3.15.79 \(\int \frac {(c+d x)^{3/2}}{(a+b x)^{11/2}} \, dx\) [1479]

Optimal. Leaf size=101 \[ -\frac {2 (c+d x)^{5/2}}{9 (b c-a d) (a+b x)^{9/2}}+\frac {8 d (c+d x)^{5/2}}{63 (b c-a d)^2 (a+b x)^{7/2}}-\frac {16 d^2 (c+d x)^{5/2}}{315 (b c-a d)^3 (a+b x)^{5/2}} \]

[Out]

-2/9*(d*x+c)^(5/2)/(-a*d+b*c)/(b*x+a)^(9/2)+8/63*d*(d*x+c)^(5/2)/(-a*d+b*c)^2/(b*x+a)^(7/2)-16/315*d^2*(d*x+c)
^(5/2)/(-a*d+b*c)^3/(b*x+a)^(5/2)

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Rubi [A]
time = 0.01, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 3, 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 {16 d^2 (c+d x)^{5/2}}{315 (a+b x)^{5/2} (b c-a d)^3}+\frac {8 d (c+d x)^{5/2}}{63 (a+b x)^{7/2} (b c-a d)^2}-\frac {2 (c+d x)^{5/2}}{9 (a+b x)^{9/2} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^(3/2)/(a + b*x)^(11/2),x]

[Out]

(-2*(c + d*x)^(5/2))/(9*(b*c - a*d)*(a + b*x)^(9/2)) + (8*d*(c + d*x)^(5/2))/(63*(b*c - a*d)^2*(a + b*x)^(7/2)
) - (16*d^2*(c + d*x)^(5/2))/(315*(b*c - a*d)^3*(a + b*x)^(5/2))

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)^{3/2}}{(a+b x)^{11/2}} \, dx &=-\frac {2 (c+d x)^{5/2}}{9 (b c-a d) (a+b x)^{9/2}}-\frac {(4 d) \int \frac {(c+d x)^{3/2}}{(a+b x)^{9/2}} \, dx}{9 (b c-a d)}\\ &=-\frac {2 (c+d x)^{5/2}}{9 (b c-a d) (a+b x)^{9/2}}+\frac {8 d (c+d x)^{5/2}}{63 (b c-a d)^2 (a+b x)^{7/2}}+\frac {\left (8 d^2\right ) \int \frac {(c+d x)^{3/2}}{(a+b x)^{7/2}} \, dx}{63 (b c-a d)^2}\\ &=-\frac {2 (c+d x)^{5/2}}{9 (b c-a d) (a+b x)^{9/2}}+\frac {8 d (c+d x)^{5/2}}{63 (b c-a d)^2 (a+b x)^{7/2}}-\frac {16 d^2 (c+d x)^{5/2}}{315 (b c-a d)^3 (a+b x)^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 73, normalized size = 0.72 \begin {gather*} -\frac {2 (c+d x)^{9/2} \left (35 b^2+\frac {63 d^2 (a+b x)^2}{(c+d x)^2}-\frac {90 b d (a+b x)}{c+d x}\right )}{315 (b c-a d)^3 (a+b x)^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^(3/2)/(a + b*x)^(11/2),x]

[Out]

(-2*(c + d*x)^(9/2)*(35*b^2 + (63*d^2*(a + b*x)^2)/(c + d*x)^2 - (90*b*d*(a + b*x))/(c + d*x)))/(315*(b*c - a*
d)^3*(a + b*x)^(9/2))

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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)^(3/2)/(a + b*x)^(11/2),x]')

[Out]

Timed out

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(240\) vs. \(2(83)=166\).
time = 0.17, size = 241, normalized size = 2.39

method	result	size
gosper	\(\frac {2 \left (d x +c \right )^{\frac {5}{2}} \left (8 b^{2} x^{2} d^{2}+36 a b \,d^{2} x -20 b^{2} c d x +63 a^{2} d^{2}-90 a b c d +35 b^{2} c^{2}\right )}{315 \left (b x +a \right )^{\frac {9}{2}} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\)	\(105\)
default	\(-\frac {\left (d x +c \right )^{\frac {3}{2}}}{3 b \left (b x +a \right )^{\frac {9}{2}}}+\frac {\left (a d -b c \right ) \left (-\frac {\sqrt {d x +c}}{4 b \left (b x +a \right )^{\frac {9}{2}}}+\frac {\left (a d -b c \right ) \left (-\frac {2 \sqrt {d x +c}}{9 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {9}{2}}}-\frac {8 d \left (-\frac {2 \sqrt {d x +c}}{7 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {7}{2}}}-\frac {6 d \left (-\frac {2 \sqrt {d x +c}}{5 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {5}{2}}}-\frac {4 d \left (-\frac {2 \sqrt {d x +c}}{3 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {3}{2}}}+\frac {4 d \sqrt {d x +c}}{3 \left (-a d +b c \right )^{2} \sqrt {b x +a}}\right )}{5 \left (-a d +b c \right )}\right )}{7 \left (-a d +b c \right )}\right )}{9 \left (-a d +b c \right )}\right )}{8 b}\right )}{2 b}\)	\(241\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(3/2)/(b*x+a)^(11/2),x,method=_RETURNVERBOSE)

[Out]

-1/3/b*(d*x+c)^(3/2)/(b*x+a)^(9/2)+1/2*(a*d-b*c)/b*(-1/4/b*(d*x+c)^(1/2)/(b*x+a)^(9/2)+1/8*(a*d-b*c)/b*(-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)^(3/2)/(b*x+a)^(11/2),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. 426 vs. \(2 (83) = 166\).
time = 2.59, size = 426, normalized size = 4.22 \begin {gather*} -\frac {2 \, {\left (8 \, b^{2} d^{4} x^{4} + 35 \, b^{2} c^{4} - 90 \, a b c^{3} d + 63 \, a^{2} c^{2} d^{2} - 4 \, {\left (b^{2} c d^{3} - 9 \, a b d^{4}\right )} x^{3} + 3 \, {\left (b^{2} c^{2} d^{2} - 6 \, a b c d^{3} + 21 \, a^{2} d^{4}\right )} x^{2} + 2 \, {\left (25 \, b^{2} c^{3} d - 72 \, a b c^{2} d^{2} + 63 \, a^{2} c d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{315 \, {\left (a^{5} b^{3} c^{3} - 3 \, a^{6} b^{2} c^{2} d + 3 \, a^{7} b c d^{2} - a^{8} d^{3} + {\left (b^{8} c^{3} - 3 \, a b^{7} c^{2} d + 3 \, a^{2} b^{6} c d^{2} - a^{3} b^{5} d^{3}\right )} x^{5} + 5 \, {\left (a b^{7} c^{3} - 3 \, a^{2} b^{6} c^{2} d + 3 \, a^{3} b^{5} c d^{2} - a^{4} b^{4} d^{3}\right )} x^{4} + 10 \, {\left (a^{2} b^{6} c^{3} - 3 \, a^{3} b^{5} c^{2} d + 3 \, a^{4} b^{4} c d^{2} - a^{5} b^{3} d^{3}\right )} x^{3} + 10 \, {\left (a^{3} b^{5} c^{3} - 3 \, a^{4} b^{4} c^{2} d + 3 \, a^{5} b^{3} c d^{2} - a^{6} b^{2} d^{3}\right )} x^{2} + 5 \, {\left (a^{4} b^{4} c^{3} - 3 \, a^{5} b^{3} c^{2} d + 3 \, a^{6} b^{2} c d^{2} - a^{7} b d^{3}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(3/2)/(b*x+a)^(11/2),x, algorithm="fricas")

[Out]

-2/315*(8*b^2*d^4*x^4 + 35*b^2*c^4 - 90*a*b*c^3*d + 63*a^2*c^2*d^2 - 4*(b^2*c*d^3 - 9*a*b*d^4)*x^3 + 3*(b^2*c^
2*d^2 - 6*a*b*c*d^3 + 21*a^2*d^4)*x^2 + 2*(25*b^2*c^3*d - 72*a*b*c^2*d^2 + 63*a^2*c*d^3)*x)*sqrt(b*x + a)*sqrt
(d*x + c)/(a^5*b^3*c^3 - 3*a^6*b^2*c^2*d + 3*a^7*b*c*d^2 - a^8*d^3 + (b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d^
2 - a^3*b^5*d^3)*x^5 + 5*(a*b^7*c^3 - 3*a^2*b^6*c^2*d + 3*a^3*b^5*c*d^2 - a^4*b^4*d^3)*x^4 + 10*(a^2*b^6*c^3 -
3*a^3*b^5*c^2*d + 3*a^4*b^4*c*d^2 - a^5*b^3*d^3)*x^3 + 10*(a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^5*b^3*c*d^2 -
a^6*b^2*d^3)*x^2 + 5*(a^4*b^4*c^3 - 3*a^5*b^3*c^2*d + 3*a^6*b^2*c*d^2 - a^7*b*d^3)*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)**(3/2)/(b*x+a)**(11/2),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 343 vs. \(2 (83) = 166\).
time = 0.18, size = 448, normalized size = 4.44 \begin {gather*} \frac {2 \left (\left (-\frac {\left (22680 b^{7} d^{10} c-22680 b^{6} d^{11} a\right ) \sqrt {c+d x} \sqrt {c+d x}}{893025 b^{8} c^{4} \left |d\right |-3572100 b^{7} d a c^{3} \left |d\right |+5358150 b^{6} d^{2} a^{2} c^{2} \left |d\right |-3572100 b^{5} d^{3} a^{3} c \left |d\right |+893025 b^{4} d^{4} a^{4} \left |d\right |}-\frac {-102060 b^{7} d^{10} c^{2}+204120 b^{6} d^{11} a c-102060 b^{5} d^{12} a^{2}}{893025 b^{8} c^{4} \left |d\right |-3572100 b^{7} d a c^{3} \left |d\right |+5358150 b^{6} d^{2} a^{2} c^{2} \left |d\right |-3572100 b^{5} d^{3} a^{3} c \left |d\right |+893025 b^{4} d^{4} a^{4} \left |d\right |}\right ) \sqrt {c+d x} \sqrt {c+d x}-\frac {178605 b^{7} d^{10} c^{3}-535815 b^{6} d^{11} a c^{2}+535815 b^{5} d^{12} a^{2} c-178605 b^{4} d^{13} a^{3}}{893025 b^{8} c^{4} \left |d\right |-3572100 b^{7} d a c^{3} \left |d\right |+5358150 b^{6} d^{2} a^{2} c^{2} \left |d\right |-3572100 b^{5} d^{3} a^{3} c \left |d\right |+893025 b^{4} d^{4} a^{4} \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 {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 )^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(3/2)/(b*x+a)^(11/2),x)

[Out]

-2/315*(4*(d*x + c)*(2*(b^7*c*d^10 - a*b^6*d^11)*(d*x + c)/(b^8*c^4*abs(d) - 4*a*b^7*c^3*d*abs(d) + 6*a^2*b^6*
c^2*d^2*abs(d) - 4*a^3*b^5*c*d^3*abs(d) + a^4*b^4*d^4*abs(d)) - 9*(b^7*c^2*d^10 - 2*a*b^6*c*d^11 + a^2*b^5*d^1
2)/(b^8*c^4*abs(d) - 4*a*b^7*c^3*d*abs(d) + 6*a^2*b^6*c^2*d^2*abs(d) - 4*a^3*b^5*c*d^3*abs(d) + a^4*b^4*d^4*ab
s(d))) + 63*(b^7*c^3*d^10 - 3*a*b^6*c^2*d^11 + 3*a^2*b^5*c*d^12 - a^3*b^4*d^13)/(b^8*c^4*abs(d) - 4*a*b^7*c^3*
d*abs(d) + 6*a^2*b^6*c^2*d^2*abs(d) - 4*a^3*b^5*c*d^3*abs(d) + a^4*b^4*d^4*abs(d)))*(d*x + c)^(5/2)/((d*x + c)
*b*d - b*c*d + a*d^2)^(9/2)

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Mupad [B]
time = 1.11, size = 268, normalized size = 2.65 \begin {gather*} \frac {\sqrt {c+d\,x}\,\left (\frac {126\,a^2\,c^2\,d^2-180\,a\,b\,c^3\,d+70\,b^2\,c^4}{315\,b^4\,{\left (a\,d-b\,c\right )}^3}+\frac {x^2\,\left (126\,a^2\,d^4-36\,a\,b\,c\,d^3+6\,b^2\,c^2\,d^2\right )}{315\,b^4\,{\left (a\,d-b\,c\right )}^3}+\frac {16\,d^4\,x^4}{315\,b^2\,{\left (a\,d-b\,c\right )}^3}+\frac {8\,d^3\,x^3\,\left (9\,a\,d-b\,c\right )}{315\,b^3\,{\left (a\,d-b\,c\right )}^3}+\frac {4\,c\,d\,x\,\left (63\,a^2\,d^2-72\,a\,b\,c\,d+25\,b^2\,c^2\right )}{315\,b^4\,{\left (a\,d-b\,c\right )}^3}\right )}{x^4\,\sqrt {a+b\,x}+\frac {a^4\,\sqrt {a+b\,x}}{b^4}+\frac {6\,a^2\,x^2\,\sqrt {a+b\,x}}{b^2}+\frac {4\,a\,x^3\,\sqrt {a+b\,x}}{b}+\frac {4\,a^3\,x\,\sqrt {a+b\,x}}{b^3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^(3/2)/(a + b*x)^(11/2),x)

[Out]

((c + d*x)^(1/2)*((70*b^2*c^4 + 126*a^2*c^2*d^2 - 180*a*b*c^3*d)/(315*b^4*(a*d - b*c)^3) + (x^2*(126*a^2*d^4 +
6*b^2*c^2*d^2 - 36*a*b*c*d^3))/(315*b^4*(a*d - b*c)^3) + (16*d^4*x^4)/(315*b^2*(a*d - b*c)^3) + (8*d^3*x^3*(9
*a*d - b*c))/(315*b^3*(a*d - b*c)^3) + (4*c*d*x*(63*a^2*d^2 + 25*b^2*c^2 - 72*a*b*c*d))/(315*b^4*(a*d - b*c)^3
)))/(x^4*(a + b*x)^(1/2) + (a^4*(a + b*x)^(1/2))/b^4 + (6*a^2*x^2*(a + b*x)^(1/2))/b^2 + (4*a*x^3*(a + b*x)^(1
/2))/b + (4*a^3*x*(a + b*x)^(1/2))/b^3)

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