*a*b^10*c*d^15 + a^2*b^9*d^16)*(d*x + c)/(b^12*c^6*abs(d) - 6*a*b^
11*c^5*d*abs(d) + 15*a^2*b^10*c^4*d^2*abs(d) - 20*a^3*b^9*c^3*d^3*abs(d) + 15*a^4*b^8*c^2*d^4*abs(d) - 6*a^5*b
^7*c*d^5*abs(d) + a^6*b^6*d^6*abs(d)) - 13*(b^11*c^3*d^14 - 3*a*b^10*c^2*d^15 + 3*a^2*b^9*c*d^16 - a^3*b^8*d^1
7)/(b^12*c^6*abs(d) - 6*a*b^11*c^5*d*abs(d) + 15*a^2*b^10*c^4*d^2*abs(d) - 20*a^3*b^9*c^3*d^3*abs(d) + 15*a^4*
b^8*c^2*d^4*abs(d) - 6*a^5*b^7*c*d^5*abs(d) + a^6*b^6*d^6*abs(d))) + 143*(b^11*c^4*d^14 - 4*a*b^10*c^3*d^15 +
6*a^2*b^9*c^2*d^16 - 4*a^3*b^8*c*d^17 + a^4*b^7*d^18)/(b^12*c^6*abs(d) - 6*a*b^11*c^5*d*abs(d) + 15*a^2*b^10*c
^4*d^2*abs(d) - 20*a^3*b^9*c^3*d^3*abs(d) + 15*a^4*b^8*c^2*d^4*abs(d) - 6*a^5*b^7*c*d^5*abs(d) + a^6*b^6*d^6*a
bs(d)))*(d*x + c) - 429*(b^11*c^5*d^14 - 5*a*b^10*c^4*d^15 + 10*a^2*b^9*c^3*d^16 - 10*a^3*b^8*c^2*d^17 + 5*a^4
*b^7*c*d^18 - a^5*b^6*d^19)/(b^12*c^6*abs(d) - 6*a*b^11*c^5*d*abs(d) + 15*a^2*b^10*c^4*d^2*abs(d) - 20*a^3*b^9
*c^3*d^3*abs(d) + 15*a^4*b^8*c^2*d^4*abs(d) - 6*a^5*b^7*c*d^5*abs(d) + a^6*b^6*d^6*abs(d)))*(d*x + c)^(7/2)/((
d*x + c)*b*d - b*c*d + a*d^2)^(13/2)

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Mupad [B]
time = 1.62, size = 459, normalized size = 3.38 \begin {gather*} \frac {\sqrt {c+d\,x}\,\left (\frac {x^2\,\left (2574\,a^3\,c\,d^5-4290\,a^2\,b\,c^2\,d^4+2938\,a\,b^2\,c^3\,d^3-742\,b^3\,c^4\,d^2\right )}{3003\,b^6\,{\left (a\,d-b\,c\right )}^4}-\frac {-858\,a^3\,c^3\,d^3+2002\,a^2\,b\,c^4\,d^2-1638\,a\,b^2\,c^5\,d+462\,b^3\,c^6}{3003\,b^6\,{\left (a\,d-b\,c\right )}^4}+\frac {x^3\,\left (858\,a^3\,d^6-286\,a^2\,b\,c\,d^5+78\,a\,b^2\,c^2\,d^4-10\,b^3\,c^3\,d^3\right )}{3003\,b^6\,{\left (a\,d-b\,c\right )}^4}+\frac {32\,d^6\,x^6}{3003\,b^3\,{\left (a\,d-b\,c\right )}^4}-\frac {x\,\left (-2574\,a^3\,c^2\,d^4+5434\,a^2\,b\,c^3\,d^3-4186\,a\,b^2\,c^4\,d^2+1134\,b^3\,c^5\,d\right )}{3003\,b^6\,{\left (a\,d-b\,c\right )}^4}+\frac {16\,d^5\,x^5\,\left (13\,a\,d-b\,c\right )}{3003\,b^4\,{\left (a\,d-b\,c\right )}^4}+\frac {4\,d^4\,x^4\,\left (143\,a^2\,d^2-26\,a\,b\,c\,d+3\,b^2\,c^2\right )}{3003\,b^5\,{\left (a\,d-b\,c\right )}^4}\right )}{x^6\,\sqrt {a+b\,x}+\frac {a^6\,\sqrt {a+b\,x}}{b^6}+\frac {15\,a^2\,x^4\,\sqrt {a+b\,x}}{b^2}+\frac {20\,a^3\,x^3\,\sqrt {a+b\,x}}{b^3}+\frac {15\,a^4\,x^2\,\sqrt {a+b\,x}}{b^4}+\frac {6\,a\,x^5\,\sqrt {a+b\,x}}{b}+\frac {6\,a^5\,x\,\sqrt {a+b\,x}}{b^5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^(5/2)/(a + b*x)^(15/2),x)

[Out]

((c + d*x)^(1/2)*((x^2*(2574*a^3*c*d^5 - 742*b^3*c^4*d^2 + 2938*a*b^2*c^3*d^3 - 4290*a^2*b*c^2*d^4))/(3003*b^6
*(a*d - b*c)^4) - (462*b^3*c^6 - 858*a^3*c^3*d^3 + 2002*a^2*b*c^4*d^2 - 1638*a*b^2*c^5*d)/(3003*b^6*(a*d - b*c
)^4) + (x^3*(858*a^3*d^6 - 10*b^3*c^3*d^3 + 78*a*b^2*c^2*d^4 - 286*a^2*b*c*d^5))/(3003*b^6*(a*d - b*c)^4) + (3
2*d^6*x^6)/(3003*b^3*(a*d - b*c)^4) - (x*(1134*b^3*c^5*d - 2574*a^3*c^2*d^4 - 4186*a*b^2*c^4*d^2 + 5434*a^2*b*
c^3*d^3))/(3003*b^6*(a*d - b*c)^4) + (16*d^5*x^5*(13*a*d - b*c))/(3003*b^4*(a*d - b*c)^4) + (4*d^4*x^4*(143*a^
2*d^2 + 3*b^2*c^2 - 26*a*b*c*d))/(3003*b^5*(a*d - b*c)^4)))/(x^6*(a + b*x)^(1/2) + (a^6*(a + b*x)^(1/2))/b^6 +
(15*a^2*x^4*(a + b*x)^(1/2))/b^2 + (20*a^3*x^3*(a + b*x)^(1/2))/b^3 + (15*a^4*x^2*(a + b*x)^(1/2))/b^4 + (6*a
*x^5*(a + b*x)^(1/2))/b + (6*a^5*x*(a + b*x)^(1/2))/b^5)

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