3.15.34 \(\int \frac {(a+b x)^5}{(c+d x)^{5/2}} \, dx\) [1434]

Optimal. Leaf size=152 \[ \frac {2 (b c-a d)^5}{3 d^6 (c+d x)^{3/2}}-\frac {10 b (b c-a d)^4}{d^6 \sqrt {c+d x}}-\frac {20 b^2 (b c-a d)^3 \sqrt {c+d x}}{d^6}+\frac {20 b^3 (b c-a d)^2 (c+d x)^{3/2}}{3 d^6}-\frac {2 b^4 (b c-a d) (c+d x)^{5/2}}{d^6}+\frac {2 b^5 (c+d x)^{7/2}}{7 d^6} \]

[Out]

2/3*(-a*d+b*c)^5/d^6/(d*x+c)^(3/2)+20/3*b^3*(-a*d+b*c)^2*(d*x+c)^(3/2)/d^6-2*b^4*(-a*d+b*c)*(d*x+c)^(5/2)/d^6+
2/7*b^5*(d*x+c)^(7/2)/d^6-10*b*(-a*d+b*c)^4/d^6/(d*x+c)^(1/2)-20*b^2*(-a*d+b*c)^3*(d*x+c)^(1/2)/d^6

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Rubi [A]
time = 0.04, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {45} \begin {gather*} -\frac {2 b^4 (c+d x)^{5/2} (b c-a d)}{d^6}+\frac {20 b^3 (c+d x)^{3/2} (b c-a d)^2}{3 d^6}-\frac {20 b^2 \sqrt {c+d x} (b c-a d)^3}{d^6}-\frac {10 b (b c-a d)^4}{d^6 \sqrt {c+d x}}+\frac {2 (b c-a d)^5}{3 d^6 (c+d x)^{3/2}}+\frac {2 b^5 (c+d x)^{7/2}}{7 d^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5/(c + d*x)^(5/2),x]

[Out]

(2*(b*c - a*d)^5)/(3*d^6*(c + d*x)^(3/2)) - (10*b*(b*c - a*d)^4)/(d^6*Sqrt[c + d*x]) - (20*b^2*(b*c - a*d)^3*S
qrt[c + d*x])/d^6 + (20*b^3*(b*c - a*d)^2*(c + d*x)^(3/2))/(3*d^6) - (2*b^4*(b*c - a*d)*(c + d*x)^(5/2))/d^6 +
(2*b^5*(c + d*x)^(7/2))/(7*d^6)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int \frac {(a+b x)^5}{(c+d x)^{5/2}} \, dx &=\int \left (\frac {(-b c+a d)^5}{d^5 (c+d x)^{5/2}}+\frac {5 b (b c-a d)^4}{d^5 (c+d x)^{3/2}}-\frac {10 b^2 (b c-a d)^3}{d^5 \sqrt {c+d x}}+\frac {10 b^3 (b c-a d)^2 \sqrt {c+d x}}{d^5}-\frac {5 b^4 (b c-a d) (c+d x)^{3/2}}{d^5}+\frac {b^5 (c+d x)^{5/2}}{d^5}\right ) \, dx\\ &=\frac {2 (b c-a d)^5}{3 d^6 (c+d x)^{3/2}}-\frac {10 b (b c-a d)^4}{d^6 \sqrt {c+d x}}-\frac {20 b^2 (b c-a d)^3 \sqrt {c+d x}}{d^6}+\frac {20 b^3 (b c-a d)^2 (c+d x)^{3/2}}{3 d^6}-\frac {2 b^4 (b c-a d) (c+d x)^{5/2}}{d^6}+\frac {2 b^5 (c+d x)^{7/2}}{7 d^6}\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 217, normalized size = 1.43 \begin {gather*} -\frac {2 \left (7 a^5 d^5+35 a^4 b d^4 (2 c+3 d x)-70 a^3 b^2 d^3 \left (8 c^2+12 c d x+3 d^2 x^2\right )+70 a^2 b^3 d^2 \left (16 c^3+24 c^2 d x+6 c d^2 x^2-d^3 x^3\right )-7 a b^4 d \left (128 c^4+192 c^3 d x+48 c^2 d^2 x^2-8 c d^3 x^3+3 d^4 x^4\right )+b^5 \left (256 c^5+384 c^4 d x+96 c^3 d^2 x^2-16 c^2 d^3 x^3+6 c d^4 x^4-3 d^5 x^5\right )\right )}{21 d^6 (c+d x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(7*a^5*d^5 + 35*a^4*b*d^4*(2*c + 3*d*x) - 70*a^3*b^2*d^3*(8*c^2 + 12*c*d*x + 3*d^2*x^2) + 70*a^2*b^3*d^2*(
16*c^3 + 24*c^2*d*x + 6*c*d^2*x^2 - d^3*x^3) - 7*a*b^4*d*(128*c^4 + 192*c^3*d*x + 48*c^2*d^2*x^2 - 8*c*d^3*x^3
+ 3*d^4*x^4) + b^5*(256*c^5 + 384*c^4*d*x + 96*c^3*d^2*x^2 - 16*c^2*d^3*x^3 + 6*c*d^4*x^4 - 3*d^5*x^5)))/(21*
d^6*(c + d*x)^(3/2))

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Mathics [A]
time = 25.16, size = 152, normalized size = 1.00 \begin {gather*} \frac {2 \left (-105 b \left (c+d x\right ) \left (a d-b c\right )^4+b^2 \left (210 a^3 d^3+70 b \left (a^2 d^2-2 a b c d+b^2 c^2\right ) \left (c+d x\right )-630 a^2 b c d^2+21 b^2 \left (a d-b c\right ) \left (c+d x\right )^2+630 a b^2 c^2 d+3 b^3 \left (c+d x\right )^3-210 b^3 c^3\right ) \left (c+d x\right )^2-7 \left (a d-b c\right )^5\right )}{21 d^6 \left (c+d x\right )^{\frac {3}{2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2 (-105 b (c + d x) (a d - b c) ^ 4 + b ^ 2 (210 a ^ 3 d ^ 3 + 70 b (a ^ 2 d ^ 2 - 2 a b c d + b ^ 2 c ^ 2) (c
+ d x) - 630 a ^ 2 b c d ^ 2 + 21 b ^ 2 (a d - b c) (c + d x) ^ 2 + 630 a b ^ 2 c ^ 2 d + 3 b ^ 3 (c + d x) ^
3 - 210 b ^ 3 c ^ 3) (c + d x) ^ 2 - 7 (a d - b c) ^ 5) / (21 d ^ 6 (c + d x) ^ (3 / 2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(293\) vs. \(2(134)=268\).
time = 0.17, size = 294, normalized size = 1.93

method	result	size
risch	\(\frac {2 b^{2} \left (3 b^{3} x^{3} d^{3}+21 d^{3} a \,x^{2} b^{2}-12 b^{3} c \,d^{2} x^{2}+70 a^{2} b \,d^{3} x -98 a \,b^{2} c \,d^{2} x +37 b^{3} c^{2} d x +210 a^{3} d^{3}-560 a^{2} b c \,d^{2}+511 a \,b^{2} c^{2} d -158 b^{3} c^{3}\right ) \sqrt {d x +c}}{21 d^{6}}-\frac {2 \left (15 b d x +a d +14 b c \right ) \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{3 d^{6} \left (d x +c \right )^{\frac {3}{2}}}\)	\(194\)
gosper	\(-\frac {2 \left (-3 b^{5} x^{5} d^{5}-21 a \,b^{4} d^{5} x^{4}+6 b^{5} c \,d^{4} x^{4}-70 a^{2} b^{3} d^{5} x^{3}+56 a \,b^{4} c \,d^{4} x^{3}-16 b^{5} c^{2} d^{3} x^{3}-210 a^{3} b^{2} d^{5} x^{2}+420 a^{2} b^{3} c \,d^{4} x^{2}-336 a \,b^{4} c^{2} d^{3} x^{2}+96 b^{5} c^{3} d^{2} x^{2}+105 a^{4} b \,d^{5} x -840 a^{3} b^{2} c \,d^{4} x +1680 a^{2} b^{3} c^{2} d^{3} x -1344 a \,b^{4} c^{3} d^{2} x +384 b^{5} c^{4} d x +7 a^{5} d^{5}+70 a^{4} b c \,d^{4}-560 a^{3} b^{2} c^{2} d^{3}+1120 a^{2} b^{3} c^{3} d^{2}-896 a \,b^{4} c^{4} d +256 b^{5} c^{5}\right )}{21 \left (d x +c \right )^{\frac {3}{2}} d^{6}}\)	\(273\)
trager	\(-\frac {2 \left (-3 b^{5} x^{5} d^{5}-21 a \,b^{4} d^{5} x^{4}+6 b^{5} c \,d^{4} x^{4}-70 a^{2} b^{3} d^{5} x^{3}+56 a \,b^{4} c \,d^{4} x^{3}-16 b^{5} c^{2} d^{3} x^{3}-210 a^{3} b^{2} d^{5} x^{2}+420 a^{2} b^{3} c \,d^{4} x^{2}-336 a \,b^{4} c^{2} d^{3} x^{2}+96 b^{5} c^{3} d^{2} x^{2}+105 a^{4} b \,d^{5} x -840 a^{3} b^{2} c \,d^{4} x +1680 a^{2} b^{3} c^{2} d^{3} x -1344 a \,b^{4} c^{3} d^{2} x +384 b^{5} c^{4} d x +7 a^{5} d^{5}+70 a^{4} b c \,d^{4}-560 a^{3} b^{2} c^{2} d^{3}+1120 a^{2} b^{3} c^{3} d^{2}-896 a \,b^{4} c^{4} d +256 b^{5} c^{5}\right )}{21 \left (d x +c \right )^{\frac {3}{2}} d^{6}}\)	\(273\)
derivativedivides	\(\frac {\frac {2 b^{5} \left (d x +c \right )^{\frac {7}{2}}}{7}+2 a \,b^{4} d \left (d x +c \right )^{\frac {5}{2}}-2 b^{5} c \left (d x +c \right )^{\frac {5}{2}}+\frac {20 a^{2} b^{3} d^{2} \left (d x +c \right )^{\frac {3}{2}}}{3}-\frac {40 a \,b^{4} c d \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {20 b^{5} c^{2} \left (d x +c \right )^{\frac {3}{2}}}{3}+20 a^{3} b^{2} d^{3} \sqrt {d x +c}-60 a^{2} b^{3} c \,d^{2} \sqrt {d x +c}+60 a \,b^{4} c^{2} d \sqrt {d x +c}-20 b^{5} c^{3} \sqrt {d x +c}-\frac {10 b \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{\sqrt {d x +c}}-\frac {2 \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}{3 \left (d x +c \right )^{\frac {3}{2}}}}{d^{6}}\)	\(294\)
default	\(\frac {\frac {2 b^{5} \left (d x +c \right )^{\frac {7}{2}}}{7}+2 a \,b^{4} d \left (d x +c \right )^{\frac {5}{2}}-2 b^{5} c \left (d x +c \right )^{\frac {5}{2}}+\frac {20 a^{2} b^{3} d^{2} \left (d x +c \right )^{\frac {3}{2}}}{3}-\frac {40 a \,b^{4} c d \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {20 b^{5} c^{2} \left (d x +c \right )^{\frac {3}{2}}}{3}+20 a^{3} b^{2} d^{3} \sqrt {d x +c}-60 a^{2} b^{3} c \,d^{2} \sqrt {d x +c}+60 a \,b^{4} c^{2} d \sqrt {d x +c}-20 b^{5} c^{3} \sqrt {d x +c}-\frac {10 b \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{\sqrt {d x +c}}-\frac {2 \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}{3 \left (d x +c \right )^{\frac {3}{2}}}}{d^{6}}\)	\(294\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d^6*(1/7*b^5*(d*x+c)^(7/2)+a*b^4*d*(d*x+c)^(5/2)-b^5*c*(d*x+c)^(5/2)+10/3*a^2*b^3*d^2*(d*x+c)^(3/2)-20/3*a*b
^4*c*d*(d*x+c)^(3/2)+10/3*b^5*c^2*(d*x+c)^(3/2)+10*a^3*b^2*d^3*(d*x+c)^(1/2)-30*a^2*b^3*c*d^2*(d*x+c)^(1/2)+30
*a*b^4*c^2*d*(d*x+c)^(1/2)-10*b^5*c^3*(d*x+c)^(1/2)-5*b*(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)/(d*x+c)^(1/2)-1/3*(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
)/(d*x+c)^(3/2))

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Maxima [A]
time = 0.27, size = 265, normalized size = 1.74 \begin {gather*} \frac {2 \, {\left (\frac {3 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{5} - 21 \, {\left (b^{5} c - a b^{4} d\right )} {\left (d x + c\right )}^{\frac {5}{2}} + 70 \, {\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} {\left (d x + c\right )}^{\frac {3}{2}} - 210 \, {\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} \sqrt {d x + c}}{d^{5}} + \frac {7 \, {\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5} - 15 \, {\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} {\left (d x + c\right )}\right )}}{{\left (d x + c\right )}^{\frac {3}{2}} d^{5}}\right )}}{21 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/(d*x+c)^(5/2),x, algorithm="maxima")

[Out]

2/21*((3*(d*x + c)^(7/2)*b^5 - 21*(b^5*c - a*b^4*d)*(d*x + c)^(5/2) + 70*(b^5*c^2 - 2*a*b^4*c*d + a^2*b^3*d^2)
*(d*x + c)^(3/2) - 210*(b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*sqrt(d*x + c))/d^5 + 7*(b^5*c
^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5 - 15*(b^5*c^4 - 4*a*b^4
*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*(d*x + c))/((d*x + c)^(3/2)*d^5))/d

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 283 vs. \(2 (134) = 268\).
time = 0.30, size = 283, normalized size = 1.86 \begin {gather*} \frac {2 \, {\left (3 \, b^{5} d^{5} x^{5} - 256 \, b^{5} c^{5} + 896 \, a b^{4} c^{4} d - 1120 \, a^{2} b^{3} c^{3} d^{2} + 560 \, a^{3} b^{2} c^{2} d^{3} - 70 \, a^{4} b c d^{4} - 7 \, a^{5} d^{5} - 3 \, {\left (2 \, b^{5} c d^{4} - 7 \, a b^{4} d^{5}\right )} x^{4} + 2 \, {\left (8 \, b^{5} c^{2} d^{3} - 28 \, a b^{4} c d^{4} + 35 \, a^{2} b^{3} d^{5}\right )} x^{3} - 6 \, {\left (16 \, b^{5} c^{3} d^{2} - 56 \, a b^{4} c^{2} d^{3} + 70 \, a^{2} b^{3} c d^{4} - 35 \, a^{3} b^{2} d^{5}\right )} x^{2} - 3 \, {\left (128 \, b^{5} c^{4} d - 448 \, a b^{4} c^{3} d^{2} + 560 \, a^{2} b^{3} c^{2} d^{3} - 280 \, a^{3} b^{2} c d^{4} + 35 \, a^{4} b d^{5}\right )} x\right )} \sqrt {d x + c}}{21 \, {\left (d^{8} x^{2} + 2 \, c d^{7} x + c^{2} d^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/(d*x+c)^(5/2),x, algorithm="fricas")

[Out]

2/21*(3*b^5*d^5*x^5 - 256*b^5*c^5 + 896*a*b^4*c^4*d - 1120*a^2*b^3*c^3*d^2 + 560*a^3*b^2*c^2*d^3 - 70*a^4*b*c*
d^4 - 7*a^5*d^5 - 3*(2*b^5*c*d^4 - 7*a*b^4*d^5)*x^4 + 2*(8*b^5*c^2*d^3 - 28*a*b^4*c*d^4 + 35*a^2*b^3*d^5)*x^3
- 6*(16*b^5*c^3*d^2 - 56*a*b^4*c^2*d^3 + 70*a^2*b^3*c*d^4 - 35*a^3*b^2*d^5)*x^2 - 3*(128*b^5*c^4*d - 448*a*b^4
*c^3*d^2 + 560*a^2*b^3*c^2*d^3 - 280*a^3*b^2*c*d^4 + 35*a^4*b*d^5)*x)*sqrt(d*x + c)/(d^8*x^2 + 2*c*d^7*x + c^2
*d^6)

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Sympy [A]
time = 29.40, size = 196, normalized size = 1.29 \begin {gather*} \frac {2 b^{5} \left (c + d x\right )^{\frac {7}{2}}}{7 d^{6}} - \frac {10 b \left (a d - b c\right )^{4}}{d^{6} \sqrt {c + d x}} + \frac {\left (c + d x\right )^{\frac {5}{2}} \cdot \left (10 a b^{4} d - 10 b^{5} c\right )}{5 d^{6}} + \frac {\left (c + d x\right )^{\frac {3}{2}} \cdot \left (20 a^{2} b^{3} d^{2} - 40 a b^{4} c d + 20 b^{5} c^{2}\right )}{3 d^{6}} + \frac {\sqrt {c + d x} \left (20 a^{3} b^{2} d^{3} - 60 a^{2} b^{3} c d^{2} + 60 a b^{4} c^{2} d - 20 b^{5} c^{3}\right )}{d^{6}} - \frac {2 \left (a d - b c\right )^{5}}{3 d^{6} \left (c + d x\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b**5*(c + d*x)**(7/2)/(7*d**6) - 10*b*(a*d - b*c)**4/(d**6*sqrt(c + d*x)) + (c + d*x)**(5/2)*(10*a*b**4*d -
10*b**5*c)/(5*d**6) + (c + d*x)**(3/2)*(20*a**2*b**3*d**2 - 40*a*b**4*c*d + 20*b**5*c**2)/(3*d**6) + sqrt(c +
d*x)*(20*a**3*b**2*d**3 - 60*a**2*b**3*c*d**2 + 60*a*b**4*c**2*d - 20*b**5*c**3)/d**6 - 2*(a*d - b*c)**5/(3*d*
*6*(c + d*x)**(3/2))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 335 vs. \(2 (134) = 268\).
time = 0.01, size = 421, normalized size = 2.77 \begin {gather*} \frac {\frac {2}{7} \sqrt {c+d x} \left (c+d x\right )^{3} b^{5} d^{36}-2 \sqrt {c+d x} \left (c+d x\right )^{2} b^{5} c d^{36}+2 \sqrt {c+d x} \left (c+d x\right )^{2} b^{4} d^{37} a+\frac {20}{3} \sqrt {c+d x} \left (c+d x\right ) b^{5} c^{2} d^{36}-\frac {40}{3} \sqrt {c+d x} \left (c+d x\right ) b^{4} c d^{37} a+\frac {20}{3} \sqrt {c+d x} \left (c+d x\right ) b^{3} d^{38} a^{2}-20 \sqrt {c+d x} b^{5} c^{3} d^{36}+60 \sqrt {c+d x} b^{4} c^{2} d^{37} a-60 \sqrt {c+d x} b^{3} c d^{38} a^{2}+20 \sqrt {c+d x} b^{2} d^{39} a^{3}}{d^{42}}+\frac {-30 \left (c+d x\right ) b^{5} c^{4}+120 \left (c+d x\right ) b^{4} c^{3} d a-180 \left (c+d x\right ) b^{3} c^{2} d^{2} a^{2}+120 \left (c+d x\right ) b^{2} c d^{3} a^{3}-30 \left (c+d x\right ) b d^{4} a^{4}+2 b^{5} c^{5}-10 b^{4} c^{4} d a+20 b^{3} c^{3} d^{2} a^{2}-20 b^{2} c^{2} d^{3} a^{3}+10 b c d^{4} a^{4}-2 d^{5} a^{5}}{3 d^{6} \sqrt {c+d x} \left (c+d x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(15*(d*x + c)*b^5*c^4 - b^5*c^5 - 60*(d*x + c)*a*b^4*c^3*d + 5*a*b^4*c^4*d + 90*(d*x + c)*a^2*b^3*c^2*d^2
- 10*a^2*b^3*c^3*d^2 - 60*(d*x + c)*a^3*b^2*c*d^3 + 10*a^3*b^2*c^2*d^3 + 15*(d*x + c)*a^4*b*d^4 - 5*a^4*b*c*d
^4 + a^5*d^5)/((d*x + c)^(3/2)*d^6) + 2/21*(3*(d*x + c)^(7/2)*b^5*d^36 - 21*(d*x + c)^(5/2)*b^5*c*d^36 + 70*(d
*x + c)^(3/2)*b^5*c^2*d^36 - 210*sqrt(d*x + c)*b^5*c^3*d^36 + 21*(d*x + c)^(5/2)*a*b^4*d^37 - 140*(d*x + c)^(3
/2)*a*b^4*c*d^37 + 630*sqrt(d*x + c)*a*b^4*c^2*d^37 + 70*(d*x + c)^(3/2)*a^2*b^3*d^38 - 630*sqrt(d*x + c)*a^2*
b^3*c*d^38 + 210*sqrt(d*x + c)*a^3*b^2*d^39)/d^42

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Mupad [B]
time = 0.08, size = 229, normalized size = 1.51 \begin {gather*} \frac {2\,b^5\,{\left (c+d\,x\right )}^{7/2}}{7\,d^6}-\frac {\left (10\,b^5\,c-10\,a\,b^4\,d\right )\,{\left (c+d\,x\right )}^{5/2}}{5\,d^6}-\frac {\frac {2\,a^5\,d^5}{3}-\frac {2\,b^5\,c^5}{3}+\left (c+d\,x\right )\,\left (10\,a^4\,b\,d^4-40\,a^3\,b^2\,c\,d^3+60\,a^2\,b^3\,c^2\,d^2-40\,a\,b^4\,c^3\,d+10\,b^5\,c^4\right )-\frac {20\,a^2\,b^3\,c^3\,d^2}{3}+\frac {20\,a^3\,b^2\,c^2\,d^3}{3}+\frac {10\,a\,b^4\,c^4\,d}{3}-\frac {10\,a^4\,b\,c\,d^4}{3}}{d^6\,{\left (c+d\,x\right )}^{3/2}}+\frac {20\,b^2\,{\left (a\,d-b\,c\right )}^3\,\sqrt {c+d\,x}}{d^6}+\frac {20\,b^3\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{3/2}}{3\,d^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^5/(c + d*x)^(5/2),x)

[Out]

(2*b^5*(c + d*x)^(7/2))/(7*d^6) - ((10*b^5*c - 10*a*b^4*d)*(c + d*x)^(5/2))/(5*d^6) - ((2*a^5*d^5)/3 - (2*b^5*
c^5)/3 + (c + d*x)*(10*b^5*c^4 + 10*a^4*b*d^4 - 40*a^3*b^2*c*d^3 + 60*a^2*b^3*c^2*d^2 - 40*a*b^4*c^3*d) - (20*
a^2*b^3*c^3*d^2)/3 + (20*a^3*b^2*c^2*d^3)/3 + (10*a*b^4*c^4*d)/3 - (10*a^4*b*c*d^4)/3)/(d^6*(c + d*x)^(3/2)) +
(20*b^2*(a*d - b*c)^3*(c + d*x)^(1/2))/d^6 + (20*b^3*(a*d - b*c)^2*(c + d*x)^(3/2))/(3*d^6)

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