3.1481 \(\int (a+b x)^{5/2} (c+d x)^{5/2} \, dx\)
Optimal. Leaf size=262 \[ -\frac {5 (b c-a d)^6 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{512 b^{7/2} d^{7/2}}+\frac {5 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)^5}{512 b^3 d^3}-\frac {5 (a+b x)^{3/2} \sqrt {c+d x} (b c-a d)^4}{768 b^3 d^2}+\frac {(a+b x)^{5/2} \sqrt {c+d x} (b c-a d)^3}{192 b^3 d}+\frac {(a+b x)^{7/2} \sqrt {c+d x} (b c-a d)^2}{32 b^3}+\frac {(a+b x)^{7/2} (c+d x)^{3/2} (b c-a d)}{12 b^2}+\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b} \]
[Out]
1/12*(-a*d+b*c)*(b*x+a)^(7/2)*(d*x+c)^(3/2)/b^2+1/6*(b*x+a)^(7/2)*(d*x+c)^(5/2)/b-5/512*(-a*d+b*c)^6*arctanh(d
^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^(7/2)/d^(7/2)-5/768*(-a*d+b*c)^4*(b*x+a)^(3/2)*(d*x+c)^(1/2)/b^3
/d^2+1/192*(-a*d+b*c)^3*(b*x+a)^(5/2)*(d*x+c)^(1/2)/b^3/d+1/32*(-a*d+b*c)^2*(b*x+a)^(7/2)*(d*x+c)^(1/2)/b^3+5/
512*(-a*d+b*c)^5*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b^3/d^3
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Rubi [A] time = 0.15, antiderivative size = 262, normalized size of antiderivative = 1.00,
number of steps used = 9, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used =
{50, 63, 217, 206} \[ \frac {5 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)^5}{512 b^3 d^3}-\frac {5 (a+b x)^{3/2} \sqrt {c+d x} (b c-a d)^4}{768 b^3 d^2}-\frac {5 (b c-a d)^6 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{512 b^{7/2} d^{7/2}}+\frac {(a+b x)^{5/2} \sqrt {c+d x} (b c-a d)^3}{192 b^3 d}+\frac {(a+b x)^{7/2} \sqrt {c+d x} (b c-a d)^2}{32 b^3}+\frac {(a+b x)^{7/2} (c+d x)^{3/2} (b c-a d)}{12 b^2}+\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*x)^(5/2)*(c + d*x)^(5/2),x]
[Out]
(5*(b*c - a*d)^5*Sqrt[a + b*x]*Sqrt[c + d*x])/(512*b^3*d^3) - (5*(b*c - a*d)^4*(a + b*x)^(3/2)*Sqrt[c + d*x])/
(768*b^3*d^2) + ((b*c - a*d)^3*(a + b*x)^(5/2)*Sqrt[c + d*x])/(192*b^3*d) + ((b*c - a*d)^2*(a + b*x)^(7/2)*Sqr
t[c + d*x])/(32*b^3) + ((b*c - a*d)*(a + b*x)^(7/2)*(c + d*x)^(3/2))/(12*b^2) + ((a + b*x)^(7/2)*(c + d*x)^(5/
2))/(6*b) - (5*(b*c - a*d)^6*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(512*b^(7/2)*d^(7/2))
Rule 50
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]
Rule 63
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 217
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] && !GtQ[a, 0]
Rubi steps
\begin {align*} \int (a+b x)^{5/2} (c+d x)^{5/2} \, dx &=\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b}+\frac {(5 (b c-a d)) \int (a+b x)^{5/2} (c+d x)^{3/2} \, dx}{12 b}\\ &=\frac {(b c-a d) (a+b x)^{7/2} (c+d x)^{3/2}}{12 b^2}+\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b}+\frac {(b c-a d)^2 \int (a+b x)^{5/2} \sqrt {c+d x} \, dx}{8 b^2}\\ &=\frac {(b c-a d)^2 (a+b x)^{7/2} \sqrt {c+d x}}{32 b^3}+\frac {(b c-a d) (a+b x)^{7/2} (c+d x)^{3/2}}{12 b^2}+\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b}+\frac {(b c-a d)^3 \int \frac {(a+b x)^{5/2}}{\sqrt {c+d x}} \, dx}{64 b^3}\\ &=\frac {(b c-a d)^3 (a+b x)^{5/2} \sqrt {c+d x}}{192 b^3 d}+\frac {(b c-a d)^2 (a+b x)^{7/2} \sqrt {c+d x}}{32 b^3}+\frac {(b c-a d) (a+b x)^{7/2} (c+d x)^{3/2}}{12 b^2}+\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b}-\frac {\left (5 (b c-a d)^4\right ) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx}{384 b^3 d}\\ &=-\frac {5 (b c-a d)^4 (a+b x)^{3/2} \sqrt {c+d x}}{768 b^3 d^2}+\frac {(b c-a d)^3 (a+b x)^{5/2} \sqrt {c+d x}}{192 b^3 d}+\frac {(b c-a d)^2 (a+b x)^{7/2} \sqrt {c+d x}}{32 b^3}+\frac {(b c-a d) (a+b x)^{7/2} (c+d x)^{3/2}}{12 b^2}+\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b}+\frac {\left (5 (b c-a d)^5\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{512 b^3 d^2}\\ &=\frac {5 (b c-a d)^5 \sqrt {a+b x} \sqrt {c+d x}}{512 b^3 d^3}-\frac {5 (b c-a d)^4 (a+b x)^{3/2} \sqrt {c+d x}}{768 b^3 d^2}+\frac {(b c-a d)^3 (a+b x)^{5/2} \sqrt {c+d x}}{192 b^3 d}+\frac {(b c-a d)^2 (a+b x)^{7/2} \sqrt {c+d x}}{32 b^3}+\frac {(b c-a d) (a+b x)^{7/2} (c+d x)^{3/2}}{12 b^2}+\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b}-\frac {\left (5 (b c-a d)^6\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{1024 b^3 d^3}\\ &=\frac {5 (b c-a d)^5 \sqrt {a+b x} \sqrt {c+d x}}{512 b^3 d^3}-\frac {5 (b c-a d)^4 (a+b x)^{3/2} \sqrt {c+d x}}{768 b^3 d^2}+\frac {(b c-a d)^3 (a+b x)^{5/2} \sqrt {c+d x}}{192 b^3 d}+\frac {(b c-a d)^2 (a+b x)^{7/2} \sqrt {c+d x}}{32 b^3}+\frac {(b c-a d) (a+b x)^{7/2} (c+d x)^{3/2}}{12 b^2}+\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b}-\frac {\left (5 (b c-a d)^6\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{512 b^4 d^3}\\ &=\frac {5 (b c-a d)^5 \sqrt {a+b x} \sqrt {c+d x}}{512 b^3 d^3}-\frac {5 (b c-a d)^4 (a+b x)^{3/2} \sqrt {c+d x}}{768 b^3 d^2}+\frac {(b c-a d)^3 (a+b x)^{5/2} \sqrt {c+d x}}{192 b^3 d}+\frac {(b c-a d)^2 (a+b x)^{7/2} \sqrt {c+d x}}{32 b^3}+\frac {(b c-a d) (a+b x)^{7/2} (c+d x)^{3/2}}{12 b^2}+\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b}-\frac {\left (5 (b c-a d)^6\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{512 b^4 d^3}\\ &=\frac {5 (b c-a d)^5 \sqrt {a+b x} \sqrt {c+d x}}{512 b^3 d^3}-\frac {5 (b c-a d)^4 (a+b x)^{3/2} \sqrt {c+d x}}{768 b^3 d^2}+\frac {(b c-a d)^3 (a+b x)^{5/2} \sqrt {c+d x}}{192 b^3 d}+\frac {(b c-a d)^2 (a+b x)^{7/2} \sqrt {c+d x}}{32 b^3}+\frac {(b c-a d) (a+b x)^{7/2} (c+d x)^{3/2}}{12 b^2}+\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b}-\frac {5 (b c-a d)^6 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{512 b^{7/2} d^{7/2}}\\ \end {align*}
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Mathematica [A] time = 2.53, size = 209, normalized size = 0.80 \[ \frac {(a+b x)^{7/2} \sqrt {c+d x} \left (-\frac {15 (b c-a d)^{11/2} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{d^{7/2} (a+b x)^{7/2} \sqrt {\frac {b (c+d x)}{b c-a d}}}+\frac {15 (b c-a d)^5}{d^3 (a+b x)^3}-\frac {10 (b c-a d)^4}{d^2 (a+b x)^2}+\frac {8 (b c-a d)^3}{d (a+b x)}+128 b (c+d x) (b c-a d)+48 (b c-a d)^2+256 b^2 (c+d x)^2\right )}{1536 b^3} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x)^(5/2)*(c + d*x)^(5/2),x]
[Out]
((a + b*x)^(7/2)*Sqrt[c + d*x]*(48*(b*c - a*d)^2 + (15*(b*c - a*d)^5)/(d^3*(a + b*x)^3) - (10*(b*c - a*d)^4)/(
d^2*(a + b*x)^2) + (8*(b*c - a*d)^3)/(d*(a + b*x)) + 128*b*(b*c - a*d)*(c + d*x) + 256*b^2*(c + d*x)^2 - (15*(
b*c - a*d)^(11/2)*ArcSinh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d]])/(d^(7/2)*(a + b*x)^(7/2)*Sqrt[(b*(c + d*x)
)/(b*c - a*d)])))/(1536*b^3)
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fricas [B] time = 0.56, size = 882, normalized size = 3.37 \[ \left [\frac {15 \, {\left (b^{6} c^{6} - 6 \, a b^{5} c^{5} d + 15 \, a^{2} b^{4} c^{4} d^{2} - 20 \, a^{3} b^{3} c^{3} d^{3} + 15 \, a^{4} b^{2} c^{2} d^{4} - 6 \, a^{5} b c d^{5} + a^{6} d^{6}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (256 \, b^{6} d^{6} x^{5} + 15 \, b^{6} c^{5} d - 85 \, a b^{5} c^{4} d^{2} + 198 \, a^{2} b^{4} c^{3} d^{3} + 198 \, a^{3} b^{3} c^{2} d^{4} - 85 \, a^{4} b^{2} c d^{5} + 15 \, a^{5} b d^{6} + 640 \, {\left (b^{6} c d^{5} + a b^{5} d^{6}\right )} x^{4} + 16 \, {\left (27 \, b^{6} c^{2} d^{4} + 106 \, a b^{5} c d^{5} + 27 \, a^{2} b^{4} d^{6}\right )} x^{3} + 8 \, {\left (b^{6} c^{3} d^{3} + 159 \, a b^{5} c^{2} d^{4} + 159 \, a^{2} b^{4} c d^{5} + a^{3} b^{3} d^{6}\right )} x^{2} - 2 \, {\left (5 \, b^{6} c^{4} d^{2} - 28 \, a b^{5} c^{3} d^{3} - 594 \, a^{2} b^{4} c^{2} d^{4} - 28 \, a^{3} b^{3} c d^{5} + 5 \, a^{4} b^{2} d^{6}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6144 \, b^{4} d^{4}}, \frac {15 \, {\left (b^{6} c^{6} - 6 \, a b^{5} c^{5} d + 15 \, a^{2} b^{4} c^{4} d^{2} - 20 \, a^{3} b^{3} c^{3} d^{3} + 15 \, a^{4} b^{2} c^{2} d^{4} - 6 \, a^{5} b c d^{5} + a^{6} d^{6}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (256 \, b^{6} d^{6} x^{5} + 15 \, b^{6} c^{5} d - 85 \, a b^{5} c^{4} d^{2} + 198 \, a^{2} b^{4} c^{3} d^{3} + 198 \, a^{3} b^{3} c^{2} d^{4} - 85 \, a^{4} b^{2} c d^{5} + 15 \, a^{5} b d^{6} + 640 \, {\left (b^{6} c d^{5} + a b^{5} d^{6}\right )} x^{4} + 16 \, {\left (27 \, b^{6} c^{2} d^{4} + 106 \, a b^{5} c d^{5} + 27 \, a^{2} b^{4} d^{6}\right )} x^{3} + 8 \, {\left (b^{6} c^{3} d^{3} + 159 \, a b^{5} c^{2} d^{4} + 159 \, a^{2} b^{4} c d^{5} + a^{3} b^{3} d^{6}\right )} x^{2} - 2 \, {\left (5 \, b^{6} c^{4} d^{2} - 28 \, a b^{5} c^{3} d^{3} - 594 \, a^{2} b^{4} c^{2} d^{4} - 28 \, a^{3} b^{3} c d^{5} + 5 \, a^{4} b^{2} d^{6}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3072 \, b^{4} d^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^(5/2)*(d*x+c)^(5/2),x, algorithm="fricas")
[Out]
[1/6144*(15*(b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*
c*d^5 + a^6*d^6)*sqrt(b*d)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 - 4*(2*b*d*x + b*c + a*d)*sqrt(b*
d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) + 4*(256*b^6*d^6*x^5 + 15*b^6*c^5*d - 85*a*b^5*c^4*d
^2 + 198*a^2*b^4*c^3*d^3 + 198*a^3*b^3*c^2*d^4 - 85*a^4*b^2*c*d^5 + 15*a^5*b*d^6 + 640*(b^6*c*d^5 + a*b^5*d^6)
*x^4 + 16*(27*b^6*c^2*d^4 + 106*a*b^5*c*d^5 + 27*a^2*b^4*d^6)*x^3 + 8*(b^6*c^3*d^3 + 159*a*b^5*c^2*d^4 + 159*a
^2*b^4*c*d^5 + a^3*b^3*d^6)*x^2 - 2*(5*b^6*c^4*d^2 - 28*a*b^5*c^3*d^3 - 594*a^2*b^4*c^2*d^4 - 28*a^3*b^3*c*d^5
+ 5*a^4*b^2*d^6)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^4*d^4), 1/3072*(15*(b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*
c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)*sqrt(-b*d)*arctan(1/2*(2*b*d*x +
b*c + a*d)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c)/(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)) + 2*(256*b^
6*d^6*x^5 + 15*b^6*c^5*d - 85*a*b^5*c^4*d^2 + 198*a^2*b^4*c^3*d^3 + 198*a^3*b^3*c^2*d^4 - 85*a^4*b^2*c*d^5 + 1
5*a^5*b*d^6 + 640*(b^6*c*d^5 + a*b^5*d^6)*x^4 + 16*(27*b^6*c^2*d^4 + 106*a*b^5*c*d^5 + 27*a^2*b^4*d^6)*x^3 + 8
*(b^6*c^3*d^3 + 159*a*b^5*c^2*d^4 + 159*a^2*b^4*c*d^5 + a^3*b^3*d^6)*x^2 - 2*(5*b^6*c^4*d^2 - 28*a*b^5*c^3*d^3
- 594*a^2*b^4*c^2*d^4 - 28*a^3*b^3*c*d^5 + 5*a^4*b^2*d^6)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^4*d^4)]
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giac [B] time = 3.53, size = 3120, normalized size = 11.91 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^(5/2)*(d*x+c)^(5/2),x, algorithm="giac")
[Out]
1/7680*(960*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)/b^2 + (b^6*c*d^3 - 13
*a*b^5*d^4)/(b^7*d^4)) - 3*(b^7*c^2*d^2 + 2*a*b^6*c*d^3 - 11*a^2*b^5*d^4)/(b^7*d^4)) - 3*(b^3*c^3 + a*b^2*c^2*
d + 3*a^2*b*c*d^2 - 5*a^3*d^3)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(
b*d)*b*d^2))*a*c^2*abs(b) - 7680*((b^2*c - a*b*d)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*
d - a*b*d)))/sqrt(b*d) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a))*a^3*c^2*abs(b)/b^2 + 40*(sqrt(b^2*
c + (b*x + a)*b*d - a*b*d)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a)/b^3 + (b^12*c*d^5 - 25*a*b^11*d^6)/(b^14*d^6
)) - (5*b^13*c^2*d^4 + 14*a*b^12*c*d^5 - 163*a^2*b^11*d^6)/(b^14*d^6)) + 3*(5*b^14*c^3*d^3 + 9*a*b^13*c^2*d^4
+ 15*a^2*b^12*c*d^5 - 93*a^3*b^11*d^6)/(b^14*d^6))*sqrt(b*x + a) + 3*(5*b^4*c^4 + 4*a*b^3*c^3*d + 6*a^2*b^2*c^
2*d^2 + 20*a^3*b*c*d^3 - 35*a^4*d^4)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/
(sqrt(b*d)*b^2*d^3))*b*c^2*abs(b) + 240*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x
+ a)/b^3 + (b^12*c*d^5 - 25*a*b^11*d^6)/(b^14*d^6)) - (5*b^13*c^2*d^4 + 14*a*b^12*c*d^5 - 163*a^2*b^11*d^6)/(
b^14*d^6)) + 3*(5*b^14*c^3*d^3 + 9*a*b^13*c^2*d^4 + 15*a^2*b^12*c*d^5 - 93*a^3*b^11*d^6)/(b^14*d^6))*sqrt(b*x
+ a) + 3*(5*b^4*c^4 + 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 + 20*a^3*b*c*d^3 - 35*a^4*d^4)*log(abs(-sqrt(b*d)*sqrt
(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b^2*d^3))*a*c*d*abs(b) + 1920*(sqrt(b^2*c + (b*x
+ a)*b*d - a*b*d)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)/b^2 + (b^6*c*d^3 - 13*a*b^5*d^4)/(b^7*d^4)) - 3*(b^7
*c^2*d^2 + 2*a*b^6*c*d^3 - 11*a^2*b^5*d^4)/(b^7*d^4)) - 3*(b^3*c^3 + a*b^2*c^2*d + 3*a^2*b*c*d^2 - 5*a^3*d^3)*
log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b*d^2))*a^2*c*d*abs(b)/b +
8*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(4*(b*x + a)*(6*(b*x + a)*(8*(b*x + a)/b^4 + (b^20*c*d^7 - 41*a*b^1
9*d^8)/(b^23*d^8)) - (7*b^21*c^2*d^6 + 26*a*b^20*c*d^7 - 513*a^2*b^19*d^8)/(b^23*d^8)) + 5*(7*b^22*c^3*d^5 + 1
9*a*b^21*c^2*d^6 + 37*a^2*b^20*c*d^7 - 447*a^3*b^19*d^8)/(b^23*d^8))*(b*x + a) - 15*(7*b^23*c^4*d^4 + 12*a*b^2
2*c^3*d^5 + 18*a^2*b^21*c^2*d^6 + 28*a^3*b^20*c*d^7 - 193*a^4*b^19*d^8)/(b^23*d^8))*sqrt(b*x + a) - 15*(7*b^5*
c^5 + 5*a*b^4*c^4*d + 6*a^2*b^3*c^3*d^2 + 10*a^3*b^2*c^2*d^3 + 35*a^4*b*c*d^4 - 63*a^5*d^5)*log(abs(-sqrt(b*d)
*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b^3*d^4))*b*c*d*abs(b) + 12*(sqrt(b^2*c + (b
*x + a)*b*d - a*b*d)*(2*(4*(b*x + a)*(6*(b*x + a)*(8*(b*x + a)/b^4 + (b^20*c*d^7 - 41*a*b^19*d^8)/(b^23*d^8))
- (7*b^21*c^2*d^6 + 26*a*b^20*c*d^7 - 513*a^2*b^19*d^8)/(b^23*d^8)) + 5*(7*b^22*c^3*d^5 + 19*a*b^21*c^2*d^6 +
37*a^2*b^20*c*d^7 - 447*a^3*b^19*d^8)/(b^23*d^8))*(b*x + a) - 15*(7*b^23*c^4*d^4 + 12*a*b^22*c^3*d^5 + 18*a^2*
b^21*c^2*d^6 + 28*a^3*b^20*c*d^7 - 193*a^4*b^19*d^8)/(b^23*d^8))*sqrt(b*x + a) - 15*(7*b^5*c^5 + 5*a*b^4*c^4*d
+ 6*a^2*b^3*c^3*d^2 + 10*a^3*b^2*c^2*d^3 + 35*a^4*b*c*d^4 - 63*a^5*d^5)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sq
rt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b^3*d^4))*a*d^2*abs(b) + 320*(sqrt(b^2*c + (b*x + a)*b*d - a*b*
d)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)/b^2 + (b^6*c*d^3 - 13*a*b^5*d^4)/(b^7*d^4)) - 3*(b^7*c^2*d^2 + 2*a*
b^6*c*d^3 - 11*a^2*b^5*d^4)/(b^7*d^4)) - 3*(b^3*c^3 + a*b^2*c^2*d + 3*a^2*b*c*d^2 - 5*a^3*d^3)*log(abs(-sqrt(b
*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b*d^2))*a^3*d^2*abs(b)/b^2 + 120*(sqrt(b^
2*c + (b*x + a)*b*d - a*b*d)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a)/b^3 + (b^12*c*d^5 - 25*a*b^11*d^6)/(b^14*d
^6)) - (5*b^13*c^2*d^4 + 14*a*b^12*c*d^5 - 163*a^2*b^11*d^6)/(b^14*d^6)) + 3*(5*b^14*c^3*d^3 + 9*a*b^13*c^2*d^
4 + 15*a^2*b^12*c*d^5 - 93*a^3*b^11*d^6)/(b^14*d^6))*sqrt(b*x + a) + 3*(5*b^4*c^4 + 4*a*b^3*c^3*d + 6*a^2*b^2*
c^2*d^2 + 20*a^3*b*c*d^3 - 35*a^4*d^4)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d))
)/(sqrt(b*d)*b^2*d^3))*a^2*d^2*abs(b)/b + (sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(4*(2*(b*x + a)*(8*(b*x + a)
*(10*(b*x + a)/b^5 + (b^30*c*d^9 - 61*a*b^29*d^10)/(b^34*d^10)) - 3*(3*b^31*c^2*d^8 + 14*a*b^30*c*d^9 - 417*a^
2*b^29*d^10)/(b^34*d^10)) + (21*b^32*c^3*d^7 + 77*a*b^31*c^2*d^8 + 183*a^2*b^30*c*d^9 - 3481*a^3*b^29*d^10)/(b
^34*d^10))*(b*x + a) - 5*(21*b^33*c^4*d^6 + 56*a*b^32*c^3*d^7 + 106*a^2*b^31*c^2*d^8 + 176*a^3*b^30*c*d^9 - 22
79*a^4*b^29*d^10)/(b^34*d^10))*(b*x + a) + 15*(21*b^34*c^5*d^5 + 35*a*b^33*c^4*d^6 + 50*a^2*b^32*c^3*d^7 + 70*
a^3*b^31*c^2*d^8 + 105*a^4*b^30*c*d^9 - 793*a^5*b^29*d^10)/(b^34*d^10))*sqrt(b*x + a) + 15*(21*b^6*c^6 + 14*a*
b^5*c^5*d + 15*a^2*b^4*c^4*d^2 + 20*a^3*b^3*c^3*d^3 + 35*a^4*b^2*c^2*d^4 + 126*a^5*b*c*d^5 - 231*a^6*d^6)*log(
abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b^4*d^5))*b*d^2*abs(b) + 5760*
(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*b*x + 2*a + (b*c*d - 5*a*d^2)/d^2)*sqrt(b*x + a) + (b^3*c^2 + 2*a*b^2*
c*d - 3*a^2*b*d^2)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*d))*a^2
*c^2*abs(b)/b^2 + 3840*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*b*x + 2*a + (b*c*d - 5*a*d^2)/d^2)*sqrt(b*x + a
) + (b^3*c^2 + 2*a*b^2*c*d - 3*a^2*b*d^2)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*
d)))/(sqrt(b*d)*d))*a^3*c*d*abs(b)/b^3)/b
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maple [B] time = 0.01, size = 1089, normalized size = 4.16 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)^(5/2)*(d*x+c)^(5/2),x)
[Out]
1/6/d*(b*x+a)^(5/2)*(d*x+c)^(7/2)+25/256*((b*x+a)*(d*x+c))^(1/2)/(d*x+c)^(1/2)/(b*x+a)^(1/2)*ln((b*d*x+1/2*a*d
+1/2*b*c)/(b*d)^(1/2)+(b*d*x^2+a*c+(a*d+b*c)*x)^(1/2))/(b*d)^(1/2)*a^3*c^3-1/16/d^2*(b*x+a)^(1/2)*(d*x+c)^(7/2
)*a*b*c+5/192/d^2*(d*x+c)^(3/2)*(b*x+a)^(1/2)*c^3*b*a+25/512/d^2*(d*x+c)^(1/2)*(b*x+a)^(1/2)*a*c^4*b+1/64/d^2*
(d*x+c)^(5/2)*(b*x+a)^(1/2)*c^2*b*a-25/512*d/b^2*(d*x+c)^(1/2)*(b*x+a)^(1/2)*a^4*c+1/192/b*(d*x+c)^(5/2)*(b*x+
a)^(1/2)*a^3+1/12/d*(b*x+a)^(3/2)*(d*x+c)^(7/2)*a+1/32/d*(b*x+a)^(1/2)*(d*x+c)^(7/2)*a^2-1/12/d^2*(b*x+a)^(3/2
)*(d*x+c)^(7/2)*b*c-5/128/d*(d*x+c)^(3/2)*(b*x+a)^(1/2)*a^2*c^2-25/256/d*(d*x+c)^(1/2)*(b*x+a)^(1/2)*a^2*c^3-5
/768/d^3*(d*x+c)^(3/2)*(b*x+a)^(1/2)*c^4*b^2-1/64/d*(d*x+c)^(5/2)*(b*x+a)^(1/2)*a^2*c+25/256/b*(d*x+c)^(1/2)*(
b*x+a)^(1/2)*a^3*c^2+5/192/b*(d*x+c)^(3/2)*(b*x+a)^(1/2)*a^3*c+5/512*d^2/b^3*(d*x+c)^(1/2)*(b*x+a)^(1/2)*a^5-5
/512/d^3*(d*x+c)^(1/2)*(b*x+a)^(1/2)*c^5*b^2-1/192/d^3*(d*x+c)^(5/2)*(b*x+a)^(1/2)*c^3*b^2-5/768*d/b^2*(d*x+c)
^(3/2)*(b*x+a)^(1/2)*a^4+1/32/d^3*(b*x+a)^(1/2)*(d*x+c)^(7/2)*b^2*c^2-75/1024*d/b*((b*x+a)*(d*x+c))^(1/2)/(d*x
+c)^(1/2)/(b*x+a)^(1/2)*ln((b*d*x+1/2*a*d+1/2*b*c)/(b*d)^(1/2)+(b*d*x^2+a*c+(a*d+b*c)*x)^(1/2))/(b*d)^(1/2)*a^
4*c^2+15/512*d^2/b^2*((b*x+a)*(d*x+c))^(1/2)/(d*x+c)^(1/2)/(b*x+a)^(1/2)*ln((b*d*x+1/2*a*d+1/2*b*c)/(b*d)^(1/2
)+(b*d*x^2+a*c+(a*d+b*c)*x)^(1/2))/(b*d)^(1/2)*a^5*c+15/512/d^2*((b*x+a)*(d*x+c))^(1/2)/(d*x+c)^(1/2)/(b*x+a)^
(1/2)*ln((b*d*x+1/2*a*d+1/2*b*c)/(b*d)^(1/2)+(b*d*x^2+a*c+(a*d+b*c)*x)^(1/2))/(b*d)^(1/2)*a*c^5*b^2-75/1024/d*
((b*x+a)*(d*x+c))^(1/2)/(d*x+c)^(1/2)/(b*x+a)^(1/2)*ln((b*d*x+1/2*a*d+1/2*b*c)/(b*d)^(1/2)+(b*d*x^2+a*c+(a*d+b
*c)*x)^(1/2))/(b*d)^(1/2)*a^2*c^4*b-5/1024/d^3*((b*x+a)*(d*x+c))^(1/2)/(d*x+c)^(1/2)/(b*x+a)^(1/2)*ln((b*d*x+1
/2*a*d+1/2*b*c)/(b*d)^(1/2)+(b*d*x^2+a*c+(a*d+b*c)*x)^(1/2))/(b*d)^(1/2)*c^6*b^3-5/1024*d^3/b^3*((b*x+a)*(d*x+
c))^(1/2)/(d*x+c)^(1/2)/(b*x+a)^(1/2)*ln((b*d*x+1/2*a*d+1/2*b*c)/(b*d)^(1/2)+(b*d*x^2+a*c+(a*d+b*c)*x)^(1/2))/
(b*d)^(1/2)*a^6
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^(5/2)*(d*x+c)^(5/2),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 details)Is a*d-b*c zero or nonzero?
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*x)^(5/2)*(c + d*x)^(5/2),x)
[Out]
int((a + b*x)^(5/2)*(c + d*x)^(5/2), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)**(5/2)*(d*x+c)**(5/2),x)
[Out]
Integral((a + b*x)**(5/2)*(c + d*x)**(5/2), x)
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