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

Optimal. Leaf size=152 \[ -\frac {10 b^4 (c+d x)^{7/2} (b c-a d)}{7 d^6}+\frac {4 b^3 (c+d x)^{5/2} (b c-a d)^2}{d^6}-\frac {20 b^2 (c+d x)^{3/2} (b c-a d)^3}{3 d^6}+\frac {10 b \sqrt {c+d x} (b c-a d)^4}{d^6}+\frac {2 (b c-a d)^5}{d^6 \sqrt {c+d x}}+\frac {2 b^5 (c+d x)^{9/2}}{9 d^6} \]

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Rubi [A]  time = 0.05, 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 = {43} \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[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 43

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)^{3/2}} \, dx &=\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*}

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Mathematica [A]  time = 0.12, size = 123, normalized size = 0.81 \begin {gather*} \frac {2 \left (-45 b^4 (c+d x)^4 (b c-a d)+126 b^3 (c+d x)^3 (b c-a d)^2-210 b^2 (c+d x)^2 (b c-a d)^3+315 b (c+d x) (b c-a d)^4+63 (b c-a d)^5+7 b^5 (c+d x)^5\right )}{63 d^6 \sqrt {c+d x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(63*(b*c - a*d)^5 + 315*b*(b*c - a*d)^4*(c + d*x) - 210*b^2*(b*c - a*d)^3*(c + d*x)^2 + 126*b^3*(b*c - a*d)
^2*(c + d*x)^3 - 45*b^4*(b*c - a*d)*(c + d*x)^4 + 7*b^5*(c + d*x)^5))/(63*d^6*Sqrt[c + d*x])

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IntegrateAlgebraic [B]  time = 0.07, size = 315, normalized size = 2.07 \begin {gather*} \frac {2 \left (-63 a^5 d^5+315 a^4 b d^4 (c+d x)+315 a^4 b c d^4-630 a^3 b^2 c^2 d^3+210 a^3 b^2 d^3 (c+d x)^2-1260 a^3 b^2 c d^3 (c+d x)+630 a^2 b^3 c^3 d^2+1890 a^2 b^3 c^2 d^2 (c+d x)+126 a^2 b^3 d^2 (c+d x)^3-630 a^2 b^3 c d^2 (c+d x)^2-315 a b^4 c^4 d-1260 a b^4 c^3 d (c+d x)+630 a b^4 c^2 d (c+d x)^2+45 a b^4 d (c+d x)^4-252 a b^4 c d (c+d x)^3+63 b^5 c^5+315 b^5 c^4 (c+d x)-210 b^5 c^3 (c+d x)^2+126 b^5 c^2 (c+d x)^3+7 b^5 (c+d x)^5-45 b^5 c (c+d x)^4\right )}{63 d^6 \sqrt {c+d x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(63*b^5*c^5 - 315*a*b^4*c^4*d + 630*a^2*b^3*c^3*d^2 - 630*a^3*b^2*c^2*d^3 + 315*a^4*b*c*d^4 - 63*a^5*d^5 +
315*b^5*c^4*(c + d*x) - 1260*a*b^4*c^3*d*(c + d*x) + 1890*a^2*b^3*c^2*d^2*(c + d*x) - 1260*a^3*b^2*c*d^3*(c +
d*x) + 315*a^4*b*d^4*(c + d*x) - 210*b^5*c^3*(c + d*x)^2 + 630*a*b^4*c^2*d*(c + d*x)^2 - 630*a^2*b^3*c*d^2*(c
+ d*x)^2 + 210*a^3*b^2*d^3*(c + d*x)^2 + 126*b^5*c^2*(c + d*x)^3 - 252*a*b^4*c*d*(c + d*x)^3 + 126*a^2*b^3*d^2
*(c + d*x)^3 - 45*b^5*c*(c + d*x)^4 + 45*a*b^4*d*(c + d*x)^4 + 7*b^5*(c + d*x)^5))/(63*d^6*Sqrt[c + d*x])

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fricas [B]  time = 1.09, size = 271, normalized size = 1.78 \begin {gather*} \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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [B]  time = 1.01, size = 350, normalized size = 2.30 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [B]  time = 0.01, size = 273, normalized size = 1.80 \begin {gather*} -\frac {2 \left (-7 b^{5} x^{5} d^{5}-45 a \,b^{4} d^{5} x^{4}+10 b^{5} c \,d^{4} x^{4}-126 a^{2} b^{3} d^{5} x^{3}+72 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}+252 a^{2} b^{3} c \,d^{4} x^{2}-144 a \,b^{4} c^{2} d^{3} x^{2}+32 b^{5} c^{3} d^{2} x^{2}-315 a^{4} b \,d^{5} x +840 a^{3} b^{2} c \,d^{4} x -1008 a^{2} b^{3} c^{2} d^{3} x +576 a \,b^{4} c^{3} d^{2} x -128 b^{5} c^{4} d x +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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/63/(d*x+c)^(1/2)*(-7*b^5*d^5*x^5-45*a*b^4*d^5*x^4+10*b^5*c*d^4*x^4-126*a^2*b^3*d^5*x^3+72*a*b^4*c*d^4*x^3-1
6*b^5*c^2*d^3*x^3-210*a^3*b^2*d^5*x^2+252*a^2*b^3*c*d^4*x^2-144*a*b^4*c^2*d^3*x^2+32*b^5*c^3*d^2*x^2-315*a^4*b
*d^5*x+840*a^3*b^2*c*d^4*x-1008*a^2*b^3*c^2*d^3*x+576*a*b^4*c^3*d^2*x-128*b^5*c^4*d*x+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)/d^6

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maxima [A]  time = 1.56, size = 267, normalized size = 1.76 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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mupad [B]  time = 0.08, size = 192, normalized size = 1.26 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 47.94, size = 243, normalized size = 1.60 \begin {gather*} \frac {2 b^{5} \left (c + d x\right )^{\frac {9}{2}}}{9 d^{6}} + \frac {\left (c + d x\right )^{\frac {7}{2}} \left (10 a b^{4} d - 10 b^{5} c\right )}{7 d^{6}} + \frac {\left (c + d x\right )^{\frac {5}{2}} \left (20 a^{2} b^{3} d^{2} - 40 a b^{4} c d + 20 b^{5} c^{2}\right )}{5 d^{6}} + \frac {\left (c + d x\right )^{\frac {3}{2}} \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 )}{3 d^{6}} + \frac {\sqrt {c + d x} \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 )}{d^{6}} - \frac {2 \left (a d - b c\right )^{5}}{d^{6} \sqrt {c + d x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b**5*(c + d*x)**(9/2)/(9*d**6) + (c + d*x)**(7/2)*(10*a*b**4*d - 10*b**5*c)/(7*d**6) + (c + d*x)**(5/2)*(20*
a**2*b**3*d**2 - 40*a*b**4*c*d + 20*b**5*c**2)/(5*d**6) + (c + d*x)**(3/2)*(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)/(3*d**6) + sqrt(c + d*x)*(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)/d**6 - 2*(a*d - b*c)**5/(d**6*sqrt(c + d*x))

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