\(\int \frac {(a+b x)^5}{(c+d x)^{3/2}} \, dx\) [1424]

Optimal result

Integrand size = 17, antiderivative size = 152 \[ \int \frac {(a+b x)^5}{(c+d x)^{3/2}} \, dx=\frac {2 (b c-a d)^5}{d^6 \sqrt {c+d x}}+\frac {10 b (b c-a d)^4 \sqrt {c+d x}}{d^6}-\frac {20 b^2 (b c-a d)^3 (c+d x)^{3/2}}{3 d^6}+\frac {4 b^3 (b c-a d)^2 (c+d x)^{5/2}}{d^6}-\frac {10 b^4 (b c-a d) (c+d x)^{7/2}}{7 d^6}+\frac {2 b^5 (c+d x)^{9/2}}{9 d^6} \]

[Out]

-20/3*b^2*(-a*d+b*c)^3*(d*x+c)^(3/2)/d^6+4*b^3*(-a*d+b*c)^2*(d*x+c)^(5/2)/d^6-10/7*b^4*(-a*d+b*c)*(d*x+c)^(7/2
)/d^6+2/9*b^5*(d*x+c)^(9/2)/d^6+2*(-a*d+b*c)^5/d^6/(d*x+c)^(1/2)+10*b*(-a*d+b*c)^4*(d*x+c)^(1/2)/d^6

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {45} \[ \int \frac {(a+b x)^5}{(c+d x)^{3/2}} \, dx=-\frac {10 b^4 (c+d x)^{7/2} (b c-a d)}{7 d^6}+\frac {4 b^3 (c+d x)^{5/2} (b c-a d)^2}{d^6}-\frac {20 b^2 (c+d x)^{3/2} (b c-a d)^3}{3 d^6}+\frac {10 b \sqrt {c+d x} (b c-a d)^4}{d^6}+\frac {2 (b c-a d)^5}{d^6 \sqrt {c+d x}}+\frac {2 b^5 (c+d x)^{9/2}}{9 d^6} \]

[In]

Int[(a + b*x)^5/(c + d*x)^(3/2),x]

[Out]

(2*(b*c - a*d)^5)/(d^6*Sqrt[c + d*x]) + (10*b*(b*c - a*d)^4*Sqrt[c + d*x])/d^6 - (20*b^2*(b*c - a*d)^3*(c + d*
x)^(3/2))/(3*d^6) + (4*b^3*(b*c - a*d)^2*(c + d*x)^(5/2))/d^6 - (10*b^4*(b*c - a*d)*(c + d*x)^(7/2))/(7*d^6) +
(2*b^5*(c + d*x)^(9/2))/(9*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*} \text {integral}& = \int \left (\frac {(-b c+a d)^5}{d^5 (c+d x)^{3/2}}+\frac {5 b (b c-a d)^4}{d^5 \sqrt {c+d x}}-\frac {10 b^2 (b c-a d)^3 \sqrt {c+d x}}{d^5}+\frac {10 b^3 (b c-a d)^2 (c+d x)^{3/2}}{d^5}-\frac {5 b^4 (b c-a d) (c+d x)^{5/2}}{d^5}+\frac {b^5 (c+d x)^{7/2}}{d^5}\right ) \, dx \\ & = \frac {2 (b c-a d)^5}{d^6 \sqrt {c+d x}}+\frac {10 b (b c-a d)^4 \sqrt {c+d x}}{d^6}-\frac {20 b^2 (b c-a d)^3 (c+d x)^{3/2}}{3 d^6}+\frac {4 b^3 (b c-a d)^2 (c+d x)^{5/2}}{d^6}-\frac {10 b^4 (b c-a d) (c+d x)^{7/2}}{7 d^6}+\frac {2 b^5 (c+d x)^{9/2}}{9 d^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.41 \[ \int \frac {(a+b x)^5}{(c+d x)^{3/2}} \, dx=\frac {2 \left (-63 a^5 d^5+315 a^4 b d^4 (2 c+d x)+210 a^3 b^2 d^3 \left (-8 c^2-4 c d x+d^2 x^2\right )+126 a^2 b^3 d^2 \left (16 c^3+8 c^2 d x-2 c d^2 x^2+d^3 x^3\right )+9 a b^4 d \left (-128 c^4-64 c^3 d x+16 c^2 d^2 x^2-8 c d^3 x^3+5 d^4 x^4\right )+b^5 \left (256 c^5+128 c^4 d x-32 c^3 d^2 x^2+16 c^2 d^3 x^3-10 c d^4 x^4+7 d^5 x^5\right )\right )}{63 d^6 \sqrt {c+d x}} \]

[In]

Integrate[(a + b*x)^5/(c + d*x)^(3/2),x]

[Out]

(2*(-63*a^5*d^5 + 315*a^4*b*d^4*(2*c + d*x) + 210*a^3*b^2*d^3*(-8*c^2 - 4*c*d*x + d^2*x^2) + 126*a^2*b^3*d^2*(
16*c^3 + 8*c^2*d*x - 2*c*d^2*x^2 + d^3*x^3) + 9*a*b^4*d*(-128*c^4 - 64*c^3*d*x + 16*c^2*d^2*x^2 - 8*c*d^3*x^3
+ 5*d^4*x^4) + b^5*(256*c^5 + 128*c^4*d*x - 32*c^3*d^2*x^2 + 16*c^2*d^3*x^3 - 10*c*d^4*x^4 + 7*d^5*x^5)))/(63*
d^6*Sqrt[c + d*x])

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.36

method	result	size
pseudoelliptic	\(\frac {\frac {2 \left (7 d^{5} x^{5}-10 c \,d^{4} x^{4}+16 c^{2} d^{3} x^{3}-32 c^{3} d^{2} x^{2}+128 c^{4} d x +256 c^{5}\right ) b^{5}}{63}-\frac {256 \left (-\frac {5}{128} d^{4} x^{4}+\frac {1}{16} c \,d^{3} x^{3}-\frac {1}{8} c^{2} d^{2} x^{2}+\frac {1}{2} c^{3} d x +c^{4}\right ) d a \,b^{4}}{7}+64 d^{2} \left (\frac {1}{16} d^{3} x^{3}-\frac {1}{8} c \,d^{2} x^{2}+\frac {1}{2} c^{2} d x +c^{3}\right ) a^{2} b^{3}-\frac {160 \left (-\frac {1}{8} d^{2} x^{2}+\frac {1}{2} c d x +c^{2}\right ) d^{3} a^{3} b^{2}}{3}+20 \left (\frac {d x}{2}+c \right ) d^{4} a^{4} b -2 a^{5} d^{5}}{\sqrt {d x +c}\, d^{6}}\)	\(206\)
risch	\(\frac {2 b \left (7 d^{4} x^{4} b^{4}+45 a \,b^{3} d^{4} x^{3}-17 b^{4} c \,d^{3} x^{3}+126 a^{2} b^{2} d^{4} x^{2}-117 a \,b^{3} c \,d^{3} x^{2}+33 b^{4} c^{2} d^{2} x^{2}+210 a^{3} b \,d^{4} x -378 a^{2} b^{2} c \,d^{3} x +261 a \,b^{3} c^{2} d^{2} x -65 b^{4} c^{3} d x +315 a^{4} d^{4}-1050 a^{3} b c \,d^{3}+1386 a^{2} b^{2} c^{2} d^{2}-837 a \,b^{3} c^{3} d +193 b^{4} c^{4}\right ) \sqrt {d x +c}}{63 d^{6}}-\frac {2 \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}{d^{6} \sqrt {d x +c}}\)	\(264\)
gosper	\(-\frac {2 \left (-7 x^{5} b^{5} d^{5}-45 x^{4} a \,b^{4} d^{5}+10 x^{4} b^{5} c \,d^{4}-126 x^{3} a^{2} b^{3} d^{5}+72 x^{3} a \,b^{4} c \,d^{4}-16 x^{3} b^{5} c^{2} d^{3}-210 x^{2} a^{3} b^{2} d^{5}+252 x^{2} a^{2} b^{3} c \,d^{4}-144 x^{2} a \,b^{4} c^{2} d^{3}+32 x^{2} b^{5} c^{3} d^{2}-315 x \,a^{4} b \,d^{5}+840 x \,a^{3} b^{2} c \,d^{4}-1008 x \,a^{2} b^{3} c^{2} d^{3}+576 x a \,b^{4} c^{3} d^{2}-128 x \,b^{5} c^{4} d +63 a^{5} d^{5}-630 a^{4} b c \,d^{4}+1680 a^{3} b^{2} c^{2} d^{3}-2016 a^{2} b^{3} c^{3} d^{2}+1152 a \,b^{4} c^{4} d -256 b^{5} c^{5}\right )}{63 \sqrt {d x +c}\, d^{6}}\)	\(273\)
trager	\(-\frac {2 \left (-7 x^{5} b^{5} d^{5}-45 x^{4} a \,b^{4} d^{5}+10 x^{4} b^{5} c \,d^{4}-126 x^{3} a^{2} b^{3} d^{5}+72 x^{3} a \,b^{4} c \,d^{4}-16 x^{3} b^{5} c^{2} d^{3}-210 x^{2} a^{3} b^{2} d^{5}+252 x^{2} a^{2} b^{3} c \,d^{4}-144 x^{2} a \,b^{4} c^{2} d^{3}+32 x^{2} b^{5} c^{3} d^{2}-315 x \,a^{4} b \,d^{5}+840 x \,a^{3} b^{2} c \,d^{4}-1008 x \,a^{2} b^{3} c^{2} d^{3}+576 x a \,b^{4} c^{3} d^{2}-128 x \,b^{5} c^{4} d +63 a^{5} d^{5}-630 a^{4} b c \,d^{4}+1680 a^{3} b^{2} c^{2} d^{3}-2016 a^{2} b^{3} c^{3} d^{2}+1152 a \,b^{4} c^{4} d -256 b^{5} c^{5}\right )}{63 \sqrt {d x +c}\, d^{6}}\)	\(273\)
derivativedivides	\(\frac {\frac {2 b^{5} \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {10 a \,b^{4} d \left (d x +c \right )^{\frac {7}{2}}}{7}-\frac {10 b^{5} c \left (d x +c \right )^{\frac {7}{2}}}{7}+4 a^{2} b^{3} d^{2} \left (d x +c \right )^{\frac {5}{2}}-8 a \,b^{4} c d \left (d x +c \right )^{\frac {5}{2}}+4 b^{5} c^{2} \left (d x +c \right )^{\frac {5}{2}}+\frac {20 a^{3} b^{2} d^{3} \left (d x +c \right )^{\frac {3}{2}}}{3}-20 a^{2} b^{3} c \,d^{2} \left (d x +c \right )^{\frac {3}{2}}+20 a \,b^{4} c^{2} d \left (d x +c \right )^{\frac {3}{2}}-\frac {20 b^{5} c^{3} \left (d x +c \right )^{\frac {3}{2}}}{3}+10 a^{4} b \,d^{4} \sqrt {d x +c}-40 a^{3} b^{2} c \,d^{3} \sqrt {d x +c}+60 a^{2} b^{3} c^{2} d^{2} \sqrt {d x +c}-40 a \,b^{4} c^{3} d \sqrt {d x +c}+10 b^{5} c^{4} \sqrt {d x +c}-\frac {2 \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}{\sqrt {d x +c}}}{d^{6}}\)	\(324\)
default	\(\frac {\frac {2 b^{5} \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {10 a \,b^{4} d \left (d x +c \right )^{\frac {7}{2}}}{7}-\frac {10 b^{5} c \left (d x +c \right )^{\frac {7}{2}}}{7}+4 a^{2} b^{3} d^{2} \left (d x +c \right )^{\frac {5}{2}}-8 a \,b^{4} c d \left (d x +c \right )^{\frac {5}{2}}+4 b^{5} c^{2} \left (d x +c \right )^{\frac {5}{2}}+\frac {20 a^{3} b^{2} d^{3} \left (d x +c \right )^{\frac {3}{2}}}{3}-20 a^{2} b^{3} c \,d^{2} \left (d x +c \right )^{\frac {3}{2}}+20 a \,b^{4} c^{2} d \left (d x +c \right )^{\frac {3}{2}}-\frac {20 b^{5} c^{3} \left (d x +c \right )^{\frac {3}{2}}}{3}+10 a^{4} b \,d^{4} \sqrt {d x +c}-40 a^{3} b^{2} c \,d^{3} \sqrt {d x +c}+60 a^{2} b^{3} c^{2} d^{2} \sqrt {d x +c}-40 a \,b^{4} c^{3} d \sqrt {d x +c}+10 b^{5} c^{4} \sqrt {d x +c}-\frac {2 \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}{\sqrt {d x +c}}}{d^{6}}\)	\(324\)

[In]

int((b*x+a)^5/(d*x+c)^(3/2),x,method=_RETURNVERBOSE)

[Out]

2/63*((7*d^5*x^5-10*c*d^4*x^4+16*c^2*d^3*x^3-32*c^3*d^2*x^2+128*c^4*d*x+256*c^5)*b^5-1152*(-5/128*d^4*x^4+1/16
*c*d^3*x^3-1/8*c^2*d^2*x^2+1/2*c^3*d*x+c^4)*d*a*b^4+2016*d^2*(1/16*d^3*x^3-1/8*c*d^2*x^2+1/2*c^2*d*x+c^3)*a^2*
b^3-1680*(-1/8*d^2*x^2+1/2*c*d*x+c^2)*d^3*a^3*b^2+630*(1/2*d*x+c)*d^4*a^4*b-63*a^5*d^5)/(d*x+c)^(1/2)/d^6

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 271 vs. \(2 (134) = 268\).

Time = 0.23 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.78 \[ \int \frac {(a+b x)^5}{(c+d x)^{3/2}} \, dx=\frac {2 \, {\left (7 \, b^{5} d^{5} x^{5} + 256 \, b^{5} c^{5} - 1152 \, a b^{4} c^{4} d + 2016 \, a^{2} b^{3} c^{3} d^{2} - 1680 \, a^{3} b^{2} c^{2} d^{3} + 630 \, a^{4} b c d^{4} - 63 \, a^{5} d^{5} - 5 \, {\left (2 \, b^{5} c d^{4} - 9 \, a b^{4} d^{5}\right )} x^{4} + 2 \, {\left (8 \, b^{5} c^{2} d^{3} - 36 \, a b^{4} c d^{4} + 63 \, a^{2} b^{3} d^{5}\right )} x^{3} - 2 \, {\left (16 \, b^{5} c^{3} d^{2} - 72 \, a b^{4} c^{2} d^{3} + 126 \, a^{2} b^{3} c d^{4} - 105 \, a^{3} b^{2} d^{5}\right )} x^{2} + {\left (128 \, b^{5} c^{4} d - 576 \, a b^{4} c^{3} d^{2} + 1008 \, a^{2} b^{3} c^{2} d^{3} - 840 \, a^{3} b^{2} c d^{4} + 315 \, a^{4} b d^{5}\right )} x\right )} \sqrt {d x + c}}{63 \, {\left (d^{7} x + c d^{6}\right )}} \]

[In]

integrate((b*x+a)^5/(d*x+c)^(3/2),x, algorithm="fricas")

[Out]

2/63*(7*b^5*d^5*x^5 + 256*b^5*c^5 - 1152*a*b^4*c^4*d + 2016*a^2*b^3*c^3*d^2 - 1680*a^3*b^2*c^2*d^3 + 630*a^4*b
*c*d^4 - 63*a^5*d^5 - 5*(2*b^5*c*d^4 - 9*a*b^4*d^5)*x^4 + 2*(8*b^5*c^2*d^3 - 36*a*b^4*c*d^4 + 63*a^2*b^3*d^5)*
x^3 - 2*(16*b^5*c^3*d^2 - 72*a*b^4*c^2*d^3 + 126*a^2*b^3*c*d^4 - 105*a^3*b^2*d^5)*x^2 + (128*b^5*c^4*d - 576*a
*b^4*c^3*d^2 + 1008*a^2*b^3*c^2*d^3 - 840*a^3*b^2*c*d^4 + 315*a^4*b*d^5)*x)*sqrt(d*x + c)/(d^7*x + c*d^6)

Sympy [A] (verification not implemented)

Time = 7.54 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.79 \[ \int \frac {(a+b x)^5}{(c+d x)^{3/2}} \, dx=\begin {cases} \frac {2 \left (\frac {b^{5} \left (c + d x\right )^{\frac {9}{2}}}{9 d^{5}} + \frac {\left (c + d x\right )^{\frac {7}{2}} \cdot \left (5 a b^{4} d - 5 b^{5} c\right )}{7 d^{5}} + \frac {\left (c + d x\right )^{\frac {5}{2}} \cdot \left (10 a^{2} b^{3} d^{2} - 20 a b^{4} c d + 10 b^{5} c^{2}\right )}{5 d^{5}} + \frac {\left (c + d x\right )^{\frac {3}{2}} \cdot \left (10 a^{3} b^{2} d^{3} - 30 a^{2} b^{3} c d^{2} + 30 a b^{4} c^{2} d - 10 b^{5} c^{3}\right )}{3 d^{5}} + \frac {\sqrt {c + d x} \left (5 a^{4} b d^{4} - 20 a^{3} b^{2} c d^{3} + 30 a^{2} b^{3} c^{2} d^{2} - 20 a b^{4} c^{3} d + 5 b^{5} c^{4}\right )}{d^{5}} - \frac {\left (a d - b c\right )^{5}}{d^{5} \sqrt {c + d x}}\right )}{d} & \text {for}\: d \neq 0 \\\frac {\begin {cases} a^{5} x & \text {for}\: b = 0 \\\frac {\left (a + b x\right )^{6}}{6 b} & \text {otherwise} \end {cases}}{c^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]

[In]

integrate((b*x+a)**5/(d*x+c)**(3/2),x)

[Out]

Piecewise((2*(b**5*(c + d*x)**(9/2)/(9*d**5) + (c + d*x)**(7/2)*(5*a*b**4*d - 5*b**5*c)/(7*d**5) + (c + d*x)**
(5/2)*(10*a**2*b**3*d**2 - 20*a*b**4*c*d + 10*b**5*c**2)/(5*d**5) + (c + d*x)**(3/2)*(10*a**3*b**2*d**3 - 30*a
**2*b**3*c*d**2 + 30*a*b**4*c**2*d - 10*b**5*c**3)/(3*d**5) + sqrt(c + d*x)*(5*a**4*b*d**4 - 20*a**3*b**2*c*d*
*3 + 30*a**2*b**3*c**2*d**2 - 20*a*b**4*c**3*d + 5*b**5*c**4)/d**5 - (a*d - b*c)**5/(d**5*sqrt(c + d*x)))/d, N
e(d, 0)), (Piecewise((a**5*x, Eq(b, 0)), ((a + b*x)**6/(6*b), True))/c**(3/2), True))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.76 \[ \int \frac {(a+b x)^5}{(c+d x)^{3/2}} \, dx=\frac {2 \, {\left (\frac {7 \, {\left (d x + c\right )}^{\frac {9}{2}} b^{5} - 45 \, {\left (b^{5} c - a b^{4} d\right )} {\left (d x + c\right )}^{\frac {7}{2}} + 126 \, {\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} {\left (d x + c\right )}^{\frac {5}{2}} - 210 \, {\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} {\left (d x + c\right )}^{\frac {3}{2}} + 315 \, {\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} \sqrt {d x + c}}{d^{5}} + \frac {63 \, {\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )}}{\sqrt {d x + c} d^{5}}\right )}}{63 \, d} \]

[In]

integrate((b*x+a)^5/(d*x+c)^(3/2),x, algorithm="maxima")

[Out]

2/63*((7*(d*x + c)^(9/2)*b^5 - 45*(b^5*c - a*b^4*d)*(d*x + c)^(7/2) + 126*(b^5*c^2 - 2*a*b^4*c*d + a^2*b^3*d^2
)*(d*x + c)^(5/2) - 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)*(d*x + c)^(3/2) + 315*(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)*sqrt(d*x + c))/d^5 + 63*(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)/(sqrt(d*x + c)*d^5))/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 350 vs. \(2 (134) = 268\).

Time = 0.32 (sec) , antiderivative size = 350, normalized size of antiderivative = 2.30 \[ \int \frac {(a+b x)^5}{(c+d x)^{3/2}} \, dx=\frac {2 \, {\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )}}{\sqrt {d x + c} d^{6}} + \frac {2 \, {\left (7 \, {\left (d x + c\right )}^{\frac {9}{2}} b^{5} d^{48} - 45 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{5} c d^{48} + 126 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{5} c^{2} d^{48} - 210 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{5} c^{3} d^{48} + 315 \, \sqrt {d x + c} b^{5} c^{4} d^{48} + 45 \, {\left (d x + c\right )}^{\frac {7}{2}} a b^{4} d^{49} - 252 \, {\left (d x + c\right )}^{\frac {5}{2}} a b^{4} c d^{49} + 630 \, {\left (d x + c\right )}^{\frac {3}{2}} a b^{4} c^{2} d^{49} - 1260 \, \sqrt {d x + c} a b^{4} c^{3} d^{49} + 126 \, {\left (d x + c\right )}^{\frac {5}{2}} a^{2} b^{3} d^{50} - 630 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} b^{3} c d^{50} + 1890 \, \sqrt {d x + c} a^{2} b^{3} c^{2} d^{50} + 210 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{3} b^{2} d^{51} - 1260 \, \sqrt {d x + c} a^{3} b^{2} c d^{51} + 315 \, \sqrt {d x + c} a^{4} b d^{52}\right )}}{63 \, d^{54}} \]

[In]

integrate((b*x+a)^5/(d*x+c)^(3/2),x, algorithm="giac")

[Out]

2*(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)/(sqrt(d*x + c)
*d^6) + 2/63*(7*(d*x + c)^(9/2)*b^5*d^48 - 45*(d*x + c)^(7/2)*b^5*c*d^48 + 126*(d*x + c)^(5/2)*b^5*c^2*d^48 -
210*(d*x + c)^(3/2)*b^5*c^3*d^48 + 315*sqrt(d*x + c)*b^5*c^4*d^48 + 45*(d*x + c)^(7/2)*a*b^4*d^49 - 252*(d*x +
c)^(5/2)*a*b^4*c*d^49 + 630*(d*x + c)^(3/2)*a*b^4*c^2*d^49 - 1260*sqrt(d*x + c)*a*b^4*c^3*d^49 + 126*(d*x + c
)^(5/2)*a^2*b^3*d^50 - 630*(d*x + c)^(3/2)*a^2*b^3*c*d^50 + 1890*sqrt(d*x + c)*a^2*b^3*c^2*d^50 + 210*(d*x + c
)^(3/2)*a^3*b^2*d^51 - 1260*sqrt(d*x + c)*a^3*b^2*c*d^51 + 315*sqrt(d*x + c)*a^4*b*d^52)/d^54

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.26 \[ \int \frac {(a+b x)^5}{(c+d x)^{3/2}} \, dx=\frac {2\,b^5\,{\left (c+d\,x\right )}^{9/2}}{9\,d^6}-\frac {\left (10\,b^5\,c-10\,a\,b^4\,d\right )\,{\left (c+d\,x\right )}^{7/2}}{7\,d^6}-\frac {2\,a^5\,d^5-10\,a^4\,b\,c\,d^4+20\,a^3\,b^2\,c^2\,d^3-20\,a^2\,b^3\,c^3\,d^2+10\,a\,b^4\,c^4\,d-2\,b^5\,c^5}{d^6\,\sqrt {c+d\,x}}+\frac {20\,b^2\,{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{3/2}}{3\,d^6}+\frac {4\,b^3\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{5/2}}{d^6}+\frac {10\,b\,{\left (a\,d-b\,c\right )}^4\,\sqrt {c+d\,x}}{d^6} \]

[In]

int((a + b*x)^5/(c + d*x)^(3/2),x)

[Out]

(2*b^5*(c + d*x)^(9/2))/(9*d^6) - ((10*b^5*c - 10*a*b^4*d)*(c + d*x)^(7/2))/(7*d^6) - (2*a^5*d^5 - 2*b^5*c^5 -
20*a^2*b^3*c^3*d^2 + 20*a^3*b^2*c^2*d^3 + 10*a*b^4*c^4*d - 10*a^4*b*c*d^4)/(d^6*(c + d*x)^(1/2)) + (20*b^2*(a
*d - b*c)^3*(c + d*x)^(3/2))/(3*d^6) + (4*b^3*(a*d - b*c)^2*(c + d*x)^(5/2))/d^6 + (10*b*(a*d - b*c)^4*(c + d*
x)^(1/2))/d^6